Geometries of Nature, Living Systems And Human Cognition: New Interactions of Mathematics With Natural Sciences And Humanities

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Author: Luciano Boi

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G e o m e t r i e s of N a t u r e , Living Systems and Human Cognition New Interactions of Mathematics with Natural Sciences and Humanities edited by

Luciano Boi

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Geometries of N a t u r e , Living Systems and Human Cognition New Interactions of Mathematics with Natural Sciences and Humanities

s i•

edited by

Luciano Boi Ecole der. Hautes Etudes en Sciences Sociales, Contra de Machematiques, and LUTH, Qbservatoire do Paris-Meudon, France

PP

G e o m e t r i e s of N a t u r e Living S y s t e m s and Human Cognition New Interactions of Mathematics with Natural Sciences and Humanities

\jjp World Scientific NEW JERSEY • LONDON

• SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

GEOMETRIES OF NATURE, LIVING SYSTEMS AND HUMAN COGNITION New Interactions of Mathematics with Natural Sciences and Humanities Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-474-8

Printed in Singapore by Mainland Press

To my brother Piergiorgio Boi, whom I honour and still love, and my daughter Vega Boi who blossomed into a living being but she was never born.

To my son Aliocha Boi, who came to life and since then has inspired me in thinking and writing. I dedicate this book to him.

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"Geometry is a magic that works" RENE THOM

"Topology unlocks the secrets of the universe" HERMANN WEYL

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Contents Foreword

xi

About the Contributors PART I

MATHEMATICAL IDEAS AND TECHNIQUES IN CLASSICAL AND QUANTUM PHYSICS

1. T-Duality, Functional Equation, and Nonconunutative String Spacetime M. Lapidus {University of California at Riverside) 2. On the Soliton-Particle Dualities F. Helein & J. Kouneiher {University of Paris 7) PART II

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KNOTS IN MATHEMATICS AND EM NATURE

3. Knots L. H. Kaujfman {University of Illinois at Chicago)

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4. Topological Knots Models in Physics and Biology L. Boi {EHESS - CAMS, Paris)

203

PART III

MATHEMATICAL AND LOGICAL MODELING IN THE NATURAL SCIENCES AND LIVING SYSTEMS

5. Beyond Modelling: A Challenge for Applied Mathematics P. T. Saunders {King's College, University of London)

281

6. Fondements Cognitifs de la Geometrie et Experience de L'espace A. Berthoz {College de France)

303

1. The Reasonable Effectiveness of Mathematics and Its Cognitive Roots G. Longo {CNRS and Ecole Normale Superieure)

351

8. Pathways of Deductions A. Carbone {University of Paris 6 and IHES)

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9. Phenomenologie et Theorie des Categories F. Patras {Lab. J.A. Dieudonne, CNRS-University Nice)

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Foreword The collection of papers forming this book is intended to provide a deeper study of some mathematical and physical subjects, which are at the core of the recent developments and far-reaching researches in the natural and living sciences, as well as to highlight the ubiquitous character and the generative power of a number of mathematical concepts, mostly discovered recently, in some apparently disparate fields. The main goal is to show that a profound movement of geometrization of the most important fields of research is in course, and that this movement has enabled the discovery of a great variety of amazing structures and behaviors in the physical reality and in the living matter. This process seems to be closely related to the action of precise geometrical and topological objects and structures. In addition, some of those objects and structures are very likely scale-invariant and act at different levels of the organization of matter. Among others, the following central questions have been raised by the authors of the nine chapter composing this volume: 1) How to reach a coherent description of the behavior of the different fundamental forces acting in nature from the properties pertaining to space-time and from its geometrical and topological structures? 2) How to explain the kind of relationships existing between some principles of geometrical and topological ordering and stability and certain physical phenomena such as those of bifurcation and of symmetry breaking in dynamical systems, such as fractals? 3) How to robustly take into account the qualitative (and morphological) changes occurring during the molecular and cellular evolution of living beings? 4) How to explain the connection between the mathematical structures of substrate spaces where phenomena unfold and the more or less complex dynamics according to which those phenomena evolve in time? An answer to those questions would very likely demand that we be able to develop new conceptual tools and especially new mathematical methods, ideas and techniques suited to work out a dynamical theory of the spatial generation of physical phenomena, as well as a theory of the emergence of natural and living processes. For example, one particularly interesting task would consist of explaining to what extent the formal structures and spatial-temporal events that constitute the natural frame of living organisms may influence its bio-chemical, physiological and cognitive growth. Some of these new mathematical methods and ideas are suggested and sometimes formally developed in such works by various authors. xi

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The book is divided in three parts, each of which comprises several chapters written by some of the most qualified scholars directly involved in the research themes addressed here, from knot theory and topological quantum field theory, to symmetry breaking in dynamical systems and developmental biology and spatial perception. Instead of trying to review such a large and rich range of contributions and corresponding topics covered by the authors, it may seem more appropriate to focus on some introductory thoughts on three key topics addressed by the authors of this collective work, namely: 1) The seemingly central role of certain topological objects such as, in particular, knots in mathematics and nature, 2) The emergence in the last two decades of several unfamiliar ideas and techniques in classical and quantum physics, 3) The possibility of new modeling in natural phenomena and living systems, and the cognitive basis of spatial perception. 1. The role of topological knots in mathematical physics and in nature. In the nineteenth century knot theory has been studied by very few scientists, remaining thus a fairly isolated subject of interest. Moreover, the study of knots was related only to physics. The German mathematician Gauss used the properties of knots in the solution of an electromagnetic problem, and the British physicist Thomson planned to describe atomic physics and chemistry in terms of knots. However, the twentieth century has known a renaissance in the interaction between physics and knot theory, which was due essentially, on the one hand, to intensive research in topology, and on the other hand, to statistical mechanics studies. A very profound interrelationship between statistical mechanics and knot theory emerged when it became clear that the Jones algebra and knot and link polynomials are formally identical to the Temperley-Lieb algebra. The latter one was introduced in the study of the Potts model, which is a generalization of the Ising model in statistical mechanics. In other words, it became apparent that statistical mechanics can be done on a link diagram and produce new invariants of links in the process. That opened the way to the discovery of a number of new invariants of knots and links that generalized the original Jones polynomial. In particular, the Yang-Baxter equation is a statistical mechanics model that leads to new knot invariants. In the last three decades, it has been understood that there are three other different ways in which physics and knot theory are related. Not only topological quantum field theory, but also the theory of quantum invariants has proved to be closely related to conformal field theories. Witten, notably, has shown that the Jones polynomial and its generalizations are related to the topological

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Chern-Simons theory. More precisely, he attempted to understand the new Jones's invariants of knots and links in a systematic way by relating them to the vacuum expectation values of Wilson loops in Chern-Simons theory. Using the invariance of the Chern-Simons action under orientation-preserving diffeomorphisms, these vacuum expectations values should be isotopy invariants. In this connection, one of the most fruitful developments in the last decades in theoretical physics is the work relating quantum field theory and threedimensional topology. Thus, Atiyah, Witten, Turaev and Kauffman were led to construct link invariants for closed three-manifolds. The chapter of Boi provides a detailed mathematical account of those subjects. The chapter of Kauffman in the present book is very illuminating in this regard. Furthermore, Kauffman gives new mathematical results relating to the invariants in knot theory, and a very interesting and original interpretation of quantum mechanics in the light of the concept of skein relations of knots and links. Its contribution is a masterpiece of imaginative and creative mathematics and mathematical physics. Even more astonishing has been the finding that there exist several fundamental connections between topological knot theory and life sciences. In fact, it became more and more clear that the spatial conformation of DNA knots are phenomena that make up part of biophysical and living matter; in other words, knots and links are microscopic and nanoscopic objects carrying a tremendous amount of precious information on the emergence of new forms and the evolution of organisms. Moreover, one might say that knotting and unknotting are "universal" transformations underlying almost all living processes at the molecular and at the macromolecular and also maybe at the cellular levels. In fact, it became recently clearer and clearer that the topological conformation of some families of enzymes, called topoisomerases, and also the threedimensional structures of proteins that act on DNA inside the space of cell are essential in order for certain vital biological functions of organisms to be accomplished. This opened totally new perspectives for fruitful collaborative work between topologists and biologists, first of all because differential geometry and knot theory can be used to describe and explain the threedimensional structures of DNA and protein-DNA complexes. In his chapter, Boi gives a detailed analysis of that subject, and also suggests some ideas that could be helpful in order to address these questions. 2. Some open problems and perspectives in classical and quantum physics. The chapters of Lapidus, Helein and Kouneiher deal with the question of the role of geometrical and (algebraic)-topological concepts in the developments of theoretical physics, especially in gauge theory as well as string theory, and they

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show the great significance of those concepts for gaining a deeper understanding of the dynamics of physics. Though starting from different paths, they lead in some sense to the same conclusion, namely that physical phenomena very likely emerge from the geometrical and topological structure of space-time. The attempts to solve one of the central problems in twentieth theoretical physics, i.e. how to combine gravity and the other forces into a unitary theoretical explanation of the physical world, depends essentially on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, this modified differential geometry still plays a fundamental role in non-Abelian gauge theories, superstring theory and non-commutative geometry, thanks to which a great variety of new mathematical structures has emerged. A very interesting hypothesis is that the global topological properties of the manifold model of spacetime play a major role in quantum field theory and that, consequently, several physical quantum effects arise from the non-local metrical and topological structures of these manifolds. In his chapter, Lapidus offer an authoritative and deep study of these questions, and present some important new mathematical results. The unification of general relativity and quantum theory requires thus some fundamental breakthroughs in our understanding of the relationship between spacetime and quantum processes. In particular the superstring theories lead to guessing that the usual structure of space-time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections and curvature in explaining the essential laws of nature. Kaluza-Klein theories, and more remarkably superstring theory, showed that spacetime symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. That essentially means that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain classes of (presumably) non-smooth manifolds. There are two key questions that are at the heart of the mathematical developments of quantum field theories, which include, among others, superstring theory, topological quantum field theory, noncommutative geometry, and loop representation (in quantum gravity). The first question concerns the meaning of quantum geometry. General relativity taught us that space-time geometry shares the same dynamical character of the electromagnetic field and the other physical fields; and we have learned with quantum theory that all

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dynamical objects exhibit quantum behavior: they can be in probabilistic superpositions of states and they manifest themselves in quanta at small scales. If we trust the general validity of these two physical discoveries, we are led to conclude that space-time geometry too must exhibit quantum behavior: it may admit probabilistic superpositions and it may manifest itself in quanta at small scales. What is then space-time geometry with such features? What kind of mathematics can describe it? What physical meaning can we give to it? We have a certain understanding of quantum field theory on a fixed geometrical background; can we understand how to describe a world in which the background geometry, space-time itself, is quantum mechanical? To illustrate the physical need of a diffeomorphism invariant quantum field theory, consider the following observations: Space-time geometry is dynamical, and thus rather similar to physical fields - indeed, space-time geometry is the gravitational field; but we can also turn this observation around and notice that the dynamics of the gravitational field is profoundly peculiar - indeed, the gravitational field is space-time geometry. In fact, this dynamics cannot be captured, as the other physical fields, by techniques that rely on the existence of a fixed background space-time. A diffeomorphism invariant quantum field theory very likely requires that we abandon local quantum field theory, in which physical operators are labeled by regions of a metric manifold, in favor of a diffeomorphism invariant quantum field theory, in which physical operators are diffeomorphism invariant objects. Thus, the definition and the construction of nontrivial diffeomorphism invariant quantum field theory is at present one of the most fascinating problems, and in our opinion it is very important in order to clarify the intimate relationship between physical intuition and mathematical conceptualization. 3. New modeling in natural phenomena and living systems, and in the cognitive basis of spatial perception. Most contributions in the third section of the collection are aimed at investigating relationships between the perceptual structures pertaining to our macroscopic world and the geometric properties which characterize the physical space and the beings that inhabit it. The main question raised by some authors, in particular by Berthoz and Longo and to some extent by Carbone is the following: What differentiates the topological and metrical structures of the 'physical' space, which is embodied into (or internalized by) the neurophysiological space in the brain, from the properties of the perceptive and representative space obtained very likely by deforming the first space according to certain general (mathematical and/or physical) laws? They try to answer this crucial question, which finds itself at the core of the present research in cognitive sciences, by raising some issues which concern: (i)

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Foreword

The mechanisms by which our complex sensorial systems such as the visual, the sensorimotor and the vestibular systems drive our perception of the surrounding (phenomenal) space and of the movements required to be performed in order to reach it. (ii) The perception of movement and its neurophysiological basis, and the role of movement in the perception of the third dimension, (iii) The geometrical and dynamical grounds of human perception and cognition and, reciprocally, the biological and cognitive roots of our mathematical knowledge. In his very interesting chapter, Berthoz supports both theoretically and experimentally the thesis that perception of space, and of things and events that are part of it, is fundamentally grounded in movements and actions of the body. Hence, geometry is less an abstract set of formal rules, and much more a complex activity which has developed phylogenetically in the course of evolution in order that we be able to reconstruct mentally the world around us through a neural allocentric processing of space, instead of an egocentric one, as it is the case for animals including primates. The goal of Longo's stimulating contribution is to stress that we organize knowledge of space and numbers by formal structures distilled in a mathematical language. In addition, he points out that the meaning of space, numbers and language is primarily grounded in mathematical intuition, which stems from the biological and historical character of human cognition. The papers of Saunders, Carbone and Patras deal with different, although related, aspects of mathematical modeling of complex living systems and formal structures. Saunders proposes a new methodology for mathematical modeling in the living and human sciences; he derives examples of that methodology from catastrophe theory and the theory of pattern formation. He points out that in conventional modeling, and especially in the physical sciences, we often construct a model and then treat it largely as an exercise in mathematics, only translating back into science at the end, whereas in both examples mentioned above, arriving at the prediction required mathematics and biology to be used together. Carbone shows, first, that replication and folding are two basic operations of complex systems, such as the brain, and further that they play a crucial role in the structural complexity and geometry of proofs. Relying on the great mathematician Alexandre Grothendieck, Patras emphasizes that dealing with meaning, that does not rely only on formal or computational tools, is necessary for internal progress of mathematics. This is proven by considering modern mathematical theories, such as category theory. Let us point out that the new relationship, of a much more intrinsic and dynamic nature than in the past, between mathematics (in particular geometry and topology) and biology appears to be one of the most significant

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developments of science in the last three decades. Relying on previous works in the correlated fields of morphology, embryology, theoretical biology and differential topology, it has been proposed recently a new view of self-organization in which topological-biological models of evolution and development of living organisms play an essential role. These attempts aim at explaining the stability and the reproduction of the global spatial-temporal structures or patterns in terms of the organization of the structure itself. For example, the self-organizational properties of morphogenesis is its capacity to reduce the symmetry order (to break symmetry) without necessarily employing any discrete external "dyssymmetrization", instead only by means of certain internal geometrical transformations bound up closely with the growth of organisms. In fact, experimental evidence strongly suggests that embryos obey some general topological principles, such as preservation of the Euler characteristic of a developmental field on the surfaces over which these fields are defined. A new mathematical approach inspired from qualitative dynamics and combinatorial-differential topology seems to provide far reaching possibilities to attack the problem of the emergence and the stability of self-reproducing structures. In our opinion, which is well exemplified in some of the papers presented here, the validity of this type of dynamic mathematical description exceeds by far the biological realm, and may concern various domains of physics (especially the problem of phase transitions), a number of morphological spaces and processes (whether animate or inanimate), as well as certain mechanisms of perception and cognition. In conclusion, we would like to stress that the main purpose of the present collection is to explore some far-reaching interfaces where mathematics, theoretical physics and natural sciences seem to interact profoundly. The volume bring a great overlap of knowledge between mathematicians, physicists and natural scientists. I hope that the long time it took to prepare the edition of this book contributes to the originality and the interest of the work presented here. Finally, I take this opportunity to thank all the contributors for their efforts, collaboration and patience; in addition, I am indebted to some of them for the several interesting conversations about many mathematical and physical questions we have had over the years of this book. I should also like to mention the editorial help I received from my students Jerome Havenel and Michel Karma in preparing the present volume. Luciano Boi Paris, February 2004

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About the Contributors Michel L. LAPIDUS has received his Ph.D., Doctorat es Sciences, and Habilitation to direct research (all in Mathematics) from the University of Paris VI (Pierre et Marie Curie) in France, in 1980, 1986, and 1987, respectively. After having taken a research position at the University of Paris VI and a postdoctoral position at the University of California at Berkeley, he has taught at the University of Southern California in Los Angeles, the University of Iowa, the University of Georgia in Athens, Yale University, and the University of California at Riverside where he is Professor of Mathematics since 1990. He has held a number of visiting positions at universities in the US and abroad, and has been a member of many research institutes, including most recently the Mathematical Sciences Research Institute in Berkeley, the Institut des Hautes Etudes Scientifiques in Paris (Bures-sur-Yvette), the Schroedinger Institute of Mathematical and Physics in Vienna, and the Newton Institute of the University of Cambridge. Since 1999, he is Deputy Governor and a honorary member of the Research Board of Advisors of the American Biographical Institute. In 2000, he was elected a Fellow of the American Association for the Advancement of Science (AAAS) for "Distinguished contributions to mathematical physics and fractal geometry". His research interests include mathematical physics, geometric analysis, partial differential equations, dynamical systems, number theory, as well as fractal and spectral geometry. Over the last year, he has published two books closely related to his research interests: the first one (joint with G. W. Johnson) entitled The Feyman Integral and Feynman's Operetional Calculus was published by Oxford University Press in May 2000. The second book (joint with M. van Frankenhuysen) was published by Birkhauser in January 2000 and is entitled Fractal Geometry and Number Theory, with the subtitle Complex dimensions offractal strings and zeros ofzeta functions. Frederic HELEIN is Professor of Mathematics at the Institut de Mathematiques Jussieu, Universite Denis Diderot Paris 7. He is the author and co-author of three books, among them, Harmonic Maps, Conservation Laws and Moving Frames (Cambridge University Press, 2002), and several research publications in differential geometry and mathematical physics. Frederic Helein is working on partial differential equations and their applications to differential geometry and to theoretical physics. His first contributions concerned the analysis of solutions XIX

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About the Contributors

of partial differential equations modeling static equilibria of liquid crystals in physics or of the harmonic map problem in geometry: he obtained several stability results by various geometrical methods (like for instance calibrations) and important regularity results on weak harmonic maps. He and F. Bethuel and H. Brezis developed a theory for the asymptotic analysis of two-dimensional solutions of the Ginzburg-Landau equation, used for modeling superconductivity and superfluidity. Later on, his interest turned to integrable systems arising in differential geometry and he proved that Willmore surfaces and (in collaboration with P. Romon) Hamiltonian stationary Lagrangian surfaces are integrable systems: this led to constructions and classifications results based on geometrical and algebraic methods. Lastly, in collaboration with J. Kouneiher, he is working on developing the multisymplectic approach to the calculus of variations, a framework suitable for a covariant Hamiltonian formalism for fields theory, and on its application to mathematical physics, for instance, the quantization of fields. Joseph KOUNEIHER is Researcher in the Laboratoire de l'Univers et de ses Theories (LUTH), CNRS and Observatoire de Paris, and is teaching in NiceSofia-Antipolis University. His research interests are in mathematical physics and in particular: quantum fields theories, quantum gravity and their topological aspects. At the same time he explore the conceptual development of the physics and the geometry of the twentieth century. His main contributions concern the cohomological aspect of quantum mechanics and the construction of the Khaler and symplectic structures underlying the foundations of the corresponding complexefied Hilbert space. In collaboration with Frederic Helein, he is now developing a covariant Hamiltonian formalism for fields and gauge theories. Louis H. KAUFFMAN is Professor of Mathematics at the University of Illinois at Chicago. He received his B.S. from MIT in 1966 and his Ph.D. from Princeton University in 1972. He has been on the faculty of the University of Illinois at Chicago since 1971, and has been a visiting professor at many universities and research institutions around the world. Professor Kauffman is the author of the book On Knots (Princeton University Press, 1987), the book Knots and Physics (World Scientific Publishing Co, 1991, 1994, 2001) and numerous books and research articles related to topology, knot theory, statistical mechanics and mathematical physics. He is the founding editor of the Journal of Knot Theory and Its Ramifications (World Scientific), and editor of the World Scientific Series on Knots and Everything.

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Luciano BOI is Associate Professor of Geometry, Scientific Theorization and Natural Philosophy at the Ecole des Hautes Etudes en Sciences Sociales, in the Centre de Mathematiques (Paris). He studied mathematics, physics and philosophy of science at the Universities of Bologna, Paris and Berlin, and received his Doctorate as well as his Habilitation from the EHESS (Paris). He has been a visiting professor and a member of several universities and research institutes, including the IHES (Bures-sur-Yvette), the IAS (Princeton), the UQAM and the University of Montreal. He has also been a visitor at the Universities of Cambridge (UK), Munich, Milan, Siena and Heidelberg, and invited lecturer in numerous international conferences and symposiums. He has received several awards, such as fellowships from the von Humboldt Foundation (1991, Berlin), from the Centre National des Lettres (Paris, 1992), and from the SSRCC (1996, Montreal), as well as a 1997 award from the Guggenheim Foundation (New York) and a 2000 award from the SingerPolignac Foundation (Paris) in recognition of the scope of his work. His main research interests include various aspects of mathematics, the geometrical foundations of theoretical physics, the interactions between topology and biology, the geometrical modeling of spatial perception, as well as the history and philosophy of sciences. He is the author of numerous books and articles on mathematical physics, theoretical biology, spatial perception, the epistemology of mathematical and physical sciences, and on natural philosophy and phenomenology. He is the editor of the "Philosophia Naturalis et Geometricalis" collection at Peter Lang (Bern), and co-director of the "Centra Internazionale di Studi e di Ricerche sulla Scienza, l'Arte e la Filosofia" Pharos at San Leo (Urbino) in Italy. Peter SAUNDERS was born in Canada and took a degree in mathematics at the University of Toronto. He came to London to do a Ph.D. in cosmology at King's College, London, where he is now Professor of Applied Mathematics. His present research interests are in mathematical and theoretical biology, including evolution theory, and in the study of complex systems. The aim of his research program is to explain the properties of organisms and other complex systems in terms of ordinary mathematics, physics and chemistry. More recently, his work is concentrated on two main areas: (a) physiological control, and (b) generic properties of complex nonlinear systems. He is the author of many research publications appeared in the Bulletin of Mathematical Biology, the Journal of Theoretical Biology, and the Journal of Physiology.

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About the Contributors

Alain BERTHOZ is Professor of the chair Physiologie de la Perception et de l'Action at the College de France, and Director of the CNRS-College de France Center of Cognitive Neuroscience. He holds Ph.D. degrees in engineering and natural sciences at the University of Paris. He has been visiting professor at the McGill University, at the Universities of Montreal, Oxford, Tubingen, Pise and Pavia. He is a member of numerous Scientific Societies and Academies, notably the Academie Internationale d'Austronatique, the Academia Europea, and the Academie des Sciences (Paris). He organized numerous scientific and interdisciplinary conferences, and has been an invited lecturer in many international congress and symposia. He was awarded by several international prizes and distinctions for its outstanding contributions to the neurosciences. He is a leading researcher in the fields of neurology and physiology of perception and movement. Professor Berthoz is the author of several books, which include (with W. Graf and P. P. Vidal) The Head-Neck Sensori-Motor System (Oxford University Press, 1991), Multisensory Control of Movement (Oxford University Press, 1993), Le sens du mouvement (Odile Jacob, 1998), and of an impressive number of research articles appeared in the leading international journals in the fields of physiology of perception and action, neurosciences, psychophysics and cognitive sciences. Giuseppe LONGO has been former Associate Professor of Mathematical Logic and, later, Professor of Computer Science at the University of Pisa. Currently, he is Directeur de Recherche CNRS (DR1) at the Ecole Normale Superieure in Paris. He worked in Mathematical Logic and at various applications of Mathematical Logic to Computer Science. In particular, Type Theory, Higher Type Recursion, Category Theory and their applications to Functional Languages have been his main research interests. Longo is editor-inchief of Mathematical Structures in Computer Science, a leading scientific journal of the Cambridge University Press. He is presently extending his research interests to Epistemology of Mathematics and Cognitive Sciences. He is the author of several books, among which, Categories, Types and Structures, MIT Press, 1991 (with A. Asperti), and published extensively, over the last years, on mathematical logic, foundations and philosophy of mathematics, and cognitive sciences. Alessandra CARBONE is Professor of Computer Science at the University Pierre et Marie Curie Paris 6 and a visitor at the Institut des Hautes Etudes Scientifiques (Bures-sur-Yvette). Her research touches on various aspects of

About the Contributors

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logic, symbolic dynamics, combinatoric and the interface between mathematics and molecular biology. She is co-author with S. Semmes of A Graphic Apology for Symmetry and Implicitness (published by Oxford University Press, 2000) and co-editor of several collection, including Pattern formation in Biology, Vision and Dynamics with M. Gromov and P. Pruzinkiewicz (published by World Scientific, 2000). Frederic PATRAS received his Ph.D. in Mathematics from the Universite Paris 7 in 1992. Currently, he is Senior Research Fellow of CNRS and Head of the "Algebra and Topology" group at the Laboratoire J.-A. Dieudonnee, Universite de Nice-Sophia Antipolis. His research interests include various aspects of algebraic topology, such as Hopf algebra, polytopes groups and homological algebra, as well as philosophy of mathematics and phenomenology. He is the author of the book La pensee mathematique contemporaine (Paris, 2001), co-editor (with L. Boi and P. Kerszberg) of the collection Rediscovering Phenomenology. Phenomenological essays concerning mathematical beings, physical reality and perception (Kluwer, 2005), and of several articles in mathematical journals, including Topology (1999), Algebraic K-Theory and its Applications (1999), Journal of Algebra (2000).

Parti

Mathematical Ideas and Techniques in Classical and Quantum Physics

"It seems reasonable to think that general relativity and quantum theory are intrinsically incompatible and that, rather than merely developing technique, what is required is some fundamental breakthrough in our understanding of the relationship between spacetime structure and quantum process." Edward Witten, Physics and Geometry, 1987.

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T-DUALITY, FUNCTIONAL EQUATION, AND NONCOMMUTATIVE STRING SPACETIME Michel L. LAPIDUS* University of California Department of Mathematics Riverside, CA 92521-0135, USA lapidus @math. ucr. edu

To the memory of Moshe FLATO, long-time friend and mentor, a brilliant scholar as well as an outstanding, compassionate, and generous human being

God made the integers, and all the rest is the work of man. Leopold Kronecker. (Quoted in [Bel, p. 477] and [Boy, p. 617].) String theory carries the seeds of a basic change in our ideas about spacetime and in other fundamental notions of physics. Edward Witten, 1996 [Wit 15, p. 24]

•Research partiallly supported by the U.S. National Science Foundation under Grants DMS-9623002 and DMS-0070497. 3

4

M. L. Lapidus

Abstract. We first give a broad (but brief) introduction to several of the main themes encountered in this work: arithmetic geometry, noncommutative geometry, quantum physics and string theory, prime number theory and the Riemann zeta function, along with fractal and spectral geometry. We then explain how string theory in a circle (or in a finite-dimensional torus)—considered from the point of view of noncommutative geometry—can be used as the starting point for a geometric and physical model for the Riemann zeta function C, = C,{s) (and other arithmetic L-series). In particular, we contend that the classic functional equation satisfied by C, corresponds to T-duality in string theory. The latter, a key symmetry that is not present in ordinary quantum mechanics, allows one to identify physically and mathematically two circular spacetimes with reciprocal radii. Furthermore, we suggest that the Riemann Hypothesis may be related to the existence of a fundamental length in string theory. This essay may serve as a relatively nontechnical introduction to the beginning of a theory presented in a book in preparation by the author and entitled In Search of the Riemann Zeros. In the latter monograph is developed, in particular, a theory of fractal membranes—viewed as suitable quantum or multiplicative analogues of the author's fractal strings. The notion of fractal membrane—interpreted heuristically as a noncommutative (adelic) infinite dimensional torus, in the spirit of the present discussion—may help in the long-term provide a unified geometric, algebraic and physical framework within which to understand number-theoretic zeta functions and their associated arithmetic geometries.

1

Introduction The recipes for quantization are a primitive manifestation of the fact that the space of internal degrees of freedom "at a single point" in vacuo is already infinite dimensional because of the virtual generation of particles. Further understanding is blocked until we relinquish the idea of space-time as the basis for all of physics. Yuri I. Manin, 1979 [Manil, p. 94]

One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation. Bernhard Riemann, 1859 [Riel], introducing his famous "Riemann Hypothesis". (Translated in [Edw, p. 301].)

T-Duality, Functional Equation

1.1

5

Arithmetic and Spacetime Geometry

I believe that at its deepest level, the geometry underlying the integers—in the old language, the 'geometry of numbers', and in modern terminology, 'arithmetic geometry', including the twin mystical notions of the 'arithmetic site' [Den3,6;Har2] and of the 'field of one element' [Mani4, So]— would have to reflect the physical and geometrical properties of what we traditionally call 'spacetime', for lack of a better word. I have held this belief, at least consciously, ever since reading in the mid-eighties the beautiful paper by Yuri I. Manin [Mani2], entitled New Dimensions in Geometry1. It was later comforted and turned into an intimate conviction by my own reflections and research experiences in developing since the late 1980's the theory of 'fractal strings' 2 and exploring its relationships with aspects of number theory, particularly the Riemann zeta function and the Riemann Hypothesis [Lapl-4,LapPol-3,LapMal-2,HeLapl-2, Lap-vFl-5]. Several years ago, I was startled to hear Alain Connes express a similar belief during a debate held at UNESCO in Paris on the occasion of the International Congress of Mathematical Physicists in July 1994. From our ensuing conversations about this subject—and from our ongoing dialogue (since the summer of 1993) about our respective approaches to the Riemann zeta-function ([BosConl-2], surveyed in [Con6,§V.ll], and [LapPol-2,LapMal-2,Lap2-4,HeLapl-2], now pursued in [Lap-vFl-4])—it appears, however, that his vision then (and probably still now in his new approach [Con9,10]) is quite different from the model I am about to propose, although key aspects of noncommutative geometry play an important role in both cases.3

1.2

Riemannian, Quantum and Noncommutative Geometry

During the course of the 20th century, and ever since the resounding success of the application of Riemannian geometry to the study of gravity in

' I am grateful to Christophe Soule for sharing with me his enthusiasm for this paper and for Arakelov theory [SoABKJ when I first met him in Berkeley in August 1984. 2 or 'fractal harps', as referred to in [Lap-vFl,2], not to be mistaken with the strings encountered in the classical string theory [Gre,GreeSWit,Kak,Mani3,Polc3-4,Schl], although part of the point of the present essay is that the two theories can be related, albeit in unexpected ways. 3 See, however, the relevant discussion towards the end of [Lap7] for some possible connections.

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Einstein's theory of general relativity, geometry has been a focal point for many mathematicians and physicists interested in apprehending aspects of physical reality. We now know that symplectic geometry is well suited to (and in fact, largely motivated by) the study of phase space in classical mechanics. As mentioned above, Riemannian geometry—in its Lorentzian version—is adopted in most models of classical physics concerned with gravitational fields. More recently, the geometry (and topology) of principal bundles over differentiable manifolds has been found to be an ideal tool to explore gauge field theories. It is to be noted, however, that most of the mathematically rigorous investigations of gauge theories have been limited so far to the classical rather than to the quantum aspects of such theories. It is much less clear, at the moment, how to determine what is "the" geometry underlying quantum mechanics, let alone quantum field theory. More generally, we do not understand what are the true mathematical foundations of quantum field theory [Witl7]. Of course, this question has been the object of many speculations and controversies. In recent years, noncommutative geometry has arisen in large part as a possible answer to such a question, although it is fair to say that we still seem to be far from having resolved this crucial problem. Beginning with the algebraic and functional analytic work of Murray and von Neumann [Mu-vN,vN], as well as of Gelfand and Naimark [GelNai], noncommutative geometry has truly emerged and flourished as an independent subject with the deep work of Alain Connes. (See, for example, the books [Con5] and [Con6]; also see [GraVarFi].) In a nutshell, the central objects of noncommutative geometry are no longer spaces of points, as in ordinary geometry, but (typically noncommutative) operator algebras, the elements of which can be thought of heuristically as representing quantum fields on the underlying 'noncommutative (or quantum) spaces'. In recent years, Connes [Con7,8] has proposed a set of axioms for noncommutative geometry that requires a much richer structure for a noncommutative space. It involves, in particular, the existence of a suitable Dirac-type operator acting on the Hilbert space on which the operator algebra is represented. (Intuitively, the noncommutative algebra itself can be thought of as the 'algebra of coordinate functions' on the associated quantum space.) This enables one, for example, to measure distances within a noncommutative space much as in a Riemannian manifold, via a formula in some sense dual to the geodesic formula. (See [Con4] and [Con6,Chapter VI].) Under appropriate assumptions, this also provides a noncommutative analogue of the de Rham complex and of aspects of differential topology and geometry (see [Conl-8] and [GraVarFi]). It is good to keep in mind, as is often stressed by Daniel Kastler [Kas2,3], that the abovementioned axioms are largely motivated by models from quantum physics, particularly the so-called 'standard model for elementary particles'

T-Duality, Functional Equation

1

(see [DV-K-M], [ConLo], [Con5,6]), as well by the long-standing problem of quantum gravity (see, e.g.,, [ChaConl,2]).

1.3

String Theory and Spacetime Geometry

Over the last twenty years, string theory—which originated as a theory of strong interactions in the early 1970's, soon to be superseded by quantum chromodynamics (QCD)—has emerged as the best candidate for unifying the four known fundamental forces (or interactions) of nature; namely, the electromagnetic force, the weak force and the strong force—all described by Yang-Mills gauge field theories—along with the gravitational force, described by Einstein's theory of general relativity. In this sense, it may eventually provide a means of reconciling quantum mechanics (or quantum field theory) with general relativity, and thereby to resolve the riddle posed by quantum gravity. Caution must be exerted, however, because despite its great beauty and mathematical power, string theory is still far from being a complete physical or mathematical theory. Moreover, due to the extremely high energies (or equivalently, the minuscule scales) involved, it has been notoriously difficult in string theory to make predictions that can be verified experimentally with our present-day technologies or even in the foreseeable future. We note, however, that although experiments involving high-energy accelerators seem to be out of the question—except to verify some of the most basic assumptions of (super)string theory, such as the existence of supersymmetry [Kan,Freu,Wein4]—interesting large-scale astronomical experiments currently under way may provide useful clues within the next ten to fifteen years. Roughly speaking, in string theory, point-particles are replaced with tiny strings (i.e., one-dimensional open strings or else closed loops) vibrating in a (target) space-time, which is assumed to be ten-dimensional in superstring theory. As time evolves, a given string sweeps out a two-dimensional world-sheet, viewed mathematically as a Riemann surface. Hence, the Feynman path integral approach to quantum mechanics ([Feyl], [FeyHi], see also [JohLap]) extends naturally to this setting, with the path integral being replaced with an integral over all possible world-sheets, or more precisely, with integrals over suitable moduli spaces of Riemann surfaces (with a given finite genus and a given number of marked points). The resulting heuristic Feynman-type integral is often referred to as a 'Polyakov integral' [Polyl-3] in the literature. (See, for example, [GreeG,GreeSWit,Kak,Polc3-4,Wit4], along with [JohLap], Chapter 20, especially Section 20.2.B.) The associated Feynman (or string) diagrams take a much simpler form than in quantum

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M. L. Lapidus

field theory and their detailed analysis provides a good understanding of perturbative string theory, at least at the physical level of rigor. The miracle is t h a t the divergences caused by the'coincidence of points in space-time (and hence the vanishing size of point-particles) in standard quantum field theory now disappear because of the extended size of the strings. In physical terms, superstring theory is said to be renormahzable or, more precisely, "finite to all orders in perturbation theory". In this respect, in concluding his plenary lecture at the International Congress of Mathematicians delivered in Berkeley in 1986, Edward Witten made the following statement ([Wit4, p. 302], 1987): / have tried to make it plausible that path integrals on Riemann surfaces can be used to formulate a generalization of general relativity. What is more, the resulting generalization is (especially in its supersymmetric forms) free of the ailments that plague quantum general relativity. If the logic has seemed a bit thin, it is at least in part because almost all we know in string theory is a trial and error construction of a perturbative expansion. [The Feynman-Polyakov path integrals over moduli spaces of Riemann surfaces] are probably the most beautiful formulas that we now know of in string theory, yet these formulas are merely a perturbative expansion ... of some underlying structure. Uncovering that structure is a vital problem if ever there was one. Such was the situation up to the late 1980's. However, during the 1990's, progress has been made towards developing a nonperturbative string theory, called M-theory, in which (one-dimensional) strings are replaced with higher-dimensional geometric objects, called 'membranes' or 'D-branes'. The associated 'dualities' (including the so-called 'S-duality' and 'T-duality') enable one to relate the five basic types of string theories, and thereby to obtain a more unified picture of string theory. (See, for example, [Witl5-17] and [GivePR,Gre, Polcl-4, Sch2-4,Val-2].) These recent developments are sometimes referred to as the "second superstring revolution" [Sch2]. Edward Witten often begins his conferences on string theory—especially when addressing himself to a mathematical audience—by stressing the following striking contrast between the historical development of string theory and general relativity (see also, for example, the introduction of [Wit4]). In ([Witl3, pp. 205-206], 1994), he writes: More fundamentally, I believe that the main obstacle [to further progress] is that the core geometrical ideas—which must underlie string theory the way Riemannian geometry underlies general relativity— have not yet been unearthed.

T-Duality, Functional Equation

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Whatever the true underlying geometric foundations of string theory (or of M-theory), there seems to be an emerging consensus among theoretical and mathematical physicists that one needs to significantly revisit the notion of spacetime, both from a geometrical and physical point of view. In particular, at extremely small scales (typically, below the Planck scale4), the classical model of spacetime as a smooth Riemannian (or Lorentzian) manifold is probably no longer valid. For example, the small-scale structure of spacetime may be discrete, or partly discrete and partly smooth. Alternately, it may be of a fractal nature. In fact, in early work on quantum gravity by Wheeler [Whe,WheFo], Hawking and others (see, for example, [GibHaw,Haw,HawIs]), there have been intriguing references to the existence of some kind of 'fractal foam' (sometimes also called 'quantum foam'). (More recently, see also [Not] in another context.) More radically, it has even been suggested to do away with the notion of spacetime altogether, at least as a primary concept. (See, for instance, Witten's article [Wit 15] entitled Reflections on the Fate of Spacetime and from which the second quote heading this essay is excerpted. Also, for a different perspective on a similar theme, see Manin's quote from [Manil] heading the present introduction.) Perhaps an appropriate modification or extension of Connes' noncommutative geometry [Con5,6] will provide clues as to how to proceed in suitably altering or replacing the concept of spacetime. Indeed, there has already been a number of attempts in this direction, several of which will be key to aspects of our present work. (See, for example, [Wit3], and more recently, [FroGa,ChaFro,Chal-2,LiSzl2,FroGrRel-2] along with [ConDouSc.LanLiSz].) Whatever the answer to these fundamental questions will turn out to be ultimately, the relationship between physics and geometry (in a broad sense) will continue to be at the center of the ongoing dialogue between physicists and mathematicians during the next few decades of the 21st century. 4 The Planck length (or scale) is the fundamental scale of quantum gravity. It is approximately equal to 1.6 x 10~ 3 3 cm (in international units) and is expressed in terms of the following three universal constants, h (Planck's constant or quantum of action), c (the speed of light), and G (Newton's gravitational constant). It is also equal to the reciprocal of the Planck mass, about 1.22 x 10 19 GeV, the natural mass (or energy) scale of quantum gravity. (It may be useful to note—as is frequently stressed by physicists—that Planck's length is about 20 orders of magnitude smaller than the size of a proton.) In much of this work, we will choose units so that Planck's length (or rather, the string length, see §2.3) is equal to one.

M. L. Lapidus

The Riemann Hypothesis and the Geometry of the Primes The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found in it a mysterious attraction impossible to resist. ... Very different results are revealed when we turn to the second principal branch of the modern theory, the theory of the average or asymptotic distribution of primes. This theory (though one of its most famous problems is still unsolved) is in some ways almost complete, and certainly represents one of the most remarkable triumphs of modern analysis. The theory centres round one theorem, the Primzahlsatz or Prime Number Theorem; and it is to the history of this theorem, which may almost be said to embody the history of the whole subject, that I shall devote the remainder of this lecture. ... The next great step was taken by Riemann in 1859, and it is in Riemann's famous memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse that we first find the ideas upon which the theory has now been shown really to rest. Riemann did not prove the Prime Number Theorem: it is remarkable, indeed, that he never mentions it. His object was a different one, that of finding an explicit expression for n(x) [the number of primes not exceeding a;], or rather for another closely associated function, as a sum of an infinite series. But it was Riemann who first recognized that, if we are to solve any of these problems, we must study the Zeta-function as a function of the complex variable s = a + it, and in particular study the distribution of its zeros. ... To these propositions [Riemann] added certain others of which he could produce no satisfactory proof. In particular he asserted that there is in fact an infinity of complex zeros, all naturally situated in the 'critical strip' 0 < a < 1; an assertion now known to be correct. Finally he asserted that it was 'sehr wahrscheinlich' that all these zeros have the real part \: the notorious 'Riemann hypothesis', unsettled to this day.

T-Duality, Functional Equation

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We come now to the time when, a hundred years after the conjectures of Gauss and Legendre [about the asymptotic distribution of the primes], the theorem was finally proved. The way was opened by the work of Hadamard on integral transcendental functions. In 1893 Hadamard proved that the complex zeros of Riemann actually exist; and in 1896 he and de la Vallee—Poussin proved independently that none of them have the real part 1, and deduced a proof of the Prime Number Theorem. It is not possible for me now to give an adequate account of the intricate and difficult reasoning by which these theorems are established. But the general ideas which underlie the proofs are, I think, such as should be intelligible to any mathematician. ... The arguments which I have advanced are not exact: I have merely put forward a chain of reasoning which seems likely to lead to the desired result. The achievement of Hadamard and de la ValleePoussin was to replace these plausibilities by rigorous proofs. It might be difficult for me to make clear to you how great this achievement was. Some branches of pure mathematics have the pleasant characteristic that what seems plausible at first sight is generally true. In this theory anyone can make plausible conjectures, and they are almost always false. Nothing short of absolute rigour counts; and it is for this reason that the Analytic Theory of Numbers, while hardly a subject for an amateur, provides the finest possible discipline in accurate reasoning for anyone who will make a real effort to understand its results. Godfrey H. Hardy, 1915 [Hard2, pp. 350-354], in his lecture on Prime Numbers The zeta-function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. ... The main interest comes from trying to improve the Prime Number Theorem, i.e., getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fanfare, and whose proof has since then become the single most desirable achievement for a mathematician. Martin C. Gutwiller, 1990 [Gut2, p. 308] It is perhaps fitting t h a t the same mathematician who brought Riemannian geometry to the world (with such an impact on physics, especially general relativity, half a century later) also proposed what later came to be known as the most famous open problem of mathematics, the so-called Riemann Hypothesis. In developing his geometry, Georg Friedrich Bernhard Riemann (1826-1866) was motivated by the work of his predecessors—including

12

M. L. Lapidus

Karl Friedrich Gauss (1777-1855) and the co-discoverers of non-Euclidean geometry, Nikolai Ivanovich Lobachevsky (1793-1856) and Johann (or Janos) Bolyai (1802-1860)—as well as by philosophical and physical considerations.5 On the other hand, Riemann's Conjecture (or Hypothesis) concerning the location of the critical zeros of the Riemann zeta function £ = £(s) (namely, C(s) = 0 with 0 < Re s < 1 implies that Re s = | ) seems to have had entirely different and purely 'internal' motivations. The Riemann Hypothesis has fascinated mathematicians since its introduction by Riemann in his famous inaugural lecture to the Berlin Academy of Sciences in 1859 (see [Riel]). Curiously, it was presented almost as a passing remark or conjecture within [Riel], the only paper by Riemann devoted to number theory. (See the second quote heading this introduction.) One of the main goals of Riemann in [Riel] (admittedly never stated overtly) seems to have been to provide the tools needed to establish the then still unproved 'Prime Number Theorem' (conjectured by Gauss and Legendre) according to which, in particular, (i.4.i)

n(x) = ^ ( i

+ 0 (i))

as x —• +oo, where o(l) denotes a function tending to zero as x —* +oo and n(:r) = E p ^ l denotes the 'prime number counting function', equal to the number of primes p not exceeding x > 0. The Prime Number Theorem6 was eventually proved almost forty years later in 1896, simultaneously and independently by Jacques Hadamard [Had2] and Charles-Jean de la Vallee Poussin [dVl]. (See also the earlier key papers [vMl,2] and [Hadl], along with the later and more precise error estimate obtained in [dV2].) We refer the interested reader to Edwards' book [Edw] or to W. Schwarz's recent survey article [Schw] for a detailed history of the Prime Number Theorem. As is wellknown (see, for example, [Edw], [In], or [Pat,§1.8]), the Riemann Hypothesis is equivalent to the statement that the prime numbers are asymptotically distributed as harmoniously as possible or, more precisely, that the error 5 Referring, in particular, to Riemann's groundbreaking Habilitationschrift—titled On the Hypotheses at the Foundations of Geometry and presented in 1854 to the University of Gottingen—Sir Arthur S. Eddington—the British astronomer whose observation of the 1919 total eclipse of the Sun first confirmed the bending of light rays grazing a massive body (like the Sun), as predicted by Einstein's theory of general relativity—made the following statement (quoted in [Ac,p.l9]): "A geometer like Riemann might almost have foreseen the more important features of the actual world." 6 either in the form (1.4.1) or in the following stronger form conjectured by Gauss, II(x) = Li(x)(l + o(l)) as x -> +oo, where Li(x) = l i m £ ^ 0 + ( / 0 '" + fi+Jukldt denotes the integral logarithm.

T-Duality, Functional Equation

13

term in the statement of the Prime Number Theorem (in the form given in the last footnote) is best possible.7 Arguably, the most beautiful and useful result obtained by Riemann in [Riel] is the so-called Riemann 'explicit formula', connecting U.(x) (or related counting functions) and the critical (or nontrivial) zeros of the Riemann zeta function ((s). (See, e.g., [Edw,Chapter3], [In], [Pat,Chapter3], [ParShal,§2.5], and [Lap-vF2,p.4 and pp.75-76].) Riemann's formula is sometimes referred to as the Riemann-von Mangoldt formula (see, e.g., [LapvF2,§4.5]) because a suitable version of it was later proved in full rigor by von Mangoldt [vMl,2] in the mid-1890's. (See Equation (2.4.20) in Section 2.4.1 below for a classic version of Riemann's formula.) We note that such an explicit formula—along with its later generalizations to other parts of number theory—has recently been extended to the setting of fractal geometry in [Lap-vF 1,2] in order to develop the theory of complex dimensions of fractal strings and to precisely describe in terms of the underlying complex dimensions the oscillations intrinsic to the geometry or the spectrum of fractals. (See [Lap-vF2], Chapter 4 and the relevant applications discussed in Chapters 5-9.) Earlier, in [LapMal,2], a geometric reformulation of the Riemann Hypothesis had been obtained in terms of a natural inverse spectral problem for the vibrations of fractal strings. Rephrased in a more pictorial language, the work of [LapMal,2] can be stated as saying that the question (a la Mark Kac [Kacl]) Can one hear the shape of a fractal drum?—suitably interpreted as the abovementioned inverse problem, connecting the geometric and spectral oscillations of a fractal string—is intimately connected with and, in fact, equivalent to the Riemann Hypothesis. This characterization of Riemann's Conjecture was extended and placed in a broader context in [Lap-vF2], especially Chapter 7. In particular, the intuitive picture of the critical strip 0 < Re s < 1 for £(s)—suggested by the work in [LapPol,2] (see especially [LapPo2,§4.4b], along with [Lap2,Figure 3.1 and §5] and [Lap3,§2.1,§2.2 and p.150]) and corroborated by the results of [LapMal,2]—has been justified rigorously in [Lap-vFl,2]. (See [Lap-vF2,Figure 7.1,p.165] and the discussion surrounding it.) In my opinion, the importance of the Riemann Hypothesis does not lie solely in the incredible multiplicity of its equivalent forms, but also in the cryptic message which it carries with it: one about the geometry of a landscape so far inaccessible to us, that underlying the prime numbers (or the integers). Once we will have found the clues needed to decode this message, 7 Namely, for every 5 > 0, Il(a;) = Li(x) + 0(x2 +<s ) as x —> +oo; see, e.g., [Pat,§1.8 and §5.8].

14

M. L. Lapidus

we should be able to discover and unify large new parts of mathematics, lying at the confluence of arithmetic and geometry. Along similar lines, one could perhaps consider the Riemann Hypothesis— together with the related information concerning the statistical distribution of the prime numbers (for example, that connecting the critical zeros of the Riemann zeta function and aspects of random matrix theory [Mon,Ber3-4, Odl-2,Gut2,KatSarl-2,BerKe])—as a mathematical analogue of the recent COBE observations regarding the extraordinary uniformity (and the tiny fluctuations) of the Penzias-Wilson cosmic background radiation [Tri,PuGis].

1.5

Motivations and Objectives of This Essay

At least from the physical point of view, our goal in the present essay is more modest than suggested earlier. Indeed, we do not plan to discuss a model of geometry that is supposed to fit physical reality at scales at which quantum gravity plays an essential role. Instead, we would like to propose a geometric and physical model that may help us to better understand aspects of number theory, particularly the set of prime numbers (or of integers) and the associated Riemann zeta function, along with their various generalizations for algebraic number fields and curves over finite fields arising naturally in arithmetic geometry (see, for example, [ParShal,2]). This new model (along with its broad extension in [Lap7]) is motivated in part by several mathematical and physical sources, including the following ones: (i) The theory of fractal strings [Lapl-4,LapPol-3,LapMal-2,HeLapl2,Lap-vFl-5,HamLap] (to be viewed here as 'fractal membranes') and the corresponding theory of (fractal) complex dimensions recently developed in the author's research monograph joint with Machiel van Frankenhuysen and entitled Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions [Lap-vF2]. (ii) More generally, the study of the vibrations of fractal drums, associated with Laplacians (or more general elliptic differential operators) on open sets with fractal boundary or on suitable (self-similar) fractals themselves. (See, for instance, [Lapl-6,LapFl,LapPol-3,LapMal-2,HeLapl-2,KiLapl2,LapPan,LapNRG,GriLap,Lap-vFl-5] and the relevant references therein related, in particular, to the "Weyl-Berry Conjecture" [Weyl-2,Berl-2].) We note that fractal strings correspond to the one-dimensional case of 'drums with fractal boundary' but also share certain features with the latter situation.

T-Duatity, Functional Equation

15

(iii) String theory from theoretical physics and its striking dualities, especially the so-called T-duality, a key symmetry not present in ordinary quantum mechanics and which enables us, for example, to identify physically two circular spacetimes with reciprocal radii. (See, e.g., [Asp,EvaGia, GivePR,Gre,Polcl-4,Sch2-4,Val-2,Witl4,16-17].) (iv) Noncommutative geometry and the recent attempts to connect it with conformal field theory and string theory. (See especially [FroGa, ChaFro,Chal-2,LiSzl-2,FroGrRel-2].) (v) Recent attempts at connecting aspects of noncommutative geometry and fractal geometry. (See especially [ConSul], [Con6,§IV.3]—particularly [Con6,§IV.3(e)], motivated in part by [LapPol-2]—as well as [Lap3,Part II], [Lap5], [Lap6] and [KiLap2].) (vi) The intriguing work of Deninger [Denl-7] on a possible cohomological interpretation of analytic number theory and on the Extended Weil Conjectures and in particular, on the (Extended) Riemann Hypothesis.8 Next, we present the contents of the rest of this work: In the main part of this paper (Chapter 2), we discuss the simple but important model of (closed) string theory on a circle (or more generally, on a finite-dimensional torus). We do so both from the standard physical point of view (in Section 2.2) and—following the work of Frohlich and Gawgdzki [FroGa], pursued by Lizzi and Sabo in [LiSzl,2]—from the point of view of noncommutative geometry (in Section 2.3). T-duality—which identifies the physics associated with string theory on toroidal spacetimes with dual lattices and in particular, on circles of reciprocal radii—is presented from each perspective in Sections 2.2 and 2.3, respectively. In Section 2.4, we suggest that in this context, the functional equation for the Riemann zeta function C, = C(s) is a natural counterpart of T-duality for string theory on a circle (or more generally, on a fractal membrane, in the sense of [Lap7]), while the Riemann Hypothesis may be connected, in particular, to the existence of a fundamental (or minimum) length in string theory, itself a consequence of T-duality. (We point out for the interested reader that in the first part of Section 2.4, we review some of the basic properties of £(s)—and of other number-theoretic zeta functions—that are used throughout much of this work; see Section 2.4.1 .) 8

See, e.g., Appendix B to [Lap7] for a brief discussion of the classic Weil Conjectures [Wei5] (and Theorem [Wei 1-4], in the case of curves over finite fields), along with some of their motivations.

M. L. Lapidus

16

This essay may serve as a relatively nontechnical introduction t o the beginning of a theory presented in a book in preparation by the author [Lap7], entitled In Search of the Riemann Zeros (and subtitled Strings, fractal membranes, and noncommutative spacetimes). In the latter research monograph is developed, in particular, a theory of fractal membranes—viewed as suitable quantum or multiplicative analogues of the author's fractal strings. The notion of fractal membrane—interpreted heuristically as a noncommutative (adelic) infinite dimensional torus 9 , in the spirit of the model presented in Chapter 2—may help in the long-term provide a unified geometric, algebraic and physical framework within which to understand number-theoretic zeta functions and their associated arithmetic geometries. More concretely, the well-known Euler product representation of the Riemann and other arithmetic zeta functions is naturally interpreted in this context in terms of the quantum partition function of the corresponding fractal (or prime) membrane. We close this introduction by pointing out t h a t in an epilogue to the present paper, entitled Fractal Membranes and Arithmetic Site, we will provide a somewhat more detailed description of the part of [Lap7] dealing with the theory of fractal membranes and its connections with zeta functions. 10 It should be made clear to the reader that the research program outlined in this essay (as well as in the forthcoming book [Lap7]) is still in a preliminary form and t h a t some of the mathematics involved or implied is rather formidable or even not yet formulated in a precise manner. We hope, nevertheless, t h a t some of the ideas presented here—once suitably modified or extended—may serve as a motivation for future research at the interface of noncommutative geometry, fractal geometry and aspects of arithmetic geometry and number theory.

2

String Theory on a Circle and T-Duality: Analogy with the Riemann Zeta Function Nature is simple if we look at it the right way. For example, I believe that God created just two dimensions—one of space and one of time. 9

or equivalently, as an adelic Hilbert cube, with opposite faces identified. We note that because in [Lap7], we consider the vibrations of fractal membranes rather than of fractal strings, the role played by the Riemann zeta function and the Riemann Hypothesis (or their generalization) is rather different than in the previous work of the author and his collaborators, Carl Pomerance, Helmut Maier, Christina He and Machiel van Prankenhuysen [Lapl-4,LapPol-3,LapMal-2,HeLapl-2,Lap-vFl-3]. 10

T-Duality, Functional Equation

What could be simpler? And then at a later epoch, there was a phase transition to four dimensions, plus six internal ones. Two is the heart of the matter in this theory and this number two will not be changed to three. Abdus Salam, 1988 [DaviBr, pp. 175-176] Recall that ... a particle's quantum wavelength is inversely proportional to its momentum, which, roughly speaking, is its energy. And so, by increasing a point particle's energy, its quantum wavelength can be made shorter and shorter—quantum smearing can be decreased further and further—and hence we can use it to probe ever finer physical structures. Intuitively, higher-energies particles have greater penetrating power and are therefore able to probe more minute features. In this regard, the distinction between point particles and strands of string becomes manifest. Just as [is] the case for plastic pellets probing the surface features of a peach pit, the string's inherent spatial extend prevents it from probing the structure of anything substantially smaller than its own size—in this case structures arising on length scales shorter than the Planck length. Somewhat more precisely, in 1988 David Gross of Princeton University and his student Paul Mende showed that when quantum mechanics is taken into account, continually increasing the energy of a string does not continually increase its ability to probe finer structures, in direct contrast with what happens for a point particle. They found that when the energy of a string is increased, it is at first able to probe shorter-scale structures, just like an energetic point particle. But when its energy is increased beyond the value required for probing structures on the scale of the Planck length, the additional energy does not sharpen the string probe. Rather, the energy causes the string to grow in size, thereby diminishing its short distance sensitivity. In fact, although the size of a typical string is the Planck length, if we pumped enough energy into a string—an amount of energy beyond our wildest imaginings but one that would likely have been attained by the big bang—we could cause it to grow to a macroscopic size, a clumsy probe of the microcosmos indeed! It's as if a string, unlike a point particle, has two sources of smearing: quantum jitter, as for a point particle, and also its own inherent spatial extent. Increasing a string's energy decreases the smearing from the first source but ultimately increases the smearing from the second. The upshot is that no matter how hard you try, the extended nature of a string prevents you from using it to probe phenomena on sub-Planck-length distances. But the whole conflict between general relativity and quantum mechanics arises from the sub-Planck-length properties of the spatial fabric. / / the elementary constituent of the universe cannot probe subPlanck-length distances, then neither it nor anything made from it

18

M. L. Lapidus can be affected by the supposedly disastrous short-distance quantum undulations. ... An electron microscope has the ability to resolve surface features to less than a millionth of a centimeter; this is sufficiently small to reveal the numerous surface imperfections [of a well-polished block of granite, for example]. By contrast, in string theory, there is no way to expose the sub-Planck-scale "imperfections" in the fabric of space. In a universe governed by the laws of string theory, the conventional notion that we can always dissect nature on ever smaller distances, without limit, is not true. There is a limit, and it comes into play before we encounter the devastating [effects of] quantum foam. Brian R. Greene, 1999 [Gre, pp. 155-156]

2.1

Quantum-Mechanical Point-Particle on a Circle

It is well-known t h a t the space of states of a quantum-mechanical pointparticle moving in a circle of radius R, S(R) = R/2irRZ, is described by the complex Hilbert space Ti = L2(S(R),ds) consisting of all square-integrable complex-valued functions on S(R), where ds denotes (normalized) Haar measure on S{R).U Equivalents, H = L2(S\ ^ ) , where S1 = 5(1) = M/27rZ denotes the unit circle equipped with the Riemannian metric ds2 = R2dx2. Note that the radius R (or equivalently 2wR, the length of the circle) is the only metric invariant in this situation. Hence the positive half-line {r : r > 0} parametrizes the space of all Riemannian circles (in the plane). The circle S1 is equipped with its natural Dirac operator D = -hsR-1-^, the square of which yields the (negative) Laplacian or (free) Hamiltonian on the circle, H = D2 = — \R~2^Moreover, from the point of view of quantum mechanics, we can recover the geometry of the circle—viewed as a set of points, or more precisely, as a smooth Riemanian manifold—from the spectral triple associated with the Dirac operator [Con5]

(2X1)

±.C-W)

or equivalently [FroGa], from the spectral triple associated with the Hamiltonian dr 1 d2 (2.1.2) (H,H, A) = (L2(S\ •—=), --R-2 — ,C°"(Si)). 11 Here and in the following, R,C and Q denote the field of real, complex and rational numbers, respectively. Further, Z = {0, ±1, ±2,...} denotes the ring of integers and N = {0,1,2,...} the set of nonnegative integers, while N* = N\{0} = {1,2,3,...}.

19

T-Duality, Functional Equation

Here, in (2.1.1) and (2.1.2), A := Cx{Sl) denotes the pre-C*-algebra of all infinitely differentiable functions on S1. Recall that the commutative algebra A is represented on the Hilbert space H via the standard representation which associates to the function / in A = C"X>(S'1) the multiplication operator Mf in C(1i) defined by (2.1.3)

(M^)(0 = /(0^(0.

for^GW

and

£ £ S1.

Thus „4 can be viewed as a subalgebra of the C*-algebra C(H) of all bounded linear operators on H. (The C*-algebra naturally associated with Cco(S1) is the commutative C*-algebra C(S1) of all continuous complex-valued functions on S1.) Both the Dirac operator D and the Hamiltonian H = D2 are unbounded self-adjoint operators on H; further, H is nonnegative and hence its spectrum is contained in [0, +oo). Moreover, D and H have a discrete spectrum, with eigenvalues respectively equal to R~xn and R~2n2, for n e Z , and associated eigenfunctions
An = |n|i?- 1 ,

withneZ.

Hence the spectral zeta function13 of \D\ on S(R), denoted Cs(i?)(s) = and defined by

GJ,|£>|(S)

oo

(2.1.5)

(S(R)(s)=TTS,ce(\D\-s)=

J2

X

n°'

for Re s » 0,

n~—oo is given b y

(2.1.6)

Cs ( *)(s)=2i? s C(s),

for

seC,

where C = C(s) denotes the Riemann zeta junction. 12

Note that all the eigenvalues of D are simple whereas 0 is a simple eigenvalue of H but the other eigenvalues of H, namely Rr2,4R~2,9R~2,..., have multiplicity two. 13 See, for example, [Lap-vF2,Appendix B] for information and references (including [See,Gil]) on the spectral zeta functions of smooth Riemannian manifolds.

20

M. L. Lapidus

Indeed, in view of (2.1.4)-(2.1.5), we have for Re s > 1, oo

oo

Cs(fl)(s) = ^ Yl M-s = 2i? s J>- s = 2irc(s),14 n=—oo

n=l

since £(s) := Y^Li n~s for Re s > 1. Thus Cs(K)(«) admit a meromorphic extension to all of C given by (2.1.6). It follows from (2.1.6) that for R = l,Cs(i)(s) essentially coincides with the Riemann zeta function; namely, (2-1-7)

Cs(i)(s)=2C(s),

for s e C .

Finally, we recall t h a t the momentum of a quantum-mechanical pointparticle on a circle is quantized, due to the compactness of S1 (or of S(R)). Indeed, mathematically, this well-known fact corresponds to the discreteness of the (normalized) momentum operator p = -y^R'1^ acting on H = L 2 (S' 1 , - 7 ^ ) . The allowed momenta—or equivalently, the eigenvalues of p— are given by (2.1.8)

k = nR-\

with n G Z .

Note t h a t the smallest nonzero value of \k\ is equal to R"1, and that this value becomes larger as the radius of the circle decreases. Therefore, the quantum wavelength of the point-particle—which is inversely proportional to |fc|—can be made arbitrarily small as R decreases to zero, thereby enabling one to probe finer and finer structures from the quantum-mechanical point of view. This is in sharp contrast with what happens in string theory due to the extended size of a string probe, as is discussed in the second quote heading this chapter as well as in the quote heading the next section.

2.2

String Theory on a Circle: T-Duality and Existence of a Fundamental Length A point particle moving in a [Garden-hose or cylindrical] universe can execute the [following] kinds of motion: It can move along the extended dimension of the Garden-hose, it can move along the curledup part of the Garden-hose, or any combination of the two. A loop

14

We use here the convention 0~s = 0 for Re s > 0. Alternately, we could adopt the common convention according to which the eigenvalue 0 is excluded from the summation defining the spectral zeta function (2.1.5).

T-Duality, Functional Equation

of string can undergo similar motion, with one difference being that it oscillates as it moves around on the surface [of the cylinder]. The oscillations of the string imbue it with characteristics such as mass and force charges. Although a crucial aspect of string theory, this is not our present focus, since we already understand its physical implications. Instead, our present interest is in another difference between pointparticle and string motion, a difference directly dependent on the space through which the string is moving. Since the string is an extended object, there is another possible configuration beyond those already mentioned: It can wrap around—lasso so to speak—the circular part of the Garden-hose universe ... The string will continue to slide around and oscillate, but it will do so in this extended configuration. In fact, it can wrap around the circular part of the space any number of times ... and again will execute oscillatory motion as it slides around. When a string is in such a wrapped configuration, we say that it is in a winding mode of motion. Clearly, being in a winding mode is a possibility inherent to strings. There is no point-particle counterpart. We now seek to understand the implications of this qualitatively new kind of string motion on the string itself as well as on the geometrical properties of the dimension it wraps. ... Strings that wrap around a circular component of space share almost all of the same properties as the [other kinds of] strings. Their oscillations, just as those of their unwound counterparts, contribute strongly to their observed properties. The essential difference is that a wrapped string has a minimum mass, determined by the circular dimension and the number of times it wraps around. The string's oscillatory motion determines a contribution in excess of this minimum. It is not difficult to understand the origin of this minimum mass. A wound string has a minimum length determined by the circumference of the circular dimension and the number of times the string encircles it. The minimum length of a string determines its minimum mass: The longer this length, the greater the mass, since there is more of it. Since the circumference of a circle is proportional to its radius, the minimum winding-mode masses are proportional to the radius of the circle being wrapped. By using Einstein's [equation] E = mc2 relating mass to energy, we can also say that the [winding] energy bound in a wound string is proportional to the radius of its circular dimension. How does the existence of wrapped strings affect the geometrical properties of the dimension around which the string winds? The answer, first recognized in 1984 by the Japanese physicists Keiji Kikkawa and Masami Yamasaki, is bizarre and remarkable. Let's consider the last cataclysmic stages of our variant on the big crunch in the Garden-hose universe. As the radius of the circular dimension shrinks to the Planck length and, in the mold of general

M. L. Lapidus

relativity, continues to shrink to yet smaller lengths, string theory insists upon a radical reinterpretation of what actually happens. String theory claims that all physical processes in the Garden-hose universe in which the radius of the circular dimension is shorter than the Planck length and is decreasing are absolutely identical to physical processes in which the circular dimension is longer than the Planck length and increasing! This means that as the circular dimension tries to collapse through the Planck length and head toward ever smaller size, its attempts are made futile by string theory, which turns the table on geometry. String theory shows that this evolution can be rephrased— exactly reinterpreted—as the circular dimension shrinking down to the Planck length and then proceeding to expand. String theory rewrites the laws of short-distance geometry so that what previously appeared to be complete cosmic collapse is now seen to be a cosmic bounce. The circular dimension can shrink to the Planck-length. But because of the winding modes, attempts to shrink further actually result in expansion. ... The new possibility of wound-string configurations implies that the energy of a string in the Garden-hose universe comes from two sources: vibrational motion and winding energy. From the legacy of Kaluza and Klein, each depends on the geometry of the hose, that is, on the radius of the curled-up component, but with a distinctly stringy twist, since point particles cannot wrap around dimensions. Our first task, then, will be to determine precisely how the winding and vibrational contributions to the energy of a string depend on the size of the circular dimension. For this purpose, it proves convenient to separate the vibrational motion of strings into two categories: uniform and ordinary vibrations. Ordinary vibrations refer to the usual oscillations [of a string]; uniform vibrations refer to even simpler motion: the overall motion of a string as it slides from one position to another without changing its shape. All string motion is a combination of sliding and oscillation—of uniform and ordinary vibrations—but for the present discussion it is easier to separate them in this manner. In fact, the ordinary vibrations will not play a central part in our reasoning, and we will therefore include their effects only after we have finished giving the gist of the argument. Here are the two essential observations. First, uniform vibrational excitations of a string have energies that are inversely proportional to the radius of the circular dimension. This is a direct consequence of the quantum-mechanical uncertainty principle: A smaller radius more strictly confines a string and therefore, through quantum-mechanical claustrophobia, increases the amount of energy in its motion. So, as the radius of the circular dimension decreases, the energy of motion of the string necessarily increases—the hallmark of an inverse propor-

T-Duality, Functional Equation

23

tionality. Second, as found [earlier], the winding mode energies are directly—not inversely—proportional to the radius. Remember, this is because the minimum length of wound strings, and hence their minimum energy, is proportional to the radius. These two observations establish that large values of the radius imply large winding energies and small vibrations energies, whereas small values of the radius imply small winding energies and large vibration energies. This leads us to the key fact: For any large circular radius of the Garden-hose universe, there is a corresponding small circular radius for which the winding energies of strings in the former universe equal the vibration energies of strings in the latter, and vibration energies of strings in the former equal winding energies of strings in the latter. As physical properties are sensitive to the total energy of a string configuration—and not to how the energy is divided between vibration and winding contributions—there is no physical distinction between these geometrically distinct forms for the Garden-hose universe. And so, strangely enough, string theory claims that there is no difference whatsoever between a "fat" Garden-hose [of radius R] and a "thin" one [with reciprocal radius, 1/R]. ... We should now include the ordinary vibrations as well, since these give additional contributions to the string's total energy and also determine the force charges it carries. The important point, however, is that investigations have revealed that these contributions do not depend on the size of the radius... We therefore conclude that the masses and the charges of particles in a Garden-hose universe with radius R are completely identical to those in a Garden-hose universe with radius 1/R. And since these masses and charges govern fundamental physics, there is no way to distinguish physically these two geometrically distinct universes. Any experiment done in one such universe has a corresponding experiment that can be done in the other, leading to exactly the same results. Brian R. Greene, 1999 [Gre, pp. 237-246]

2.2.1

String T h e o r y on a Circle

We now consider (bosonic, closed) string theory (or the associated conformal field theory) on a circle, S(R), assumed to be of radius R and centered at the origin. We could similarly discuss string theory on a tube or a cylinder, corresponding to a target space with one compact circular dimension and one (or several) noncompact extended dimensions(s) (see [Polc3, §8.2]), as in the quote heading the present section. Also, we could consider in much the same way the case where the circle is replaced by a (higher-dimensional) torus, a situation often referred to as 'toroidal compactification' in string theory

24

M. L. Lapidus

(see, for example, [Polc3,Chapter 8], [FroGa,§2], and [LiSz2]). In fact, we will discuss this easy generalization later on in this chapter. (See Remark 2.2.2 following Equation (2.2.33), as well as Section 2.3.) For now, however, we restrict ourselves to a circular spacetime for notational simplicity and for conceptual clarity. We will closely follow the exposition given in [Polc3,§8.2], [FroGa,§2], and [LiSz2]. Because of its basic physical importance and because it is prototypical of what happens for more complex geometric situations, 'circle duality' or the T-duality associated with string theory on a circle is discussed in many physics references, including [Asp,EvaGia,GivePR,Gre,Polcl-4,Sch2-4,Val2,Witl5-17]. The reader interested in a detailed exposition may wish to consult the classic survey article [GivePR], entitled Target Space Duality in String Theory. We hope that the reader who was not previously familiar with string theory or with the associated mathematics may still acquire an intuitive feeling for some of the physical considerations involved by reading the quotes heading this chapter and several of its sections (including the present one). In fact, these quotes should be considered as an integral part of our discussion. We also strongly recommend reading the very nice popular exposition of string theory given in Brian Green's recent book for a general scientific public [Gre], entitled The Elegant Universe and from which most of these quotes are excerpted. For an early expository account of string theory (pre-dating T-duality and its various ramifications) putting it in the context of quantum field theory and of the standard model for elementary particles [Wein2-4], we point out Steven Weinberg's beautifully written book [Weinl], entitled Dreams of a Final Theory. In particular, the philosophically inclined reader may find illuminating some of Weinberg's comments on quantum mechanics, viewed as a fundamental physical theory, in contrast to quantum field theory, which is viewed as an effective theory that is a 'mere' (albeit very important) low-energy approximation to a more fundamental theory, perhaps a suitable version of string theory. (Some of these comments are quoted from [Weinl] in the last few pages of Chapter 20 of [JohLap].) Let us first briefly discuss the classical counterpart of the abovementioned situation. Thus we consider the simplest type of linear 'sigma model' when the 'target space' is a circle S(R). Then the action of this sigma model is given by (2.2.1)

S(x(. , •)) = T - [[(%*

~ %x)dadt,

25

T-Duality, Functional Equation

where x = x(a,r) denotes the angle variable along the circle S(R), viewed as S1 equipped with the Riemannian metric ds2 = R2dx2 as in Section 2.1. Here, E is the world-sheet of the closed string (or loop), the twodimensional surface spanned by the string as it moves with time r in the circle target space; it is assumed to be a cylinder S = K x S1, parametrized by the time and space variables r £ l and a £ S1, respectively. (More precisely, a is the variable parametrizing the oriented closed string.) Later on, it will also be convenient to use the so-called light-cone coordinates r ± a or equivalent^, the holomorphic and anti-holomorphic coordinates z± = e~^T±
(2.2.2)

This is the classical equation of motion in this context. Using Fourier series, it is easy to check that the general solution of (2.2.2) is given by (2.2.3) x(a,

T)=X°

+

R~2kT + wa + Y

-7^—a+ein{c7+T)

+ V —^—o^e"^"^,

where the summation in (2.2.3) is taken over all nonzero integers n in Z, i := y/— 1, and where w g Z i s called the winding number. Note (as in [FroGa]) that w is a homotopy invariant which labels the connected components of the space of solutions of the wave equation (2.2.2). Further, in (2.2.3), x° denotes the position of the center-of-mass of the closed string (or loop) and k stands for its linear momentum. If {. , .} denotes the standard Poisson bracket, the (classical) canonical (equal-time) commutation relations (2.2.4)

{x(a1,0),dTx{a2,0)}

= 2irR'25(al

- a2)

imply that (2.2.5)

{x°,fc} = l, {a+,a+} = m£ n+roi0

and

{a~,a^}

=in5n+m<0,

with all the remaining Poisson brackets vanishing. (Here and thereafter, we use the standard symbol Kronecker delta; that is, Sab = 1 or 0 according to whether a = 6 or a =fi b, respectively.)

26

M. L. Lapidus

Finally, from the classical point of view, this model has two types of symmetries: (i) Invariance by reparametrization of the chiral variables T±CT, viewed as a conformal symmetry, and (ii) the transformations X(O,T)^X{O,T)±X±X{T±G),

(2.2.6)

where \

± xare

'arbitrary' periodic functions.

We next consider the quantum version of this model: conformal field theory or (bosonic) closed string theory on a circle. First, the quantum field corresponding to x is a scalar field X satisfying (2.2.7)

X ~X +

2TTR.

Physically, X represents the quantum state of a closed string moving in the circle. The periodicity required by (2.2.7) has two key consequences: (i) First, the operator exp(27rzi?p) which translates strings around the circle must leave the states invariant; hence, as is the case for a point-particle moving around a circle (see Section 2.1 above), the momentum of the centerof-mass of a string is quantized: (2.2.8)

k = ^ , with v eZ. R (Above, p denotes the corresponding momentum operator.)

(ii) Second, and this is characteristic of string theory with circular (or more generally, toroidal) target space, a closed string (or loop) may wind around the circle an integral number of times (clockwise or counterclockwise); that is, (2.2.9)

X(a + 2?r) = X(a) + 2irRw,

with

w € Z.

The integer w is called the winding number (associated with the state X of the string). We stress that this does not have a counterpart for the situation considered in Section 2.1 of a quantum-mechanical point-particle moving in a circle. Indeed, such a motion of wrapped strings around the circular (or compact) dimension is only possible because of the extended nature of the strings and the particular geometry of the target space. (See, for instance, the quotes heading the present section and the next one. Also see [Polc3,Figure 8.1,p.236] which gives various examples of closed strings on a cylinder with different winding numbers.) This fact is crucial to understand T-duality and some of our later arguments.

T-Duality, Functional Equation

27

For clarity, we will refer to v as the vibration number (as in [Gre]). (Note that v was denoted by n in Section 2.1.) In the present situation, which is essentially that of a free field theory, quantization amounts to replacing the Poisson brackets in Equations (2.2.4)(2.2.5) by —i times the commutators of the corresponding operators. We postpone until the next section the precise description of the resulting Hilbert space of quantum states. (See Section 2.3.2.) For now, we mention that the states of the quantum model are labeled by \v;w), where w € Z is the winding number of the string (as in Equation (2.2.9)) and v = vw is the vibration number, the quantum number associated with the momentum of the center-of-mass (as in Equation (2.2.8)). Equivalently, the quantum states are labeled by \p+\p~), where p+ denotes the left-momentum and p~ is the right-momentum, given (up to normalization) by (2.2.10a)

p+ =

_L(l

+ WjR )

and (2.2.106)

p-

=

_L(^_WjR).i5

Physically, in (2.2.10) (or in (2.2.23) below), the term v/R corresponds to the contribution of a uniform-vibration mode (in the sense of the quote from [Gre] heading this section), associated with the overall motion of the sting without changing its shape; further, the term wR corresponds to the contribution of a winding mode, associated with the closed string wrapping \w\ times around the circle S(R), clockwise or counterclockwise, depending on whether w < 0 or w > 0, respectively. The set of left-right momenta {(p+,p~)}, form (2.2.11)

Q((P+,p-))

equipped with the quadratic

:= (p+)2 - (p~)2 = 2vw € 2Z,

is an integral, even self-dual Lorentzian lattice A, of rank 2 and of signature (1,1). (Hence, A = A*, the dual lattice of A, and the values of the quadratic 15

Frequently, the normalization adopted in the literature is such that p * = R~xv ± wR or in dimensional units, if the string tension {2na')~1 is taken into account, as in [Polc3], for example, p * = vR~x ±wR(a')~v; in the latter case, the string scale is equal to (a') 1 ^ 2 whereas in our units, it is equal to one. We caution the reader that we may not always be entirely consistent in our various choices of normalization, which should have only minor consequences.

28

M. L. Lapidus

form Q are even integers.) In the physics literature, A is often called a Narain lattice [Nar]. The even and self-duality properties of A provide an elegant mathematical translation of the modularity of the associated conformal field theory on the world-sheet [SeiWit]. In other words, they play a key role in expressing mathematically the T-duality symmetry satisfied by string theory on circles. The chiral (multi-valued) quantum fields X± of the string theory—also called the Fubini-Veneziano fields—can be described by a suitable counterpart of Equation (2.2.3):

(2.2.12)

X±(T ±O-) = X°±+P±(T±O-)

+ J2

- Q n ein(T±CT),16

njtO

where the sum runs over all nonzero integers n € Z and where the zero-modes (i.e., the center-of-mass coordinates of the string) X± and the center-of-mass, left-right momenta operators p* are canonically conjugate variables; that is, they satisfy the commutation relations (2.2.13)

[X°,p ± ] = - i ,

with the remaining commutators vanishing. (Here and in the following, [a, b] := ab — ba denotes the commutator of the operators a and b.) Moreover, the oscillatory modes a„ (n e Z) in (2.2.12) obey the commutation relations (2.2.14)

[a±,a±]=n<W m , 0 ,

with all other commutators vanishing. (Observe that the formulas (2.2.13) and (2.2.14) are the quantum analogue of the commutation relations (2.2.5).) The oscillatory modes a* correspond to (bosonic) creation and annihilation operators (for n < 0 and n > 0, respectively) acting on a commuting pair of Fock spaces, to be further discussed in Section 2.3.2. (Physically, they represent the internal oscillations of the string, referred to as 'ordinary vibrations' in the quote from [Gre] heading this section.) Also, (a*)* = Q^n, where u* denotes the adjoint of the operator u, and cvj := p^. The quantum counterpart of the symmetry (2.2.6) is generated by a commuting pair of u(l) current algebras with the currents17 oo

(2.2.15)

f{T ± a) := 8aX±(r ± a) = £

j ^

T ±

' \

____^_ n=—oo 16 Here and thereafter, the + signs should be taken together, and similarly for the — signs. Hence, for example, the left-hand side of (2.2.12) reads X+{T + a) or X-(r — a), respectively. 17 Here, u(l) = K denotes the Lie algebra of S1.

29

T-Duality, Functional Equation

where (2.2.16)

Jo := p±

and

j± := a±

for n =£ 0, n € Z.

Remark 2.2.1. In the physics literature and in papers on vertex algebras, one often adopts complex coordinates z = a + IT, Z = a — IT (or rather, z = e~l<-T+a\ z = e -, ( T- °')), and consider that the quantum field X is given by X = X(z, z). Then, for example, the wave equation (2.2.2) becomes ddX(= BdX) = 0 and so dX must be holomorphic while dX is anti-holomorphic (i.e., holomorphic in z); hence the notation dX(z) and dX(z).18 One then uses interchangeably the terminology holomorphic or left-moving field, and similarly, anti-holomorphic or right-moving field. Further, in view of (2.2.12) (and since formally, r + a = i log z while r — a — i log z), the last two equalities of (2.2.15) can be written as the Laurent series expansions oo

(2.2.17)

-idX(z)=

oo

JnZ~(n+l)

Y,

and -idX(z)=

n=—oo

^

j-f-("+1).

n=—oo

(It is useful to keep these facts in mind when comparing the model proposed in [Lap7] with aspects of the work in [Denl-7].) See, for instance, [Polc3,Chapters 2 and 8], where somewhat different notation are used. The quantum analogue of the abovementioned classical conformal symmetry (see (i) just before (2.2.6))19 is given by a commuting pair of Virasoro algebras generated by (2.2.18)

1 °° L± = - ^ :j£?±_m:,

with n e Z,

m=—oo

where the symbol : : stands for the Wick (or normal) ordering that puts the creation-annihilation operators a„ with n < 0 to the left of the ones with n > 0. (More generally, : ct^a^ : is equal to a^a^ if k < m and to a^a^ if k > m.) The Virasoro generators L^ satisfy the commutation relations (2.2.19)

[Z±, L^ = (n - m)L± +m + g ( n 3 - n)<5n+m,0,

with central charge C := 1, the dimension of the circular space-time S1 (or S(R))18 Later on, we will also use the alternate notation z± = e 1 (
30

M. L. Lapidus

It follows from (2.2.16) and (2.2.18) that

^ = ^(P ± ) 2 + E "-»"»•

(2-2.20)

Z

n=l

The significance of L j and LQ is that they enable us to construct the "global space-time symmetry generators of the Poincare algebra" [LiSz2,p.680]. In fact, the momentum operator, generating classically the space translations of S1, is defined by P = LQ — LQ. Moreover, the Hamiltonian or energy operator (generating classically the time translations of spacetime) is defined

by # = L + + L 0 - - f | . In view of (2.2.20), the momentum operator P is given explicitly by -

(2.2.21)

..

+ 2

P = L+- L~0 = - ( p ) - -{p-f

oo

oo

+ £>!„«+ - $>:»«». n=l

n=l

while the Hamiltonian operator is given by f~l

(2.2.22) H = LZ+L;-^

1

1

°°

°Q

p

= -(p+)2+_(p-)2+£a+na++£a:na-_^. n=\

n=l

In Section 2.3, we will introduce the Hilbert space on which P and H act. (See especially §2.3.2.) 2.2.2

Circle Duality (T-Duality for Circular Spacetimes)

Next, we discuss T-duality (also called target space duality) in this context. In a nutshell, circle duality consists in stating that from the point of view of string theory, all physical processes associated with a circular spacetime of radius R are identical with those associated with a circular spacetime of radius 1/R.20 (See, for example, the quotes heading the present section and Section 2.3.) In other words, the corresponding 'stringy spacetimes' are physically indistinguishable, even though the original spacetimes S(R) and S(l/R) are not isometric, as Riemannian manifolds. In the next section, we will see one mathematical embodiment of T-duality—expressed in the language of noncommutative geometry—but for now, we will limit ourselves to a more standard physical discussion. 20

If the underlying spacetimes also have noncompact—but identical—dimensions, then the T-duality principle is still valid and is applied to the compactified circular (or more generally, toroidal) dimensions.

T-Duality, Functional Equation

31

First, we consider what is probably the simplest (but still quite physical) illustration of circle duality. The energy of a string state with vibration number w e Z and winding number t o e Z i s given by (2.2.23)

Ev,w =

v2

-^+w2R2.

Therefore, the totality of allowable energies is obviously invariant under the replacement of R by l/R. (See, e.g., [Polc3,Chapter 8,esp.§8.3] for a physically more accurate discussion.) Similarly, when R is replaced by l/R (i.e., R <-> 1/i?), the winding modes and the vibration (or momentum) modes are exchanged. More precisely, in view of (2.2.10), v <-> — w and w <-> v, or equivalently, p+ <-> p~ and p~ <-> — p+, when i? <-> l/R. It follows from the above discussion that the only self-dual radius is given by i?Seif-duai = 1, since it is equal to its own reciprocal. Note that in dimensional units as specified in footnote (19) above, i?seif-duai = (a1)1^2, the string length (or scale). As was alluded to earlier, a mathematical translation of circle duality is the fact that the Narain lattice of left-right momenta, A = {(p+ ,P~)}, is even and self-dual, relative to the quadratic form Q given by (2.2.11). (Observe that Q <-» — Q under the above transformation.) In particular, A coincides with its dual lattice (relative to the nondegenerate bilinear form associated with Q): A = A*. Thus, it is clear to someone familiar with some of the mathematics and the physics involved that circle duality should be understood in terms of the Poisson Summation Formula21, recalled in Section 2.4 below in Equation (2.4.11).22 21 Indeed, this insight struck the author while he was discussing the physical aspects of T-duality with the theoretical physicists Giovanni Landi and Fedele Lizzi during the Conference on Noncommutative Geometry held in Lisbon in September 1997. Along with the work in [Lap-vF 1,2], it is in part at the origin of the model for the Riemann zeta function proposed in this work. (See especially Section 2.4.2 and [Lap7].) 22 In quantum physics, a classic example related to T-duality can be found in statistical mechanics (on a lattice), in which the exchange of the (absolute) temperature T with its reciprocal 1/T leaves invariant the partition function of the system. The latter property follows from the modularity (or the functional equation) of the theta function of the lattice (see Equation (2.4.13) below), itself a consequence of the classic Poisson Summation Formula (2.4.11). (Despite this well-known analogy, the term 'T-duality' seems to come from 'target space duality'.) We note that another important precursor of the duality symmetries of string theory is provided by the electric/magnetic exchange symmetries of Maxwell's theory of electromagnetism; see, e.g., [Witl4,16].

32

M. L. Lapidus

We will now close our technical discussion of T-duality in this section by presenting one such standard interpretation, following the exposition given in [Polc3, esp.§7.2, pp.208-215, and §8.2, pp.237-238]. (We note t h a t at the end of Remark 2.2.2 and in the higher-dimensional case, we will provide a more general discussion in terms of the self-duality of the Narain lattice A.) In the present situation, by analogy with quantum statistical physics, the partition function of the model is given by (2 2 24)

Zst(u)

= ZstR(u))

= Tracefexp(27rio;iP — 2TTW2H)\ = (qq)~l/24

Trace^V0"),

where w = u>i + icu2 G C, with w2 > 0, q := exp(27rzw), q = exp(—2iriuj), P and H denote the momentum operator and the Hamiltonian operator given by (2.2.21) and (2.2.22), respectively, while the operators L j and L^ are the Virasoro generators given by (2.2.20). In light of [Polc3,§7.2], formula (2.2.24) becomes successively (with the left-right momenta p± given by (2.2.10)): oo

2

z«A») = \riH\-

E

?(p+)2/¥p")2/2

V W

(2.2.25)

- Z-°°

= |?7(a;)|"

2

E exp <-TTLJ2(^ v,w=—oo ^

where r\ = r?(w) is the Dedekind eta function

+ UJ2R2)

+ 2TTIUJ1VW \ , '

defined by

oo

(2.2.26)

7/(0,):= ql'2i

\{{l-qn). 71=1

Physically, in (2.2.25) (and in (2.2.30)-(2.2.31) below), the term \T](UJ)\-2 can be viewed as the contribution of the oscillatory modes while the 'moment u m sum' represents the contribution of the vibrational and winding modes. Recall that n satisfies the following identities (see, e.g., [Polc3,Eq.(7.2.44), p.216] or [Ser,§VII.4-5]): (2.2.27a)

7](to + 1) = exp(wr/12)r?(w),

(2.2.276)

»?(-l/w) = ( - i w ) 1 / 2 r ? H ; 2 3

23 The function n(u) is closely related to a well-known modular form, H(w) := (rj(u))24 = J3^_ 1 (l — gn)24- The function H(u>) is a modular form of weight 12, a property equivalent to (2.2.27). Further, S(w) is a common eigenfunction of all the Hecke operators and is wne given by the Fourier series expansion: £(u;) = Y^=\T{n)ln^ r e n i—• r(n) is called the Ramanujan function; for example, r ( l ) = l , r ( 2 ) = - 2 4 , T ( 3 ) = 252,... We refer, for instance, the interested reader to [Ser,Chapter VII,esp.§4 and §5] for information about this classic subject.

33

T-Duality, Functional Equation

so that (2.2.28a)

Mw + 1)| - 2 = \v(")\~2,

(2.2.286)

\r)(-l/u)\-2 = M"^^)!" 2 -

To establish the modular invariance of the partition function (i.e., Zst,R(u + 1) = Z$t>R(uj) and Zst,R{-\/u) = Zst,R(uj)), we now apply a special case of the Poisson Summation Formula (2.4.11), namely (for t > 0 and tiel), oo

(2.2.29)

oo

exp(-7rtv2 + 2niuv) = t~1/2

^ V= — OO

7r(m — u)

J ^ exp

t

T71= —OO

Writing the double sum, ^2V w as J^ w ^Z m the ^as^ e Q u a hty of (2.2.25) and letting, for each fixed w € Z, t = u^/R2 and u = wiw in formula (2.2.29), we obtain after some elementary manipulations (and noting that \m — UJW\2 = m2 +UJ\W2 + w2w2 — 2mwuJi):

(2.2.30)

ZstM(uj) = RZ,(u)

TTR

J2 exp

\m — wu>\ UJ2

m,w=—oo

where Z» = (2.2.31)

Z„{UJ)

is independent of Z? and is defined by Z.(u>) :=

\r,(w)\-W/2,

with 77 = r/(a;) given by (2.2.26). Since u 2 = I m w = I m ( u + l), we have Z*(UJ) = Zt(uj+1) by (2.2.28a), and since Im(—1/w) = LU2/\U}\2, we also have Zt(—l/w) = Z»(<j) by (2.2.28b). Hence, Z* is modular invariant. Further, the double sum on the right-hand side of (2.2.30) is easily seen to be invariant under u> <-> u> + 1 (by making the change of variables m' = m + w, w1 = w). Moreover, it is also seen to be invariant under w <-> —1/ui (by making the change of variables m' = —w, w' = m).2i Therefore, in light of (2.2.30), we conclude that the partition function, Zst = ZstR, associated with string theory on a circle of radius R, is modular invariant, as desired: (2.2.32a) 24

ZsttR(uj + 1) = Zst,R(u)

which, not coincidentally, is identical to the transformation v <-> — w and w «-» v (exchanging vibration and winding numbers) in our earlier discussion of circle duality preceding Equation (2.2.24).

34

M. L. Lapidus

and (2.2.326)

= ZsttR(u;).25

Zst,R(-l/cj)

Since the translation w *—• w + 1 together with the inversion u — i > — \/uj generate SL(2, Z), 26 this means that Zst = Zst,R is invariant under the action of the modular group SL(2, Z). Finally, we close this part of our discussion of circle duality by noting that, in light of the last equality of (2.2.25), the partition function Zst = ZstR is clearly invariant under the transformation R <-» l/R which exchanges a circular target space of radius R with one of reciprocal radius: (2.2.33)

ZsttR(oj) =

Zstil/R(co).

Remark 2.2.2 (String Theory Compactified on a Torus). All the results of this section extend without any difficulty to higher-dimensional toroidal spacetimes, as we now explain. (We will make use of this extension in Section 2.3. We refer to [FroGa,§2.2] and [LiSzl-2] for more details). We thus consider (bosonic) closed string theory on a fiat, d-dimensional torus Td = Rd/2TrL = (5 1 ) d , where L is a lattice of rank d (and of Euclidean signature). The space Td (or rather, the lattice L) is equipped with the Riemannian metric g = { ^ J J ^ j , along with the associated inner product; that is, ds2 = ^glivdx>1dxv{= | Sui/=i g^dx^dx", according to Einstein's summation convention), where {x^}d=1 are the angular coordinates on Td. As in [FroGa,§2.2] and [LiSzl,2], we allow for the possibility of torsion, due to the presence of an anti-symmetric (topological) instanton, namely a 2-form (3 = (3lil/dxfldx1' on Td satisfying appropriate integrality conditions. Then the (nonsingular) matrices d± = {rf* }d „ =1 are defined by (2.2.34)

d% =

9liv

± /V

At the quantum level, the center-of-mass coordinates {X^i}dl and the left-right momenta operators {pt}t=\ satisfy the following commutation relations (a higher-dimensional analogue of (2.2.13)): (2-2.35) 25

[X¥,fi] = -i&i,

At least at the physical level of rigor, the expression (2.2.30) for Zst—along with the modular invariance (2.2.32) of Zst—can be obtained from a Feynman-Polyakov path integral representation for Zst (and a standard application of the method of stationaryphase involving infinite dimensional Gaussian integrals), along with a suitable change of variables. See, for example, [Polc3,§8.2,pp. 237-238]. 26 Recall that 5L(2,Z) is the group of 2 x 2 matrices (la) with determinant one and coefficients in Z, acting on the Poincare upper-half plane {u> = U\ + iu>2 6 C : W2 > 0} via the Mobius transformations u — i > ?"fS.

T-Duality, Functional Equation

35

with all the remaining commutators vanishing. Further, the oscillatory modes { a * " : n G Z}£ =1 , with (a±-")* = Q^M, a j ' " = g»up±, and acting as creation (n < 0) and annihilation (n > 0) operators on a commuting pair of Fock spaces, satisfy the following commutation relations (the counterpart of (2.2.14)): (2-2.36)

[a^,a^]=ng^Sn+mfi,

with all other commutators vanishing and where {g,iVY^,v=\ denotes the inverse matrix of ( v l ^ i Moreover, the left-right momenta, also denoted by {pt}, are given by (2.2.37)

p± = - ^ ( ^ ± d > " ) ,

where {fcM}^=1 £ L* and {w M }^ =1 € L stand for the space-time (or uniformvibrations) momenta and the winding modes, respectively. (Compare with Equation (2.2.10) above, where the notation are somewhat different.) Here, following standard notation in Riemannian geometry, we have k^ = g^k"', for example. Further, L* is the dual lattice of L, relative to the inner product induced by {g^}- [Physically, the dual lattice L* represents the set of allowed (spacetime) momentum modes while the lattice L itself represents the set of allowed winding modes. Also, w^ corresponds to the number of times the closed string wraps around the /i-th circle (or cycle) of the torus Td = {Sl)d.} Hence, the Narain lattice is the set of left-right momenta A = {(p+,p~)}, with p± := {pt}t=i> equipped with the following quadratic form (an analogue of (2.2.11)): (2.2.38)

Q((P+,P-))

= p+gTPt - P»9TPZ = 2 * ^ " € 2Z.

It is an integral, even self-dual Lorentzian lattice, of rank 2d and of signature (d, d), given as the direct sum of the dual lattices L* and L (equipped with dual metrics): (2.2.39)

A = L* © L.27

Recall that if ( • , • )A denotes the (Lorentzian) inner product associated with Q by polarization, then, the dual lattice A* is defined to be the set of points 27

Note that if d = 1,0 = 0 and Td = S(R) = R/2TTLZ, as in the rest of this section, then L = BZ, L* = R~1Z and so A = RL 0 R~1Z. Alternately, much as in Section 2.1, one could let L = Z and L* = Z, each equipped with the appropriate dual metrics and associated inner products.

36

M. L. Lapidus

q = (q+,
(2.2.40)

L± := - £

9iaf

: a^a^m

: ,

m=—oo

still satisfy the commutation relation (2.2.19) except for the value of the central charge C, which is now equal to d, the dimension of the target space Td. Moreover, P and H, the generators of the Poincare algebra, are defined exactly as before, except that their concrete expressions are slightly more complicated than in (2.2.21) and (2.2.22), where d = 1. More precisely, the momentum operator P (the generator of space translations) and the energy or Hamiltonian operator (the generator of time translations) are still defined by P = LQ — LQ and H — LQ+LQ—J^ (with C :— d). Thus the Hamiltonian operator (from which we can recover the Laplace-Beltrami operator on the "effective spacetime" Td, see [FroGa] and Section 2.3 below) is given by the counterpart of (2.2.22): (2.2.41) H = L$ + Lg - — -]

1

= ^Vlvt

°°

+ ^ " W +£

OO

V-a-n

a

,-,

y + £5/u/a-"a--" - - .

Moreover, the momentum operator P is given by the analogue of (2.2.21): P = L0 — L0

(2.2.42)

=

l_rp+p+ _ 1 rp-p-

+

J- g^etiar

" £ ^«=»
n=l

n=l

Finally, we point out that the T-duality (or target-space duality) is expressed as follows in this context: Td^{Td)\

where

(Td)* = Rd/2irL*

denotes the dual torus of Td. Furthermore, the d-dimensional torus (Td)* (or rather, the dual lattice L*) is equipped with the 'dual metric', g = {^I/}^ I V = 1 , defined by (2.2.43)

-9ia,

=

(d+r«gKXtd-)x».

T-Duality, Functional Equation

37

(Note that when d = 1,/? = 0 and Td = S(R), a circle of radius R, then (Td)* — S(R~1), a circle of reciprocal radius i ? - 1 , and we recover the usual circle duality discussed in the rest of this section.) Moreover, under T-duality, the left-right momenta and the winding modes are exchanged; more precisely, \p+;p~) <-•

(2.2.44a)

c(p+,p-)\p+;p-),

where c{P+,p-):={-l)k"w,i

(2.2.446) and (2.2.44c)

p+ •- (d+y1p+,

p~ :=

-(d~)-lp-.

We note that the above expression in (2.2.44) provides an explicit realization of 'the' duality isomorphism T to be discussed in Section 2.3; see especially §2.3.1. In closing this remark, we point out that the modular invariance of the partition function Zst = ZstTd(= Zst^Tdy)—still defined by (2.2.24), except with the exponent —1/24 replaced by — d/24—can be established much as in the discussion preceding Equation (2.2.32); see [Polc3,§8.4,pp.251-253]. Conceptually, this is seen most clearly by writing Zst in the form Zst = ZstA, where A is the Narain lattice [Nar] defined just before and after Equation (2.2.38). Indeed, we have for all UJ = ui\ + iu>2 6 C with uii > 0: Zst(io) = ZstA(u>) M

=hHr E

exp

M"(P+)2

-*(p-) 2 )]

peA

2d

= ir?Mr E

ex

p [-™*((p+)2 + ip-f)+^i((p+)2

- (p-f)],

peA

where p = (p+,p~) G A and (p*) 2 := g^ptpt(Note that in the displayed equation just above, the exponent of |/7(w)| is now —2d instead of —2. The reader can easily check that when d = 1, as in the rest of this section, the last expression for Zst(u>) = Zati\(u>) just above reduces to the last one for Zst(u) = ZstiR(uj) occurring in Equation (2.2.25).) Then, using the Poisson Summation Formula (see (2.4.11) in Section 2.4 below) along with the modular transformation (2.2.27) of the Dedekind eta function r\ = rj(uj), we obtain (see [Polc3,Eq.(8.4.23),p.252]) Z s t,A*(-l/w) =

vAZstA(ui),

M. L. Lapidus

38

where VA denotes the volume of the fundamental domain (or unit cell) of A. The self-duality of A (i.e., A* = A) implies t h a t v\ = 1 (because VA = v^}) and hence (2.2.45)

ZstA{-l/w)

=

ZstA(u),

which together with the easily verified identity (using (2.2.27a)) (2.2.45a)

Zrf,A(w + 1) =

ZstA(w),

implies the modular invariance of the partition function Zst = ZstA(= Zst^), in the present case of (bosonic, closed) string theory on a d-dimensional torus. (As is observed in [Polc3,p.252], the modular invariance of ZstA is actually equivalent to the self-duality of A, since (2.2.32') must hold for all w = uj\ +iu)2 e C with CJ2 > 0.) We refer the interested to [Polc3,§8.4,pp.251-255] for more details and for an interesting discussion of the consequences in the present higher-dimensional context of the key properties of the Narain lattice A for the T-duality symmetry (viewed as an 'enhanced gauge symmetry', see [Polc3,§8.3]), which is much enlarged from the one-dimensional case. Remark 2.2.3 ( T - D u a l i t y a n d Mirror S y m m e t r y ) . As was alluded to earlier, T-duality has many other aspects, including nonperturbative ones. (In fact, as is pointed out in [Polc3,4], T-duality is an exact symmetry of string theory, not just one valid at the perturbative level.) We will not discuss them here but refer instead the interested reader to Polchinski's second volume on String Theory, [Polc4]. Moreover, as is explained by Cumrun Vafa in his interesting recent survey article on Geometric Physics, [Va2], "the notion of duality [is] in some sense a non-linear infinite dimensional generalization of the Fourier transform" [Va2,p.537]. Circle duality—or T-duality on circular space-times—is only the simplest example of such dualities, or mirror symmetry. Further on, Vafa writes ([Va2, pp. 541-542], 1998): One of the most interesting aspects of string theory is that strings moving on one manifold may behave identically with strings moving on a different manifold. Any pair of manifolds M\ and M2 which behave in this way are called mirror pairs. Of course, this would be a trivial duality if M\ and M2 are isomorphic Riemannian manifolds. The interesting dualities arise when M\ and M2 are distinct Riemannian manifolds. In some cases M\ and M2 are topologically the same [as for circle (or torus) duality], but in some cases they are distinct even topologically ... The simplest example of mirror symmetry corresponds to choosing M\ to be a circle [of radius R] and M2 to be a circle [of radius 1/R]. This is a case that can be rigorously proven ...

T-Duality, Functional Equation

39

Earlier on in his paper ([Va2, pp. 539-540], 1998) Vafa gives the following general and enlightening definition of the notion of physical duality, inspired by string theory: Consider a physical system Q (which I will not attempt to define). And suppose this system depends on a number of parameters [Aj]. Collectively, we denote the space of the parameters A, by M, which is usually called the moduli space of the coupling constants of the theory. The parameters Aj could for example define the geometry of the space the particles propagate in, the charges and masses of particles, etc. Among these parameters there is a parameter Ao which controls how close the system is to being a classical system (the analog of what we call h in quantum mechanics). For Ao near zero, we have a classical system and for Ao > 1 quantum effects dominate the description of the physical system. [In the case of circle duality, Ao is proportional to 1/R, the reciprocal of the radius; hence, the "classical limit" when Ao —> 0 corresponds to large scales. Similarly, for more general mirror symmetries, Ao can be taken to be the reciprocal of the volume of the manifold.] Typically, physical systems have many observables which we could measure. [The momentum operator P and the Hamiltonian operator H are two such observables in the above discussion of circle duality.] Let us denote the observables by Oa. Then we would be interested in their correlation functions which we denote by (Oai

. • • Oan)

= /Ql...Qn(Aj).

Note that the correlation functions will depend on the parameters defining Q. The totality of such observables and their correlation functions determines a physical system. Two physical systems Q[M,Oa], Q[M,Oa] are dual to one another if there is an isomorphism between M and M and O <-> O respecting all the correlation functions. Sometimes this isomorphism is trivial and in some cases it is not. We are interested in the cases where this isomorphism is non-trivial. In such cases typically what happens is that a parameter which controls quantum corrections Ao on one side gets transformed to a parameter Afc with k ^ 0 describing some classical aspect of the dual side. This in particular implies that quantum corrections on one side have the interpretation on the dual side as to how correlations vary with some classical concept such as geometry. This allows one to solve difficult questions involved in quantum corrections in one theory in terms of simple geometrical concepts on the dual theory. This is the power of duality in the physical setup. Mathematics parallels the physics in that it turns out that the mathematical questions involved in computing quantum corrections in certain cases [are] also very difficult and the questions involved on the dual side are mathematically simple. Thus

40

M. L. Lapidus

non-trivial duality statements often lead to methods of solving certain difficult mathematical problems. One should note, however, that very rarely can one prove (even in the physics sense of this word) that two given physical systems are dual to one another. Often the existence of dualities between two systems is guessed at based on some physical consistency arguments. Testing many non-trivial consequences of duality conjectures leads us to believe in their validity. In fact we have observed that duality occurs very generically, for reasons we do not fully understand. This lack of deep understanding of duality is not unrelated to the fact that it leads to solutions of otherwise very difficult problems. At the mathematical level, evidence for duality conjectures amounts to checking validity of proposed solutions to certain difficult mathematics problems.

2.2.3

T - D u a l i t y a n d t h e E x i s t e n c e of a F u n d a m e n t a l L e n g t h

The quote heading this section and that heading the next one, Section 2.3 (as well as that heading this chapter) explain in detail the physical considerations t h a t lead to deduce from T-duality (in the present case, circle duality), the existence of a fundamental (or minimum) length in string theory. For this reason, we will keep our present discussion rather brief. In these quotes from [Gre], the fundamental length is referred to as Planck length (or scale), for simplicity. However, if the string tension (2na')~1 is taken into account, the fundamental (or minimum) length is equal to [a1)1!2 and is called the string length or scale. (See the footnote following Equation (2.2.10).) The string length is often taken to be of the order of Planck length ( « 10~ 3 3 cm, see footnote (4)) but—depending on the value of the string tension, the basic parameter of string theory—it may also be significantly different from it. (See, for example, [GreeSWit],[Kak], and [Polc3,4].) We assume here t h a t we have chosen units so t h a t the string length is equal to 1. We begin by citing an excerpt from [FroGa] t h a t describes this topic in more technical details. Referring to circle duality (discussed in Section 2.2.2 above 2 8 ), Frohlich and Gawedzki write ([FroGa, p. 67], 1994): 29 This phenomenon distinguishes the stringy geometry of circles from their point-set Riemannian geometry. While the space of Riemannian circles was the half line parametrized by the radius R, that of the sigma 28

and reinterpreted from the point of noncommutative geometry, as in Section 2.3 below ([FroGa], [LiSzl,2]). 29 We note that we have slightly adapted the notation of [FroGa] to our present ones.

T-Duality, Functional Equation

41

models with circle targets is an orbifold of the latter obtained by the identification of R with R~l. The above is an example of a more general duality phenomenon responsible for the appearance of a fundamental length in string theory. This is certainly one of the most promising features of the latter. In order to understand how such a length scale arises, let us think of how one probes the effective space(-time) geometry in string theory. As mentioned before, this should be done by looking at the low energy states of the string in which the stringy internal modes are not excited and one effectively sees quantum mechanics of point-like objects. Let us suppose that the string vacuum is described by a sigma model with the Sl target. The oscillatory modes of the string created by [the] operators a*, n < 0, have energies quantized to integer values, so we should look at the states with energies C 1. / / the radius R of the circle is much bigger than 1 (or than the Planck length [really, the string length] in dimensional units) then the low-energy spectrum of the sigma model is given by the states |po;0), Po £ Z, describing the barycentric degree of freedom x° [the position of the center-of-mass of the string]. We effectively obtain the quantum mechanics of a particle moving on the circle of radius R. If R is much smaller than 1, then the low-energy spectrum of the sigma model is given by the states \0;w) corresponding to the winding modes of the string. But this is exactly like the spectrum in the quantum mechanics of a particle moving on the circle of radius R~l. As a result, we never see circles of small radii as effective geometries! Notice that this is a quantum phenomenon as is signaled by the presence of h in the expression for the Planck length.

We close this subsection by briefly mentioning t h a t the entire discussion of T-duality—and hence, its crucial physical consequence, the existence of a fundamental length—applies just as well to toroidal (rather than circular) spacetimes of arbitrary finite dimension. (See Remark 2.2.2 just after Equation (2.2.23) for the easy extension of the setting of Section 2.2 to toroidal spacetimes.) Intuitively, this is to be expected and can be verified in much the same way as for one-dimensional tori (i.e., circles). However, the curious reader may wonder for which types of spacetimes analogous conclusions can be drawn. There are beautiful formulations of (and partial results for) the socalled "Mirror Symmetry Conjectures" for spacetimes t h a t are Calabi-Yau manifolds, for example, and these are mathematically very challenging. (See, [Va2], [Witl2], and the relevant references therein.) We do not wish to discuss further this fascinating topic, however. Instead we would like to quote once again Brian Greene [Gre, pp. 254-255] who pursues in a more general context the train of thought presented in the first quote (from [Gre,pp.249252]) heading this section (i.e., Section 2.2):

M. L. Lapidus What if the spatial dimensions are not circular in shape? Do these remarkable conclusions about minimal spatial extent in string theory still hold? No one knows for sure. The essential aspect of circular dimensions is that they permit the possibility of wound strings. As long as the spatial dimensions—regardless of the details of their shape— allow strings to wind around them, most of the conclusions we have drawn shoud still apply. But what if, say, two of the dimensions are in the shape of a sphere? In this case, strings cannot get "trapped" in a wound configuration, because they can always "slip off" much as a stretched rubber band can pop off a basketball. Does string theory nevertheless limit the size to which these dimensions can shrink? Numerous investigations seem to show that the answer depends on whether a full spatial dimension is being shrunk [as in the case of circles or tori] ... or ... [whether] an isolated "chunk" of space is collapsing. The general belief among string theorists is that, regardless of shape, there is a minimum limiting size, much as in the case of circular dimensions, so long as we are shrinking a full spatial dimension. Establishing this expectation is an important goal for further research because it has a direct impact on a number of aspects of string theory, including its implications for cosmology.

Noncommutative Stringy Spacetimes and T-Duality Distance is such a basic concept in our understanding of the world that it is easy to underestimate the depth of its subtlety. With the surprising effects that special and general relativity have had on our notions of space and time, and the new features arising from string theory, we are led to be a bit more careful even in our definition of distance. The most meaningful definitions in physics are those that are operational—that is, definitions that provide a means, at least in principle, for measuring whatever is being defined ... How can we give an operational definition of the concept of distance? The answer to this question in the context of string theory is rather surprising. In 1988, the physicists Robert Brandenberger of Brown University and Cumrun Vafa of Harvard University pointed out (in [BrVa]) that if the spatial shape of a dimension is circular, there are two different yet related operational definitions of distance in string theory. Each lays out a distinct experimental procedure for measuring distance and is based, roughly speaking, on the simple principle that if a probe travels at a fixed and known speed then we can measure a given distance by determining how long the probe takes to traverse

T-Duality, Functional Equation

it. The difference between the two procedures is the choice of probe used. The first definition uses strings that are not wound around a circular dimension, whereas the second definition uses strings that are wound. We see that the extended nature of the fundamental probe is responsible for there being two natural operational definitions of distance in string theory. In a point-particle theory, for which there is no notion of winding, there would be only one such definition. How do the results of each procedure differ? The answer found by Brandenberger and Vafa is as surprising as it is subtle. The rough idea underlying the result can be understood by appealing to the uncertainty principle. Unwound strings can move around freely and probe the full circumference of the circle, a length proportional to R [the radius of the circle]. By the uncertainty principle, their energies are proportional to 1/R (recall [from quantum mechanics] the inverse relation between the energy of a probe and the distance to which it is sensitive). On the other hand, we have seen that wound trings have minimum energy proportional to R; as probes of distances the uncertainty principle tells us that they are therefore sensitive to the reciprocal of this value, 1/R. The mathematical embodiment of this idea shows that if each is used to measure the radius of a circular dimension of space, unwound string probes will measure R while wound string probes will measure 1/.R, where, as before, we are measuring distances in multiples of the Planck length. The result of each experiment has an equal claim to being the radius of the circle—what we learn from string theory is that using different probes to measure distance can result in different answers. In fact, this property extends to all measurements of lengths and distances, not just to determining the size of a circular dimension. The results obtained by wound and unwound string probes will be inversely related to one another. [Italics added.] If string theory describes our universe, why have we not encountered these two possible notions of distance in any of our day-to-day or scientific endeavors? Any time we talk about distance, we do so in a manner that conforms to experience of there being one concept of distance without any hint of there being a second notion. Why have we missed the alternative possibility? The answer is that although there is a high degree of symmetry in our discussion, whenever R (and hence 1/R as well) differ significantly from the value 1 (meaning again, 1 times the Planck length), then one of our operational definitions proves extremely difficult to carry out while the other proves extremely easy to carry out. In essence, we have always carried out the easy approach, completely unaware of there being another possibility. The discrepancy in difficulty between the two approaches is due to the very different masses of the probes used—high-winding-energy/low-

M. L. Lapidus

vibration-energy, and vice versa—if the radius R (and hence 1/R as well) differs significantly from the Planck length (that is, R = 1). "High" energy here, for radii that are vastly different from the Planck length, corresponds to incredibly massive probes—billions and billions times heavier than the proton, for instance—while "low" energy corresponds to probe masses at most a speck above zero [in Planck units]. In such circumstances, there is a monumental difference in difficulty between the two approaches, since even producing the heavy-string configurations is an undertaking that, at present, is beyond our technological prowess. In practice, then, only one of the two approaches is technologically feasible—the one involving the lighter of the two types of string configurations. This is the one used in all of our discussions involving distance encountered to this point. This is the one that informs and hence meshes with our intuition. Putting issues of practicality aside, in a universe governed by string theory one is free to measure distances using either of the two approaches. When astronomers measure the "size of the universe" they do so by examining photons that have traveled across the cosmos and happened to enter their telescopes. No pun intended, photons are the light string modes in this situation. The result obtained is [equal to] 1061 times the Planck length [or about 15 billion light-years] ... If the three familiar spatial dimensions are in fact circular and string theory is right, astronomers using vastly different (and currently nonexistent) equipment, in principle, should be able to measure the extent of the heavens with heavy wound-string modes and find a result that is the reciprocal of this huge distance. It is in this sense that we can think of the universe as being huge, as we normally do, or terribly minute. According to the light string modes, the universe is large and expanding; according to the heavy modes it is tiny and contracting. There is no contradiction here; instead, we have two distinct but equally sensible definitions of distance. We are far more familiar with the first definition due to technological limitations, but, nevertheless, each is an equally valid concept ... Since we now have two concepts of distance—using light or heavy string probes—we must compare measurements made in the same manner ... If one does stick to measuring distances "the easy way"— that is, using the lightest of the string modes instead of the heavy ones—the results will always be larger than the Planck length. To see this, let's think through the hypothetical big crunch for the three extended dimensions, assuming them to be circular. For argument's sake, let's say that at the beginning of our thought experiment, unwound string modes are the light ones and by using them it is determined that the universe has an enormously large radius and that it is shrinking in time. As it shrinks, these unwound modes get heav-

T-Duality, Functional Equation

ier and the winding modes get lighter. When the radius shrinks all the way to the Planck length—that is, when R takes on the value 1—the winding and vibration modes have comparable mass. The two approaches to measuring distances would become equally difficult to carry out, and, moreover, each would yield the same result since 1 is its own reciprocal. As the radius continues to shrink, the winding modes become lighter than the unwound modes and hence, since we are always opting for the "easier approach", they should now be used to measure distances. According to this method of measurement, which yields the reciprocal of that measured by the unwound modes, the radius is larger than one times the Planck length and increasing. This simply reflects that as R—the quantity measured by unwound strings—shrinks to 1 and continues to get smaller, 1/R—the quantity measured by wound strings—grows to 1 and gets larger. Therefore, if one takes care to always use the light string modes—the "easy" approach to measuring distance—the minimum value encountered is the Planck length [which explains the existence of a fundamental length or "minimum size" in string theory]. Brian R. Greene, 1999 [Gre, pp. 249-252] A quantum theory of point particles on [the spin, Riemannian manifold] M naturally supplies [the Hilbert space of spinors] H. = L 2 (spin(M)) and can therefore be thought of as describing an ordinary spacetime M. In the case of string theory then, we can anticipate that the full stringy spacetime is described by a noncommutative geometry, i.e., a spectral triple [denoted by T] with a noncommutative algebra A. In this case the notion of a "space" breaks down, as it is in general impossible to speak of points. In the low-energy limit of the string theory, the vibrational modes are negligible and the model describes ordinary pointlike particles corresponding to an ordinary (commutative) spacetime. Thus in the low-energy limit of the string theory, T should contain a subspace TQ [= (L 2 (spin(M),C(M),D), where D is the Dirac operator on M] representing an ordinary spacetime manifold M at large distance scales. The symmetries of the quantum spacetime T then determine the stringy effects on the geometry of the spacetime 7"o represented by ordinary quantum field theory. In the following we construct the spectral triple T associated with the stringy geometry and show that the full duality group of spacetime is determined in a very simple fashion by the group of automorphisms of the appropriate algebra A. The duality symmetries of the spacetime emerge as the possibility of assigning two independent Dirac operators to the spectral triple T associated with strings, so that duality is simply a consequence of a change of

45

46

M. L. Lapidus

metric representing a redefinition of distances in the string spacetime. These automorphisms leave the spectral triple T invariant and correspond to internal symmetries of the quantum geometry. We shall also construct the low-energy projections [onto] TQ and illustrate how the duality isomorphisms of T correspond to the more familiar notions of target space duality [i.e., T-duality] in string theory. These projections then illuminate the full structure of the stringy modification of classical general relativity. Fedele Lizzi and Richard J. Szabo, 1997 [LiSzl, pp. 3581-3582]

2.3.1

Target S p a c e D u a l i t y a n d N o n c o m m u t a t i v e G e o m e t r y

We continue considering closed string theory on a circular spacetime of radius R, as in the previous two sections, viewed as S(R) or as the unit circle M := S1 equipped with the Riemannian metric ds2 = R2dx2 and Haar measure dx/\^2n. Because the setting is identical and the formulas involved are very similar, we will also consider higher-dimensional target spaces, as in [FroGa,LiSzl-2]; namely, toroidal space-times M = Td := Rd/2irL, where L is a (Euclidean) lattice of rank d (for example, M may be a finite product of circles). This should help better understand the special case of circular space-times and place it in a broader framework. (We refer to Remark 2.2.2 where we have indicated the simple changes to be made in order to extend the discussion of Section 2.2 from circular to toroidal target spaces.) We could also work with spacetimes t h a t are Calabi-Yau manifolds, as is suggested in [LiSz2]; we will refrain from doing this, however, since our main focus—from the point of view of the applications—will be circular (and possibly, toroidal) target spaces. Our goal is to provide a geometric description of the circular (or toroidal) spacetime from the point of view of string theory. The underlying quantum spacetime is often referred to as a stringy spacetime (or string 'spacetime') in the physics and mathematical physics literature. A very interesting noncommutative geometric description of a circular (or toroidal) stringy spacetime has been provided by Frohlich and Gaw§dzki in 1994 in [FroGa] and pursued in a number of papers, including those of Chamseddine and Frohlich [ChaFro], Chamseddine [Chal,2], as well as Lizzi and Szabo ([LiSz2], announced in [LiSzl]). A more systematic treatment—giving, in particular, an abstract noncommutative differential geometric version of Witten's classic approach to supersymmetric quantum mechanics (see, for example, [Witl,2,7])—can be found in the interesting recent papers by Frohlich, Grandjean and Recknagel [FroGrRel,2]. References to other approaches—

T-Duality, Functional Equation

Al

motivated by (nonperturbative) M(atrix) or M-theory rather than by ordinary (super)string theory—are provided in the above papers (e.g., [LiSz2]); some of these—particularly the work of Ho and Wu [HoWu] as well as that of Michael Douglas and his collaborators (see, for example, [Dou,DouKPS])— along with aspects of [ProGa] and [LiSzl-2], have been synthetized recently in the paper by Connes, Douglas and Schwarz [ConDouSc] and pursued in a number of works, including the paper by Landi, Lizzi and Szabo [LanLiSz], entitled String Geometry and the Noncommutative Torus. We will follow here especially the approach of the paper [ProGa], Conformal Field Theory and Geometry of Strings, pursued and expanded in the work [LiSz2], Duality Symmetries and Noncommutative Geometry of String Spacetimes. From the point of view of Connes' noncommutative geometry, a noncommutative space (or 'quantum space') is described by a spectral triple T = (H, A, D), where H is a complex Hilbert space, A is a suitable (normed or topological) *-algebra30 represented on H, and D is a suitable unbounded self-adjoint operator (acting on H), viewed as an abstract analogue of the Dirac operator, see [Con5,6].31 (As was alluded to in Section 1.2, further axioms for noncommutative geometry are required by Connes in [Con7,8]. It is noteworthy that they are all satisfied here, as was recently shown in [LanLiSz], but we will not be concerned by those for now.) One original aspect of [FroGa] (along with [LiSzl,2] and [FroGrRel,2], for example), is that from the point of view of noncommutative geometry, a string spacetime (underlying a circular or more generally, toroidal spacetime, say) should be described by two distinct spectral triples; namely, T = (TC, A, D) and T = (H, A, D), where the algebra A and the Hilbert space Ji are the same for 3" and T, but the 'Dirac operators' D and D are distinct (although they are constructed according to a very similar procedure). A crucial fact is that these two spectral triples are isomorphic, as noncommutative spaces. The above setup is ideally suited to give a natural noncommutative geometric description of T-duality—also called target space duality, and specialized to circle duality in the setting of Section 2.2. It can also be used to 30 For example, a C*-algebra or a von Neumann algebra (or often, in practice, a dense subalgebra thereof); see, for instance, [Con6] and [KadRi] for the definition of these terms. 31 Here and thereafter, as in [Con6], we assume that for any spectral triple T = (H,A,D), A,li and D satisfy the following two compatibility conditions: (i) the commutator [D, a] (a e A) is bounded for all elements a in a dense subalgebra of A. (Here, [D, a] is defined via the representation of the elements of A as bounded linear operators of the Hilbert space H.), (ii) D has compact resolvents; that is, for all A £ C\K, (D — A ) - 1 is a compact operator on H. (We note that these conditions are satisfied by all the spectral triples, T,T,TQ and 2~o—constructed in [LiSzl,2]—which will be used in this section.)

48

M. L. Lapidus

describe more general examples of mirror symmetries as well as other types of symmetries encountered in string theory; we refer to [FroGa, LiSzl-2] for more detailed information. Remark 2.3.1. In [FroGa], the two spectral triples are defined as (7i,A,H) and (H,A,H), where H and H are the Hamiltonian operators instead of 9

the Dirac operators D and D, with H = D2 and H = D . We will prefer, however, the approach of [LiSzl,2]—relying on the use of Dirac operators and suggested in part by the work in [Wit 1-3,7], [SeiWit] and [ChaFro,Chal-2], in addition to [FroGa]—because it is more flexible (enabling one, for example, to study a suitable noncommutative analogue of the de Rham complex, see Section 2.3.3) and is closer to Connes' original point of view. This approach is abstracted and developed in [FroGrRe2]. We postpone until later on in this section a more precise description of the Hilbert space 7i, the noncommutative algebra A and the abstract Dirac operator D (in the case when M = S1 or, more generally, M = Td = Rd/27rL is a d-dimensional torus; see Section 2.3.2. We only mention that the definition of Ji involves the use of Fock spaces and that A is the algebra of vertex operators, representing physically—the space of string interactions or the algebra of observables on the stringy space-time. Further, D (along with D) is a suitable version of Witten's Dirac-Ramond operator (acting on loop space). We assume for now that M is a compact, Riemannian (d-dimensional) spin manifold, with Riemannian metric g = {g^}^=i and real-valued gamma matrices {7M}f=i generating the Clifford algebra (associated with the spin structure of M) and satisfying the standard anti-commutation relations (2.3.1)

{7M,7,} = 2
where {7,2,71/} := 7^7i/ — lulu denotes the anti-commutator of 7^ and 7,,. (See, however, Remark 2.3.2 below.) The usual Dirac operator, Do, on the spin manifold M is then given (up to normalization) by (2.3.2)

D0 =

igTipVu,

where V = {V„}j5=1 is the standard covariant derivative constructed from the spin connection on M. The Dirac operator Do acts on the complex Hilbert space Ho = L 2 (spin(M)), the space of square-integrable spinors on M (i.e., the L2-sections of spin(M), the spin bundle of M). Heuristically, according to [Con3-4,6], D0 can be thought of as "the inverse of the infinitesimal [ds] which determines geodesic distances in the

T-Duality, Functional Equation

49

space-time M" [LiSzl,p.3581]. Further, physically, D0 can be viewed as "the propagator of the fermions" [Con7,8]. We can now describe the spin Riemannian manifold M as a space of "points" (that is, as a commutative space) by means of the spectral triple (2.3.3)

T0 =

{Ho,Ao,D0),

where Ho = L 2 (spin(M)) and D0 is the natural Dirac operator on M, as above, and where Ao : = C(M) is the commutative C*-algebra of all continuous complex-valued functions f on M (equipped with the supremum norm H/lloo : = suPy6Ml/(2/)l)> acting on the Hilbert space Ho as multiplication operators. The spectral triple T0 also determines M as a metric space (see [Con4] and [Con6,Chapter VI]). It is perfectly adequate to describe quantum mechanics or, more generally, (aspects of) quantum field theory on M. However, from the point of view of string theory on M, we must work with a much larger noncommutative space, represented mathematically by the aforementioned isomorphic spectral triples T = (H, A, D) and T = (H, A, D). Remark 2.3.2. We stress that the construction of the noncommutative spectral triple T = (H, A, D) is valid in the present generality of a spin Riemannian manifold while that of the dual spectral triple T = (H, A, D) along with its interpretation in terms of target space duality requires much more restrictive assumptions, for example, a toroidal space-time or else, a CalabiYau manifold (although the construction of T in the latter case does not yet seem to have been explicited in the mathematical physics literature). (Other types of target spaces amenable to this treatment are also considered in [ProGa], where S1 or Td is replaced by a compact Lie group.) A similar comment applies to the construction of the dual pair of commutative subspaces T0 = (Ho, Ao, D0) and T0 = (Tio, Ao, D0) discussed below. As a noncommutative space, T = (H, A, D) contains the spectral triple TQ — (Ho, Ao, DQ) as a (commutative and much smaller) subspace. Roughly speaking, this means that Ho — L 2 (spin(M)) C H,Ao = C(M) C ^4, and that the restriction of D to the smaller Hilbert space Ho coincides with Do, the natural Dirac operator on M. Physically, TQ represents M, and therefore corresponds (commutative) spacetime M. this section.) In [LiSzl,2], TQ

the low-energy limit of the string theory on to point-like particles moving in the ordinary (See the second quote, from [LiSzl], heading is also referred to as the effective low-energy

50

M. L. Lapidus

spectral triple, much for the same reason that quantum field theory is considered as an effective theory, a low-energy limit of a more fundamental unified theory, perhaps string theory [Weinl]. Similarly the 'dual' noncommutative space T = (H, A, D) contains T0 = (Ho,Ao,D0) as a (commutative) subspace. Hence, Ho C. H,Ao Q A, and Do is the restriction of D to the closed subspace Ho of H. As is pointed out in [LiSzl,2], the existence of the two independent Dirac operators D and D is directly related to the chiral structure of the world-sheet conformal field theory underlying the string theory on M. Moreover, "the existence of both D and D severely restricts the effective spacetime geometry and is intimately related to the fact that the quantum spacetime determined by the string is not [M, a rf-dimensional torus, for example], but rather its quotient under the action of the duality group G of the string theory" [LiSzl,p.3583]. (Indeed, many truly noncommutative spaces are obtained as the quotient of an ordinary space by a suitable group or equivalence relation; see, for instance, [Con6,Chapter II, esp.§11.3]. We will consider such an example towards the end of this paper when discussing our mathematical model for the notion of 'arithmetic site' and its possible connections with our 'moduli space of fractal strings'. In earlier work [Lap5,6], we have suggested that self-similar fractals should also be naturally considered as noncommutative spaces because they can be viewed as quotients of an ordinary geometric space under the action of a suitable semigroup of transformations.) Here, "the duality group [G] appears naturally as a subgroup of the automorphism group of the vertex operator algebra" [LiSzl,p.3581]. We can now discuss target space duality (or T-duality) 32 from the present perspective of noncommutative geometry. It turns out that there are several unitary transformations T : H —> H sending the spectral triple T = (H, A, D) isomorphically onto its dual counterpart, the spectral triple T = (H,A,D): (2.3.4)

T = (H,A,D)^T={H,A,D).

In particular, this means that T sends D to D, (2.3.5)

D =

TDT~\

and that T induces an (inner) automorphim of the vertex algebra (which, as we recall, is represented on the Hilbert space H), (2.3.6)

TAT~l = A.

32 of which the circle duality R <-» 1/R studied in Section 2.2 is a special case, corresponding to M := S(R).

T-Duality, Functional Equation

51

As is pointed out by Lizzi and Szabo in [LiSz2, p. 687], [The] isomorphism (2.3.4) *s « special case of the spectral action principle [ChaConl,2] which describes Riemannian manifolds which are isospectral (i.e., their Dirac K-cycles have the same spectrum) but not necessarily isometric. It states that the noncommutative string spacetime determined by the spectral triple T [associated with D] is identical to that [determined by T , the spectral triple associated with D\. As such a change of Dirac operator in noncommutative geometry simply represents a change of metric structure on the spacetime, the isomorphism (2.3.4) *5 simply the statement of general invariance of the noncommutative string spacetime represented as an isometry of T.

It turns out that suitable operators Vo and VQ can be constructed t h a t project the full spectral triples T and T onto their respective low-energy subspaces T 0 and T 0 . (See [LiSzl] or [LiSz2,pp.688-691] for the construction of these projections.) As is pointed out in [LiSz2], target space duality can be represented symbolically by the following commutative diagram: T

(2.3.7)

——*

P0| T0(Td)

T ^ T

JPO = T0 -5-> T0 =

T0((TdY)

As is noted in [LiSz2, p. 687], From the point of view of classical general relativity, [the two spacetimes To and To] are inequivalent. However, the isomorphism T in the top line of (2.3.7) makes the diagram commutative, so that TQVO = VtyT and To is an isomorphism of subspaces of the noncommutative spacetime. Thus as subspaces of the full quantum spacetime, the commutative spacetimes To and To are equivalent. This is the essence of the stringy modification of classical general relativity.

The triple T 0 is the commutative spacetime (2.3.3) given by (2.3.8)

To = (L2(spin(Td)),C(Td),A)),

with D 0 = ig'"'7 M Vi / , the Dirac operator on the d-dimensional torus M = rpd _ ^12-KL, whereas the dually transformed triple T 0 (also obtained as

52

M. L. Lapidus

the low-energy projection of T ) 3 3 is isomorphic to the commutative spacetime associated with the dual (d-dimensional) torus M* = (Td)* = M.d/2wL*, where L* is the dual lattice of L: (2.3.9)

T0 =

(L2(spm((Tdy)),C((Tdy),D0),

with D0 = i5 / i "7 / J V„, the Dirac operator on (Td)*, dual metric on L*, given by (2.2.43). 34

where g = {g^}

is the

As is stated in [LiSz2, p. 691], According to the T-duality symmetry (2.3.4) °f ^ e effective string spacetime, as subspaces of the quantum spacetime we have the isomorphism35 [T0 = T 0 , and in particular, C{Td) ^ C{{Td)*)} which is the usual statement of the T-duality Td <-> (Td)* of string theory compactified on [a g^v [where {jj^v} is the inverse matrix of {5^,1/}] and the symmetry under interchange of momentum and winding numbers in the compactified string spectrum arise very naturally from the point of view of noncommutative geometry. It appears as a discrete TLisymmetry of the noncommutative geometry. Remark 2.3.3 (Brandenberger—Vafa M e t r i c s a n d C o n n e s M e t r i c s ) . It is interesting to compare the above quote (from LiSz2])—along with the second quote (from [LiSzl]) heading this section—with the discussion in [Gre,pp.249-252] of the two different metrics 'used' to measure distances in string theory. (See the first quote, from [Gre], heading the present section.) Even though this does not seem to be pointed out explicitly in [LiSzl,2] (or in [FroGa]), it is clear t h a t the two metrics introduced in 1988 by the physicists Brandenberger and Vafa in [BrVa] 36 t o understand conceptually and physically T-duality and the existence of a fundamental length in string theory could be interpreted within the framework of noncommutative geometry as the metrics on the noncommutative stringy spacetime respectively associated with the Dirac operators D and D. 33

For simplicity, we omit some technical details in the description of the low-energy subspaces To and T 0 . For example, the Hilbert spaces in (2.3.8) and (2.3.9) consist of anti-chiral (square-integrable) spinors; see [LiSzl,2]. 34 In (2.3.8) and (2.3.9), one could prefer to work with the pre-C*-algebras C°°(Td) and Cx{{TdY), consisting of smooth functions on the torus Td or (Td)*, respectively, much as we did in Section 2.1 above. 35 via the isomorphism To on the bottom line of the commutative diagram (2.3.7). 36 Interestingly, their article, [BrVa], is entitled Superstrings in the Early Universe and also discusses some of the cosmological implications of their proposed interpretation.

T-Duality, Functional Equation

53

More precisely, according to the prescription given in [Con4], the metric associated with the spectral triple T = (H, A, D) is defined by (2.3.10)

%>i,¥>2) = s u p { | V l ( a ) -v> 2 (a)| : ||[D,o]|| < l } , aeA

where the norm of the commutator of D and a is defined via the representation of A on Ti, and 1. In other words, we must use the first BV metric, interpreted here as being associated with the Dirac operator D. From the point of view of the above discussion, we are in the effective low-energy subspace, given formally by the spectral triple T 0 = (L 2 (spin(M),C(M),A>)—with D0 = - ^ and M = S{R), viewed as S1 with the Riemannian metric ds2 = R2dx2—and the string theory on M effectively reduces to the quantum mechanics of point-particles on the circle of radius R, as in Section 2.1 above. (See the second quote, from [LiSzl], heading the present section.) On the other hand, in an hypothetical 'big bang universe', with energies larger than the string mass scale, we would have to use 'rulers' made of winding modes strings, since these are now the probes with the smallest

54

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possible amount of energy. In other words, we must use the second BV metric, that associated with the other Dirac operator D. 2.3.2

Noncommutative Stringy Spacetimes and Vertex Algebras

We now try to give a relatively nontechnical overview of the construction of the spectral triple T = (H, A, D) (along with the dual spectral triple T = (H,A,D)) representing the noncommutative space underlying the (bosonic, closed) string theory on the torus M = Td. This will provide, in particular, a mathematical realization of the noncommutative stringy spacetime (or 'quantum spacetime') in this context and will entail giving an outline of the construction of the complex Hilbert space H, the algebra A (represented on 7i), and of the two Dirac operators D and D (acting as unbounded operators on Ti). The mathematics involved may be unfamiliar to many readers and since going into depth would take us too far afield, we prefer to refer the interested reader to [ProGa] and [LiSzl,2] for the detailed construction (and for additional physical motativations), along with the book in preparation [Lap7], while also giving a few general references that may be helpful in this context. The construction of H, A (and the Hamiltonian H = D2 given in (2.2.41) above) is due in this context to Frohlich and Gaw§dzki, while that of D and D is due to Lizzi and Szabo [LiSzl,2] who also adapted the construction of H and A to spinors and further investigated the symmetries of the underlying quantum space. An axiomatization of and further results about noncommutative spaces admitting a dual pair (or more generally, JV dual pairs) of Dirac operators— corresponding to N = 1 (or N > 1) supersymmetry—is provided by Frohlich, Grandjean and Recknagel in two consecutive long papers [FroGrRel,2] (see also the earlier work [ChaFro], [Chal,2]) which are well worth reading for gaining a more conceptual understanding of the present situation. We work (as in Remark 2.2.2) with a d-dimensional toroidal space-time M = Td = M.d/2irL, where L is a Euclidean lattice of rank d (and with dual lattice L*). Further, we use freely the notation introduced in this setting in Remark 2.2.2 (just after Equation (2.2.33)). We begin by discussing the Hilbert space H. First, we mention a smaller Hilbert space Jix (used in [FroGa]), which is the natural Hilbert space of the bosonic closed string theory on Td: Hx = L2(Td)L T+ ® J7', where T+ and T~ are a pair of commuting (bosonic) Fock spaces, and where we have set L2(Td)L = ®^w^eiL2{Td), the orthogonal direct sum of L2-spaces d indexed by the winding modes {w'*} l=1 € L.

T-Duality, Functional Equation

55

Next, we indicate how to construct the noncommutative algebra Ax, which is the essential ingredient in the definition of the algebra A of the spectral triples T = (H, A, D) and T = (H, A, D). The algebra Ax is represented on the Hilbert space Tix, while A is represented on H. Further, both TL and A are simple 'augmentations' of Jix and Ax, respectively, that suitably take into account the spinorial structure needed to define the Dirac operators D and D. In short, the algebra Ax is (essentially) the natural vertex algebra associated with the bosonic closed string theory (or the underlying world-sheet conformal field theory), compactified on the torus Td. Physically, it represents the algebra of quantum fields of the string spacetime, that is, the algebra of observables of the quantum theory. It can also be thought of as the space of interacting strings. (In fact, this physical situation—already encountered in the early days of string theory—was the first example of what later came to be known as a 'vertex (operator) algebra'; see, e.g., [Kac-V].) Mathematically, leaving aside some analytical technicalities, Ax is the lattice vertex algebra of L (or of L*, really of the Narain lattice A). In some sense, it yields a suitable quantization of the 'loop space', the space of maps from S1 to the torus Td. (See, e.g., [Kac-V,§5.5,esp.Theorem 5.5 and Example 5.5a, pp.106-108], [Geb,§4], [Polc3,Chapter 2], [LiSz2] and [LanLiSz,§2] for more information about the algebraic construction of the lattice algebra, along with additional references, including Dong's key paper [Don] on the algebraic construction of lattice vertex algebras, valid for Euclidean as well as for Lorentzian lattices, in particular.) The reader unfamiliar with the notion of vertex (operator) algebra may find it helpful to consult Appendix A of [Lap7] for a brief review of the definition and some of the basic properties of this beautiful algebraic concept, abstracted by Borcherds in [Borl-3]. We note that Appendix A of [Lap7] gives the simpler (but equivalent) axioms found in [Geb], for example, rather than in [Borl-4] or [Kac-V]. Finally, we mention that the Dirac operators D and D respectively associated with the spectral triples T and T are obtained as suitable linear combinations of "low energy limit[s] of Witten's Dirac-Ramond operator for the full superstring theory corresponding to the N = 1 supersymmetric sigmamodel [Witl-3]"; see [LiSz2]. Remark 2.3.4 (Poincare Duality, T-Duality, and de Rham Complex). (a) It is noteworthy that in [LanLiSz] are constructed (full) noncommutative spaces, T = (H, A, D, J, K) and its dual 7", representing string theory

56

M. L. Lapidus

on Td and satisfying all the axioms of noncommutative geometry, as stated in [Con8]. The operator J on H represents physically the charge conjugation operator and yields mathematically a real structure for the underlying quantum space, which can be used to implement the counterpart of Poincare duality in this context. (See [Con6-8].) Furthermore, the grading operator K satisfies K2 = I and decomposes H = H+ © H- into its chiral and anti-chiral eigenspaces 'H±. It defines an orientation of the quantum geometry, along with a Hodge duality operator for the associated noncommutative de Rham complex. (b) Another way of implementing Poincare duality in this context is provided in [LiSz2,pp.702-706].37 There, noncommutative 'cohomology spaces' are defined in terms of the noncommutative de Rham complex associated with the spectral triple T = (H, A, D) and its dual T = (H, A, D). (The latter is defined by means of the representation pD induced by pD{da) = [D, a], the commutator of D and the vertex operator on TL associated with a € A.)38 These cohomology spaces are computed to be of the form H°, if 1 ,..., i/ 2 d _ 1 , H2d, with W and H2d~i (for j = 1 , . . . , d) exchanged via T-duality; in particular, Hj ^ H2d-i. (See especially [LiSz2,Eq.(5.312),p.704] for the algebraic expression of these spaces in terms of A, D and D.) This remarkable 'doubling' of the number of nontrivial cohomology spaces (compared to the standard situation of an ordinary d-dimensional torus Td) is attributed physically to the stringy modifications brought upon the topology of the classical spacetime. In this regard, Lizzi and Szabo [LiSz2,p.705] state that "higher-spin oscillatory modes of the strings 'excite' the cohomology groups, leading to generalized, infinitely-many connected components in the string spacetime." From our present perspective, it would be very interesting to gain a better physical and mathematical understanding of the true nature of these cohomology spaces (or of their if-theoretic counterpart). (See Section 2.4.2 below and especially the relevant parts of [Lap7] for a suitable motivation.) 37 The procedure used in [LiSz2] is admittedly not rigorous and somewhat ad hoc but the result and its interpretation are still illuminating. As is noted in [LiSz2], a suitable AT-theoretic reformulation remains to be carried out. 38 We refer, e.g., to [Kasl], [Con5,Appendix 2,pp.86-93] or [Con6,Chapter III] for an introduction to the noncommutative de Rham complex and the associated theory of cyclic cohomology.

57

T-Duality, Functional Equation

2.4

Analogy with t h e Riemann Zeta Function: Functional Equation and T-Duality

In this section, we provide some background about the Riemann zeta function Q = £(s) (and other number-theoretic zeta functions) which may be helpful to a reader unfamiliar with this aspect of number theory. (See Section 2.4.1.) Then, in Section 2.4.2, we propose an analogy between two key properties of £ (or of its natural generalizations to arithmetic geometry)—namely, the functional equation and the (still unproved) Riemann Hypothesis—and two key physical consequences of string theory, namely, T-duality and the existence of a minimum length, respectively (see Sections 2.2 and 2.3 above). We will draw and further elaborate on these analogies in [Lap7]. In particular, we will propose in [Lap7] a geometric interpretation of the Euler product in terms of the vibrations of suitably chosen fractal membranes.

2.4.1

Key Properties of the Riemann Zeta Function: Euler Product and Functional Equation

We first recall several of the basic properties of the Riemann zeta function C = C(s) that will be useful to us in the paper. For more information on this classical but beautiful subject, we refer, for example, to [Da], [Edw], [In], [Pat], or [Ti]. In particular, the interested reader will find in Edward's book [Edw] a thorough historical discussion of topics related to the Riemann zeta function, the Prime Number Theorem and the Riemann Hypothesis, up to the early 1970's. We note that towards the end of the present subsection, we shall briefly review some of the properties of other number-theoretic zeta functions, such as the Dedekind zeta function of a number field. By definition, £ is given by the Dirichlet series oo

(2.4.1)

C(s) = X! n ~ S '

forRes>1

-

71=1

Moreover, it follows easily from the unique prime factorization of integers and the identity S~ = 0 p~ms = (1 - p " s ) _ 1 that (2.4.2)

<(s) = n ^ 1 ~ P~S)~\ p

fo

r

Re s > 1,

58

M. L. Lapidus

where p runs over the prime numbers 2 , 3 , 5 , 7 , 1 1 , . . . The infinite product on the right-hand side of (2.4.2) is commonly referred to as an Euler product. Hence, for Re s > 1, ((s) is given both by the Dirichlet series (2.4.1) and the Euler product (2.4.2). An immediate consequence of (2.4.2) is that ((s) =fi 0 for Re s > 1 since a convergent infinite product cannot vanish. (Although for s real, s > 1, this property also follows from the infinite series representation (2.4.1), it is no longer the case for complex values of s.) If we differentiate logarithmically Euler's product representation (2.4.2), we obtain for Re s > 1, (2-4.3)

- ^ ^

= £ '

(logp)p-"".

m>l, p

We do not wish to go into the details here, but the representation (2.4.3) (or its counterpart for log ((s)) is very useful to show that £(s) does not have any zero on the vertical line Re s = 1 ([Hadl,2], [dVl,2]), and hence to prove the Prime Number Theorem. (See, for example, [Edw,§4.2 and §5.2].) It also plays an important role in the derivation of the Riemann-von Mangoldt explicit formula mentioned in Section 1.4. (See, for example, [Edw,§4.3] or [Lap-vF2,§4.5.1].) Another key property of £(s) is the fact that it admits an analytic continuation to the whole complex plane, with a single (and simple) pole at s = 1 (with residue 1), and that it satisfies the following functional equation:

(2.4.4)

7r-«/2r(i)c(*) = ^ - ^ r x i ^ K a - *),

for all s € C, where T = T(s) denotes the classical gamma function. Both the analytic continuation and the functional equation of £ were established in his memoir [Riel] by Riemann, who also showed that "the key to the deeper investigation of the distribution of the primes lies in the study of £(s) as a function of the complex variable s" [Da,p.59]. We note that much earlier, Euler had already introduced the 'Riemann zeta function' as a function of a real variable and had given (by a very different method) an equivalent form of the functional equation satisfied by C(s) for real values of s. Since it is important for our later discussion, we will now derive formula (2.4.4), the functional equation of C(s), following Riemann's second proof given in [Riel]. (See, for example, [Da,Chapter 8], [Edw,§1.7] or [Ti,§2.6].)

T-Duality, Functional Equation

59

This proof is remarkable for its conceptual simplicity and for its emphasis on the modularity or the functional equation of the Jacobi theta function oo

(2.4.5)

e(t):=

e-" 2 ^ = ^ e - " 2 r t ,

^ n=—oo

for t > 0;

neZ

namely,39 Q(t) = r ^ e f t - 1 ) ,

(2.4.6)

for t > 0.

Recalling that T(s) = JQ e~vys~ldy stitution y = n2irx, we easily obtain

for Re s > 0 and making the sub-

n-V-s/2r(J)= r

n2

e-

™xs'2

—,

for Re s > 1. We then sum this identity over all integers n > 1, interchange sum and integral, and by using (2.4.1), deduce that 7r'/2r(|)<(S) = r 2 Jo

(2.4.7)

z(x)x'/*^, z

for Re s > 1, where oo

(2.4.8)

e

1

z(x) := V " "

2,ra

= o ( 9 ( x ) - !)>

for x

> °-

2

n=i

Next, in view of (2.4.8), we can rewrite (2.4.6) in the form (2.4.6a)

1 + 2z{x) = (1 + 2z(x~1))x-1/2,

for x > 0.

We then split the integral on the right-hand side of (2.4.7) over the intervals (0,1) and (l,oo), make the substitution u = x~l in the resulting integral over the interval (0,1), to deduce successively that 7r- s / 2 r(|)C(s)

= f z{x)xs'2 — + fX x Jo Ji dx

z{x) xs/2 — X

I

(2.4.9)

. . sli /9 dx

z(x)x

—

z(x) (xs'2 + x^~s)l2) •

/

1 r°° + i / (x-l>-W

-

2 ££ s/2\ )

x-°l

X 39

The functional equation (2.4.6) satisfied by B is attributed by Jacobi to Poisson and is probably one of the first applications of the celebrated Poisson Summation Formula, to be discussed below; see the second footnote of [Edw,p.l5].

60

M. L. Lapidus

(Note that we have used formula (2.4.6a) to deduce the third equality of (2.4.9).) Thus, for Re s > 1, (2.4.10)

n-^T(S-)as) = f

z(x) (x°V + x^'2)

^

-

- ^ —

Since the function z(x) = J ^ L i e " 7ra decays exponentially fast as x —> 4-oo, the integral on the right-hand side of (2.4.10) is convergent for all s G C. Hence, taking into account the (simple) pole of T(s) at s = 0, we deduce that £(s) has a meromorphic continuation to all of C with a single, simple pole at s = 1 (with residue 1). In particular, (2.4.10) continues to hold for all s G C. Moreover, since the right-hand side of (2.4.10) is clearly invariant under s <-> 1 — s, we conclude that for s G C, £(s) satisfies the functional equation (2.4.4) as desired. We now show that the modularity of the Jacobi theta function (see (2.4.6)) is a special case of the famous Poisson Summation Formula (see [Ser,§6.1] and [Schwa,Eq.(VII,7;5)]): If A C Rd is a lattice, with dual lattice denoted by A*, then for every smooth rapidly decreasing (or Schwartz) function / on M.d, we have

(2.4.H)

E / ( i ) = u " 1 E/(»)-

where / denotes the Fourier transform of / and v is the volume of a fundamental domain of A. (Here, A* := {y G Kd : (x, y)Rd G Z, for all x G A} and f(y) := Jmd exp(—2Tri(x,y)^d)f(x)dx, where (x,y)md denotes the inner product in Rd.) If we define the theta function of the lattice A by (2.4.12)

©A(t) = 5^e- 1rt<x,!r) "- )

fort>0,

xsA

we deduce from (2.4.11) the following identity: (2.4.13)

9A(<) = t-d,2v-lQK,{rl),

for t > 0,

of which (2.4.6), the functional equation for the Jacobi theta function 0(f), is just the special case when A = A* = Z c K (and so d = 1 and v = 1). (Note that in view of (2.4.5) and (2.4.12), we have 0 z ( i ) = 0(f).) To prove (2.4.13), simply apply the Poisson Summation Formula (2.4.11) to the function f(x) = e~1T^x'x^»d (so that f(y) = f(y) itself) and to the lattice fx/2A (with dual lattice t~l'2A*); see [Ser,§VII.6.2,Proposition 16].

T-Duality, Functional Equation

61

Recall that T(s) is meromorphic in all of C, with no zeros and with simple poles at s — 0, —1, - 2 , . . . (See, for example, [Da,Chapter 10].) Since £(s) ^ 0 for Re s > 1, it thus follows from the functional equation (2.4.4) that £(s) has (simple) zeros at —2, —4, —6,..., the even negative integers. These are called the trivial zeros of £ and are the only zeros of £ for Re s < 0. Hence, the zeros of £ within the critical strip 0 < Re s < 1 are the only interesting ones; they are called the nontrivial or critical zeros of £• We will also sometimes refer to them in this work as the Riemann zeros. Since £(s) = ((s), where s denotes the complex conjugate of s, they come in complex conjugate pairs. Moreover, in view of the functional equation (2.4.4), if p is a critical zero off the critical line Re s = | , then so are J>, 1 — p and 1 — p. The Riemann Hypothesis asserts that all the critical zeros of C He on the critical line Re s = |. 4 0 Remark 2.4.1. In the present case of the Riemann zeta function, it is also known that £ does not have any zero on the real interval [0,1]; see, for example, [Ti]. For this reason, the critical zeros are also called the complex zeros of £• Moreover, it is conjectured that all those zeros are simple, in addition to being located on the critical line. Remark 2.4.2. (a) It has been shown that infinitely many critical zeros of C lie on the critical line Re s = | . This is a result originally due to Hardy [Hardl] in 1914, later improved by a number of authors (including Littlewood [Lit], Selberg, and Levinston [Lev]) to deduce that a positive proportion of the critical zeros of £ lie precisely on the critical line. (See, for example, [Ti] and [Schw,pp.601-602].) (b) Moreover, with the help of a computer, it has been rigorously verified in 1986 that the first one-and-a-half billion critical zeros of C lie exactly on the critical line. (See [vL-tR-W]; also see [Od2]. This type of numerical result is, of course, reassuring. It must be considered with caution, however, since many conjectures in number theory which had been initially verified computationally for very large ranges have eventually been disproved. In this regard, as was pointed out in [Schw,p.602], Lehmer already wrote in 1956: UA tendency for the behaviour of the zeta-function to become more and more capricious as [the imaginary part] £ increases is evident from the values so far computed ..." 40

This proportion is increasing slightly with each new result, but is still very far from the conjectured value of one. For example, Levinson [Lev] has shown in 1974 that it is at least equal to | .

62

M. L. Lapidus

The completed Riemann zeta function, £(s), is defined by the left-hand side of (2.4.4); namely, (2.4.14)

£( s ) = 7r- s / 2 r(|)C(s),

for s e C .

By construction, it satisfies the functional equation (2.4.15)

CM = ? ( ! - * ) .

forseC.

Clearly, £(s) is a meromorphic function in all of C, with (simple) poles at s = 0 and s = 1. Moreover, it does not have any zero for Re s < 0 or Re s > 1; in fact, the zeros of £(s) are precisely the critical zeros of C(s), and hence, according to the Riemann Hypothesis, should all be located on the critical line. From a contemporary perspective (especially since the 1967 publication of Tate's celebrated thesis [Tat]), the significance of £(s) is that, in view of the Euler product (2.4.2) for £(s), it admits the following 'adelic representation':

(2.4.16)

^ ) (=c(s)CooOO) = ncp(*)> p

for Re s > 1, where the infinite product now runs over all 'finite' and 'infinite' primes p. More specifically, in the present situation corresponding to the field of rational numbers Q, for p = p, a (finite) prime, we have (2.4.17)

CP(s):=(l-p-T\

the p-th Euler factor, and for p = oo (the only 'infinite' prime for the ring Z of rational integers), (2.4.18)

Us) := T T ^ r ^ ) .

In other words, in (2.4.16), the product over the finite (or ordinary) primes p corresponds to the non-archimedean valuations of Q, whereas that involving the infinite (or 'real') primes (here, only p = oo) corresponds to the archimedean valuations of Q. In the present case, it is well-known that the non-archimedean valuations of Q are the p-adic valuations (associated with the p-adic norms and thus indexed by the rational primes p), while Q has a single archimedean valuation, associated with its standard absolute value (induced by the metric of R). The p's over which the product is taken in (2.4.16) are also referred to as the (finite and infinite) 'places' of Q in the literature. (In algebraic number theory, a place is an equivalent class of valuations on a given field K, with two valuations said to be equivalent if they

63

T-Duality, Functional Equation

induce the same topology on K; see, for example, [ParSha2,§I.4] for a good introduction to this subject.) A long-standing problem in number theory and in arithmetic geometry is to obtain a deeper geometric and analytic understanding of the way the finite and infinite primes (or places) coexist on an equal footing within the same geometric space. This fascinating problem is the major theme of Haran's forthcoming book [Har2], The Mysteries of the Real Prime. (See also [Harl,3].) It is also at the heart of Deninger's 'cohomological approach to analytic number theory' and notion of 'arithmetic site' [Denl-7], which will be discussed in [Lap7], in conjunction with our own geometric model. It is difficult to imagine that a satisfactory resolution of Riemann's Conjecture would not throw new light on this matter. We close this overview of some of the key properties of the Riemann zeta function by stating the Riemann-von Mangoldt explicit formula, mentioned in Section 1.4 above. (See, for example, [Da], [In], [JorLang],[ParShal], [Wei6,8], [Edw,Chapter 3,esp.§3.2], [Harl], [Den2], [Lap-vF2,Chapter 4, including pp.75-76 and esp.§4.5.1] and the relevant references therein, for a proof of this formula and for alternate forms as well as pointwise or distributional extensions to other number-theoretic zeta functions.) Consider the 'prime power counting function': (2.4.19)

*(x) = ^2 logp,

for x > 0,

pm<x

where the summation runs over all prime powers pm(m > 1) not exceeding x > 0.41 Then, for x > 1, (2.4.20)

*

( a ;

)

= x

_^^_|M_I

l o g (

i_

: r

-2

) i

4

2

where p runs over all the critical zeros of C-43 The last term on the righthand side of (2.4.20), - | l o g ( l - x~2) = E ^ i ^ " 2 " / 2 " ) . corresponds to 41 If x is a prime power and (2.4.20) is interpreted pointwise, then the 'weight' logp in (2.4.19) must be replaced by | logp. 42 As was alluded to in Section 1.4, formula (2.4.20) was established in 1895 by von Mangoldt [vMl,2], who also proved Riemann's original explicit formula for the 'prime number counting function' II(x), denned just after Equation (1.4.1). (See [Edw,Chapter 3,esp.§3.2 and §3.7].) 43 If (2.4.20) is interpreted pointwise rather than distributionally, then the terms involving p and p are to be taken together and summed over increasing values of |Im p\. See, e.g., [Lap-vF2,§4.5] for a distributional interpretation of (2.4.20).

64

M. L. Lapidus

the trivial zeros of (, located on the real axis at —2, — 4 , . . . . Further, —C'(0)/C(0) = log(27r). Intuitively, the sum ^2pxp/p involving the complex zeros of ( provides detailed information about the oscillations (or fluctuations) of 5f(x) — x. Before turning our attention to the connections with T-duality, we briefly mention that CK(S), the L-series associated with an algebraic number field K, has an Euler product similar to (2.4.2), a functional equation analogous to (2.4.4), and is also expected to satisfy the counterpart of the Riemann Hypothesis. (Recall that an algebraic number field is a finite field extension of Q lying in C; further, £AT(S) ls called the Dedekind zeta function of K— or more generally, when nontrivial characters are allowed, a Hecke L-series. Moreover, when K = Q, the field of rational numbers, we have C,K{S) = C(s)> the classical Riemann zeta function.) It is noteworthy that the functional equation for CK(S)—or equivalently, for the completed zeta function £K(S), as defined in [Tat] or in [Lap-vF2,Eq.(A.12),p.223], for instance—is derived by means of the Poisson Summation Formula. (See [Tat] or [ParSha2,§6.2].) Moreover, £R-(S) satisfies an analogue of the 'adelic representation' (2.4.16), with typically several 'infinite' primes (or archimedean places), and the explicit formula (2.4.20) has a suitable counterpart in this situation. Finally, a direct analogue of the Riemann Hypothesis—often referred to as the Extended Riemann Hypothesis (ERH)—is conjectured to hold in this context. We refer, for example, to [Lap-vF2,Appendix A] for a brief introduction, and to [Lang], [Ko,Chapters 7-8] or [ParShal,2] for a detailed discussion of the main properties of Hecke L-series. We only recall the analogue of the Dirichlet series (2.4.1),

(2.4.21)

(K{s) = J2 N(&)~S>

for

Re s > 1,

a

and the counterpart of the Euler product (2.4.2), (2.4.22)

CK(S) = f j ( l - N(y)-sy\

for Res > 1.

Here, in (2.4.21), 21 runs over the nonzero ideals of £>, the ring of integers of K\ further, AT(21), the norm of the ideal 21, is defined (possibly by multiplicativity) as the cardinality of the quotient ring D/21. (Naturally, JV(2l) is finite for each nonzero 21.) Moreover, in (2.4.22), ty runs over the prime ideals of O and hence, N(ty) is equal to the number of elements of the quotient field £)/^3, also called the residue field of <}3. Remark 2.4.3. When K = Q and hence D = Z, we note that since the ideals of Z are principal ideals, of the form nZ, with n S N, (2.4.21) reduces to the

T-Duality, Functional Equation

65

usual Dirichlet series (2.4.1) for the Riemann zeta function ((s) = CQ(S). Also, since the prime ideals of Z are of the form pZ, with p a rational prime, (2.4.22) reduces to the standard Euler product for C(s). Remark2.4.4 (Analogy Between Number Fields and Function Fields). We have briefly discussed in Remark 2.4.2 above some theoretical and numerical evidence in support of the Riemann Hypothesis. However, perhaps the strongest evidence for the Extended Riemann Hypothesis (ERH) is provided by the analogy between number fields and function fields or, more precisely, between the zeta functions of algebraic number fields and of curves (or varieties) over finite fields. In fact, as is well-known, the analogue of the Riemann Hypothesis was proved by Andre Weil in the 1940's for curves (of arbitrary genus) over finite fields. (See Weil's 1948 monograph [Wei4] for a complete proof of the result announced in 1941 in [Wei2] and one year earlier, in the Note aux Comptes Rendus [Weil].) Weil's 1948 proof in [Wei4] required the development of new chapters in algebraic geometry—in particular, putting on a firm footing and significantly extending some of the results of the famed Italian school of algebraic geometry. (See [Wei3].) 'Elementary proofs'—based on a direct proof of the analogue of the Prime Number Theorem with (a suitable) Error Term—were later given in the early 1970's by Stepanov [Ste], W. M. Schmidt [Sc-W] and Bombieri [Bom]. In 1949, motivated in part by his earlier work in [Weil2,4] (as well as by that of his predecessors, see, e.g., [Die,Kat,FreKie,Hou]), Andre Weil formulated in [Wei5] a series of beautiful conjectures concerning the zeta functions of (higher-dimensional) varieties over finite fields. These celebrated conjectures are now known as the Weil Conjectures in the literature on number theory; they include, in particular, the analogue of the Riemann Hypothesis for varieties over finite fields. (See, for example, Appendix B to [Lap7].) They have stimulated in part a number of new developments in algebraic geometry in the 1950's, 1960's and 1970's, notably by Alexandre Grothendieck. Some of Grothendieck's seminal work on the modern foundations of algebraic geometry and on the so-called 'etale and £-adic cohomologies' [Gro] was instrumental in Deligne's complete proof during the 1970's of the Weil Conjectures—and hence, in particular, of the analogue of the Riemann Hypothesis—for (projective, nonsingular) varieties as well as, more generally, for (suitable) schemes over finite fields. (See [Dell,2].) Deligne's work is remarkable for its ingenuity and insight. Along with that of Weil (and Grothendieck, among others), it has opened-up the door to the application of cohomologicai methods to tackle problems in number theory and arithmetic geometry and, in particular, to try to solve the Riemann Hypothesis for algebraic number fields. (Recall from the preceding discussion that the original Riemann Hypothesis for the Riemann zeta function corre-

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sponds to the special case when the number field is chosen to be the field of rational numbers.) This line of research is being actively pursued by some of the current leaders of arithmetic geometry, such as Deninger [Denl-7], and will no doubt continue to play an important role in the future. (See, for example, the survey articles [Den3] and [Den6] where a suitable analogue of the Weil Conjectures is formulated and discussed for number fields—rather than for function fields.) It is also a key motivation for our own work in this paper and especially in the book in preparation [Lap7]. We refer, for example, to [Die], [Kat], [FreKie], [Wei9,vol.I] and [Hou] for the historical developments and results that have led to the formulation of Weil's Conjectures and for an overview of Deligne's proof of these conjectures. In these same references—and especially in [Die] and [Hou]—the interested reader will also find an exposition of some of the key aspects of Weil's proof of the Riemann Hypothesis for curves over finite fields, along with a fascinating account of the results preceding it. We also refer to Koch's recent book [Ko] where the parallel between algebraic number fields and function fields is developed in a comprehensive manner. Finally, we point out that in Appendix B to [Lap7], we briefly discuss the zeta function of a curve (or, more generally, a variety) over a finite field, as well as the corresponding Weil Conjectures. Moreover, in [Lap7], we propose a geometric and physical model that would enable us to treat in parallel the 'arithmetic geometries' associated with curves over finite fields (i.e., with function fields) and those associated with number fields. 2.4.2

Functional Equation, T-Duality, and Riemann Hypothesis

As was alluded to just above, we will propose in [Lap7] a richer noncommutative model for the Riemann and other arithmetic zeta functions enabling us to interpret naturally the associated Euler product representation. It is really in that context that one should consider the main themes of the present section. For the purpose of the present paper, however, we confine ourselves to the model considered earlier in this chapter, namely, that of string theory in a circle (of radius R, say) viewed from the perspective of noncommutative geometry (see especially Section 2.3). In fact, it may be useful to consider simultaneously the noncommutative spaces T{= T) associated with the orbifold of circles S{R) of radius R (identified with S(R~l), the circle of reciprocal radius), as R varies in the positive half-line. (See the derivation of Equation (2.4.9) above.) We wish to stress (much as in [Gre], but in the present context of noncommutative string geometry) that, 'in practice', we do not have the choice of which realization of the noncommutative space—either the spectral triple

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T = (H, A, D) or the dual spectral triple T = (W, A, D)—we can use in a given situation, for example, when probing the metric aspects of the stringy spacetime (see Remark 2.3.3). Indeed, the choice should be dictated by physical or mathematical considerations. Nevertheless, the isomorphism between T and T reveals a deep (and often hidden) identity of structure between the two descriptions of the noncommutative space. This is an important point because we contend here, in particular, that the functional equation for £(s) (see Equation (2.4.4) or (2.4.15)) is a mathematical analogue of target space duality and that the Riemann Hypothesis may be related to the existence of a fundamental length in string theory. We also suggest44 that the lefthalf plane Re s < | corresponds to the 'nonarchimedean world',45 while the right-half plane Re s > \ corresponds to the 'archimedean world',46 and that these two worlds meet at the 'critical line' Re s = | , where the critical zeros of £(s) should all be located according to the Riemann Hypothesis. In the author's view, T-duality and its companion, the existence of a minimum length, is one of the deepests insights brought upon by string theory or the rest of theoretical physics over the latter half of the 20th century. Thanks to the work in [FroGa] and [LiSzl-2], in particular, discussed in Section 2.3, we now have an elegant mathematical translation in the language of noncommutative geometry of these (yet unverified) new physical laws. It would be tempting to think that one of the most cherished secrets of mathematics—namely, the geometric structure of the primes (or of the integers), the symmetries of £(s) (as well as of other number-theoretic zeta functions), and the (extended) Riemann Hypothesis—are but a mirror reflection of these recently discovered physical laws. That this may indeed be the case is one of the central messages of this essay, although we certainly do not pretend to provide a compelling rigorous proof here, only hopefully suggestive analogies along with a possible mathematical model47 which may serve as a starting point for future work in this direction. Another deep insight brought upon by theoretical physics over the last fifty to sixty years (in this case, originally in the setting of quantum mechanics and quantum electrodynamics) is the notion of Feynman path integral. (See, for example, [Feyl], [FeyHi] and [JohLap].) In fact, it has provided a key tool for applied and theoretical physics, chemistry, and biology, ranging from the dynamics of macromolecules to quantum physics, quantum gravity and string theory. As Mark Kac wrote, with his characteristic enthusiasm, in [Kac2,p.51]: "Still the intuitive appeal of Feynman's definition is enormous, 44

following a proposal made in joint work in progress [Lap-vF7], going beyond [Lap-vF2]. i.e., to the p-adic numbers. 46 i.e., to the real numbers. 47 progressively enriched and extended in [Lap7]. 45

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and it gave birth to a dream that perhaps all of Physics could be rewritten in terms of 'sums over histories'.'" Using heuristic Feynman-type integrals as a guide, Edward Witten has made wonders in his own work over the last twenty years (see, e.g., [Witl-17]) on supersymmetric quantum mechanics, quantum field theory, gauge theory and string theory, as well as on a number of topics lying at the border of contemporary mathematics and physics, such as knot theory and low-dimensional topology [Wit8,10,ll]. (See, for example, [JohLap,§20.2].) On the mathematical side, Feynman integrals have been an endless source of research challenges. As Gian-Carlo Rota wrote somewhat pessimistically in [Rot,p.229]: "T/ie Feynman path integral is the mathematician's pons asinorum. Attempts to put it in on a sound footing have generated more mathematics than any subject in physics since the hydrogen atom. To no avail. The mystery remains, and it will stay with us for a long time. "is Returning to the physical side now, it is well-known that heuristic Feynman-type integrals—often called Polyakov integrals in this context [Polyl-3]—play a major role in perturbative string theory. (See, for example, [Wit4], [GreeSWit], [Kak], [Polc3,4], and [JohLap,§20.2.B,pp.688-696].) In particular, they can be used to understand aspects of T-duality and to make manifest the modular invariance of the partition function without using the Poisson Summation Formula. (See, for example, [Polc3,§8.2,pp.237-238] and the footnote following Equation (2.2.32) above.) Hence, it would perhaps not be so surprising if a suitable form of heuristic Feynman path integral were to play eventually a significant role in unraveling the secrets underlying the prime numbers and the Riemann Hypothesis.49 At present, physicists in their work on M(atrix) Theory and nonperturbative string theory are engaged in a search for concepts and mathematical tools that go beyond Feynman path integration. Since aspects of M-theory have recently been translated in the language of noncommutative geometry, particularly by using 'noncommutative tori' (see, for exam48

See, however, several of the mathematically rigorous theories of Feynman integrals presented in the recent book [JohLap], mostly in a quantum-mechanical context. Admittedly, these fall far short from what physicists would like, but this is a common problem in the constant dialogue between mathematics and physics. 49 At this point, we do not have a concrete model to propose involving such a path integral. However, a reasonable guess is that it would be closely related to a suitable dynamical interpretation of the noncommutative version of fractal membranes discussed in [Lap7]. The desired Feynman integral would then provide a counterpart of the dynamical trace formula searched for by Deninger in [Denl-7] (and, more recently, in a noncommutative geometric context, by Connes in [Con9,10]). It would be a suitable analogue of the Gutzwiller Trace Formula for quantum chaos [Gutl,2] (see, for example, [Lap-vF2,§10.4.3] and the relevant references therein).

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pie, [Dou,DouKPS,ConDouSc,HoWu,LanLiSz] and the relevant references therein), it is also possible that a suitable modification of our own model along those lines would be helpful in the future. We will not, however, pursue this direction in the present work. We now return more closely to the main topic of this section: the conjectured connections between functional equation and T-duality. In light of the classic derivation of the functional equation50 and of the Weil Conjectures [Wei4] (see Appendix B of [Lap7])51, it is widely expected that in a suitable dictionary, Poisson Summation Formula, functional equation, and Poincare Duality should appear on the same line (or row). (See, for example, [Denl-6], [Die], [FreKie], [Lap-vF2,§10.5], as well as the introduction of [ConlO], for different perspectives on this theme.) We would like here to propose to add a fourth entry (or column) to this row; namely, T-duality for string theory in circular spacetimes (or finite-dimensional tori) 52 , interpreted from the point of view of noncommutative geometry as in Section 2.3 above. As was briefly discussed, in Remark 2.3.4 at the end of Section 2.3, 'cyclic cohomology' (or rather, A"-theory) provides a way of encoding the "stringy modifications" brought upon the topology of a circular (or, more generally, toroidal) spacetime. In the model of [LiSz2]—based on an admittedly heuristic (and ad hoc) noncommutative version of the de Rham complex—two interesting and closely related new phenomena occurring in this context are the 'doubling' of the cohomology spaces along with what might clearly be interpreted as Poincare Duality (in the sense of noncommutative geometry as, e.g., in [Con6,8]). Mathematically, this corresponds to the existence of a dual pair of Dirac operators, D and D, while physically, it is naturally associated with the chirality of the two-dimensional conformal field theory underlying string theory. Consider now the d = 1 case of the discussion of Section 2.3.1; that is, consider (closed) string theory in a circular spacetime. Then, at the cohomological level, the isomorphism T = T (reflecting T-duality) between the noncommutative stringy spacetime T and its dual 7" induces an isomorphism between H° and H2 as well as between H1 and itself.53 The latter isomor50 as recalled, for example, in Section 2.4.1, and extended very elegantly to a large class of L-series in Tate's thesis [Tat], via the Poisson Summation Formula (2.4.11) and Fourier transform. 51 along with the well-known analogy between number fields and function fields, as briefly discussed in Remark 2.4.4 above. 52 which would need to be suitably adapted to string theory in adelic infinite dimensional tori, according to the (noncommutative) model of 'fractal membranes' and arithmetic geometries proposed in [Lap7]. 53 Here, as in Remark 2.3.4, H°, H1 and H2 denote the cohomology spaces of the quantum space T = (H, A, D).

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phism of H1 with itself can be defined via the intersection form, expressed in terms of the notion of Dixmier trace (viewed as a 'noncommutative integral'); see, for example, [ProGrRel,2] (or in the more rigorous language of if-theory, [Con8]). According to the philosophy of the Weil Conjectures, the poles of the (completed) Riemann zeta function (or of another arithmetic L-series under consideration) corresponds to H°(= H2) while its zeros are associated with H1. More specifically, there should be a suitable analogue of the Frobenius operator t h a t induces on the cohomological space H° or H1 a linear operator whose eigenvalues are respectively the poles or the zeros of the (completed) zeta function; see the end of Appendix B to [Lap7]. Consequently, the natural symmetries of the zeros (resp., the poles) implied by the functional equation follow from the above isomorphism of H1 with itself (resp., of H° with H2). Furthermore, we would like to suggest t h a t the fact t h a t , conjecturally, the zeros 'bounce off' the half-planes to the right or to the left of the critical line—and hence must be located on the critical line itself, as stated in the Riemann Hypothesis—corresponds to the existence of a fundamental length in string theory. Finally, the critical line would represent the locus where the archimedean and the nonarchimedean worlds meet, as was alluded to above (following [Lap-vF7]), while T-duality (a physical counterpart of the functional equation, according to our proposal) would exchange archimedean and nonarchimedean valuations (in the case of number fields).54

Epilogue: Fractal Membranes and Arithmetic Site Rien n'est plus fecond, tous les mathematiciens le savent, que ces obscures analogies, ces troubles reflets d'une theorie a une autre, ces furtives caresses, ces brouilleries inexplicables; rien aussi ne donne plus de plaisir au chercheur. Un jour vient oil l'illusion se dissipe; le pressentiment se change en certitude; les theories jumelles revelent leur source commune avant de disparaitre; comme 1'enseigne la Gita on atteint a la connaissance et a l'indifference en meme temps. La metaphysique est devenue mathematique, prete a former la matiere d'un traite dont la beaute froide ne saurait plus nous emouvoir. 55 Andre Weil, 1960 [Wei7, p. 52] 54

For example, in the case of the field of rational numbers corresponding to C, = ((s), it would 'exchange' the real and p-adic places. 55 This beautifully written French text is best left untranslated.

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As was mentioned towards the end of the introduction (§1.5), the work presented in this paper is pursued and expanded in a book in preparation by the author [Lap7], entitled In Search of the Riemann Zeros: Strings, fractal membranes, and noncommutative spacetimes. In this epilogue, we now give a brief overview of the part of [Lap7] dealing with the theory of 'fractal membranes' and its potential applications to aspects of number theory and arithmetic geometry. In [Lap7], we propose an extension of the model discussed in Chapter 2 to string theory on an (adelic) infinite dimensional torus, or on a Riemann surface with infinite genus. This yields a geometric model for the vibrations of a fractal membrane, viewed as a multiplicative (or quantum) analogue of a fractal string.56 For a suitable choice of data—directly expressed in terms of the sequence of prime numbers—the quantum partition function of such a model then coincides with the Riemann zeta function £ = ((s). We therefore obtain a new solution to a problem posed by the physicist Bernard Julia [Jul,2] about the possible connections between aspects of prime number theory and quantum statistical physics, and consisting of finding a mathematical model of the so-called 'free Riemann gas'. (An earlier solution of a different nature was obtained by Bost and Connes in [BosConl,2]; see also [Con6,§V.ll].) More specifically, after having recalled some basic facts concerning the theory of fractal strings (e.g., [Lapl-3,LapPol-3,LapMal-2,HeLapl-2,LapvFl-4]) and of their associated complex dimensions [Lap-vFl-4], we introduce in the second part of [Lap7] the new notion of fractal membrane, along with its self-similar counterpart, the notion of self-similar membrane. This enables us, in particular, to provide a mathematical model57 of many arithmetic geometries and to obtain a natural interpretation of the Euler product of the associated zeta function—viewed as the (spectral) partition function Z = Z(s) of the corresponding fractal membrane. (The special case of the so-called prime membrane yields the classic Riemann zeta function: Z(s) = C(s)i a s discussed in the previous paragraph.) We also develop and deepen significantly the analogy between arithmetic and self-similar geometries pointed out in earlier work of the author and his collaborators, particularly in [Lap3] and [Lap-vF2]. 56 Alternately, rather than an (adelic) infinite torus, a fractal membrane can be thought of as a vibrating (adelic) Hilbert cube, with opposite faces identified. By 'adelic', in this context, we mean that each normal mode of vibrations of such an object involves only finitely many circles (or faces). 57 recently made rigorous in a joint work in preparation with Ryszard Nest [LapNeJ, where fractal membranes are shown to be, in a suitable sense, the second quantization of fractal strings.

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Moreover, in the latter part of [Lap7], which is of a more speculative nature, we discuss the possibility of further extending this model to the noncommutative setting. We are thus led to introduce a vertex algebra and the corresponding Dirac operator(s) associated with each hole (or 'circle') in the Riemann surface with infinite genus (or the 'adelic torus'). (Recall from Section 2.3.2 that vertex algebras are algebraic structures used to describe the quantum fields and their interactions in conformal field theory and in string theory; see, for example, [Geb], [Kac-V] or [Polc3], and in the context of noncommutative geometry, [FroGa] and [LiSz2]; also see Appendix A to [Lap7].) Mathematically, this yields a sheaf of vertex algebras—or more generally, of noncommutative spaces—providing an algebraic and geometric model for the quantum geometry underlying string theory in a fractal membrane. For a suitable choice of data, the resulting 'noncommutative spacetime' may be an interesting model for exploring and trying to understand the geometry underlying the prime numbers.58 Furthermore—since our proposed construction can easily be adapted to every algebraic number field as well as to curves (or varieties) over finite fields, for example—the resulting family (or 'moduli space') of quantum geometries may provide a natural algebraic and physical model for Deninger's (heuristic) notion of 'arithmetic site' actively explored in his beautiful work [Denl-7] on 'cohomological number theory' and the Weil Conjectures; see, e.g., Appendix B to [Lap7] for an introduction to these conjectures. (We note that the latter notion of arithmetic site is central to arithmetic geometry in connection with the Extended Riemann Hypothesis.) As will be discussed in [Lap7], this should be closely related to the notion of 'moduli space of fractal strings' introduced several years ago by the author in order to provide a natural receptacle for many of the zeta functions arising in arithmetic and fractal geometry and to classify the various types of (one-dimensional) fractal geometries occurring in his theory of fractal strings and of their vibrations. As will be explained in [Lap7], the moduli space of fractal strings—along with its 'quantized' version, the moduli space of fractal membranes—is a highly noncommutative space that can be thought of as a broad generalization of the set of all Penrose tilings [Pen] (or quasiperiodic tilings of the plane), as interpreted in [Con6,§II.3]. In closing, we mention that in the latter part of [Lap7] will also be proposed several conjectures59—regarding fractal membranes, their (dynamical) 58

as well as the integers, which viewed multiplicatively, coincide with the frequencies or 'energy levels' of the membrane. 59 one of which concerns the possible relationship between the recent work in [Con9,10] and the author's (highly noncommutative) moduli space of fractal strings; see the end of [Lap7].

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complex dimensions, and moduli spaces of fractal strings (or membranes)— that would yield new insight into the nature of the Riemann zeros60 and into the possible algebraic and geometric structures underlying the Riemann Hypothesis. The interested reader may wish to consult [Lap7] (when it is available) in order to obtain more detailed information about the theory of fractal membranes and its potential connections with the Riemann and other (Beurlingtype [Beul,2]) zeta functions, as well as with the still mysterious notion of arithmetic site. As will be clear from the latter part of [Lap7]—which discusses increasingly rich and more noncommutative models of fractal membranes—a lot more mathematical and physical research at the confluence of string theory, noncommutative geometry, and arithmetic geometry, remains to be carried out before the notions presented in the present essay and in the forthcoming book [Lap7] can be properly understood. In the long-term, we hope that, even in small measure, this work will help motivate some of the future investigations in this fascinating arena of human knowledge—which is still largely to us, terra incognita.

Acknowledgements I wish to express my deep gratitude to Professor Luciano Boi, the editor of the volume Geometries of Nature, Living Systems and Human Cognition [Boi] (Springer-Verlag), in which this essay is due to appear. Indeed, since he asked me to contribute to this pluridisciplinary volume about five years ago, he has been extremely generous with his time, patience and helpful advice. Although some of the ideas presented here (or in [Lap7]) have been haunting me for many years, I have not started writing them down systematically until a little over a year ago, when I finally accepted his challenge. Further, the fact of being faced with the daunting task of attempting to explain them to a broader and diverse scientific public—composed of mathematicians, physicists, and even philosophers of science (or biologists)—has been helpful to me in surprising ways. Moreover, when it became clear that the text was reaching the proportions of a book, Dr. Boi's enthusiastic support, gentle proding and counseling was again instrumental in helping me reach a suitable compromise, out of which both this essay and the forthcoming book [Lap7] were born.

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as well as more generally, a suitable spectral and cohomological interpretation of the zeros and the poles of arithmetic zeta functions.

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B. M. Hambly and M. L. Lapidus, Complex dimensions of random fractal strings, Work in preparation.

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C. Q. He and M. L. Lapidus, Generalized Minkowski content and the vibrations of fractal drums and strings, Mathematical Research Letters 3 (1996), 31-40.

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C. Q. He and M. L. Lapidus, Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zetafunction, Memoirs Amer. Math. Soc. No. 608, 127 (1997), 1-97.

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G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 2000 (771 + (xviii) pages). (Paperback edition and corrected reprinting, Jan. 2002.)

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N. M. Katz and P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 1-26.

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J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Commun. Math. Phys. 158 (1993), 93-125.

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J. Kigami and M. L. Lapidus, Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals, Commun. Math. Phys. No. 1, 217 (2001), 165-180.

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M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), 465-529.

[Lap2]

M. L. Lapidus, Spectral and fractal geometry: From the WeylBerry conjecture for the vibrations of fractal drums to the Riemann zeta-function, in: Differential Equations and Mathematical Physics (C. Bennewitz, ed.), Proc. Fourth UAB Internat. Conf. (Birmingham, March 1990), Academic Press, New York, 1992, pp. 151-182.

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M. L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture, in: Ordinary and Partial Differential Equations (B. D. Sleeman and R. J. Jarvis, eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, Longman Scientific and Technical, London, 1993, pp. 126-209.

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M. L. Lapidus, Fractals and vibrations: Can you hear the shape of a fractal drum?, Fractals 3, No. 4 (1995), 725-736. (Special issue in honor of Benoit B. Mandelbrot's 70th birthday. Reprinted in [EvePeVo, pp. 321-332].)

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M. L. Lapidus, Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions, Topological Methods in Nonlinear Analysis, 4 (1994), 137-195.

[Lap6]

M. L. Lapidus, Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals, in: Harmonic Analysis and Nonlinear Differential Equations, Contemporary Mathematics, vol. 208, Amer. Math. Soc, Providence, R. I., 1997, pp. 211-252.

[Lap7]

M. L. Lapidus, In Search of the Riemann Zeros. (Strings, fractal membranes, and noncommutative spacetimes.) Book in preparation, 2001-03. (Expected length: approx. 320 pages.)

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M. L. Lapidus and J. Fleckinger-Pelle, Tambour fractal: vers une resolution de la conjecture de Weyl-Berry pour les valeurs propres du laplacien, C. R. Acad. Sci. Paris Ser. I Math. 306 (1988), 171175.

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M. L. Lapidus and H. Maier, Hypothe.se de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiee, C. R. Acad. Sci. Paris Ser. I Math. 313 (1991), 19-24.

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M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. (2) 52 (1995), 15-34.

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M. L. Lapidus and R. Nest, Fractal membranes as the second quantization of fractal strings (tentative title), Work in preparation.

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M. L. Lapidus, J. W. Neuberger, R. J. Renka and C. A. Griffith, Snowflake harmonics and computer graphics: Numerical computation of spectra on fractal domains, Internat. J. Bifurcation & Chaos 6 (1996), 1185-1210.

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M. L. Lapidus and M. M. H. Pang, Eigenfunctions of the Koch snowflake drum, Commun. Math. Phys. 172 (1995), 359-376.

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M. L. Lapidus and C. Pomerance, Fonction zeta de Riemann et conjecture de Weyl-Berry pour les tambours fractals, C. R. Acad. Sci. Paris Ser. I Math. 310 (1990), 343-348.

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M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), 41-69.

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M. L. Lapidus and C. Pomerance, Counterexamples to the modified Weyl-Berry conjecture on fractal drums, Math. Proc. Cambridge Philos. Soc. 199 (1996), 167-178.

[Lap-vFl]

M. L. Lapidus and M. van Prankenhuysen, Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic, in: Spectral Problems in Geometry and Arithmetic (T. Branson, ed.), Contemporary Mathematics, vol. 237, Amer. Math. Soc, Providence, R. I., 1999, pp. 87-105. (Also: Preprint, IHES/97/34, 38 & 85, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France, 1997.)

[Lap-vF2]

M. L. Lapidus and M. van Frankenhuysen, Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions, Research Monograph, Birkhauser, Boston, 2000 (268 + (xii) pages). [2nd rev. enl. edn to appear in 2006; approx 400 pages.]

[Lap-vF3]

M. L. Lapidus and M. van Frankenhuysen, A prime orbit theorem for self-similar flows and Diophantine approximation, Contemporary Mathematics, vol. 290, Amer. Math. Soc, Providence, R. L, 2001, pp. 113-138. (Appeared in [Lap-vF6].)

[Lap-vF4]

M. L. Lapidus and M. van Frankenhuysen, Complex dimensions of self-similar fractal strings and Diophantine approximation, MSRI

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M. L. Lapidus and M. van Frankenhuysen, Fractality, Selfsimilarity and complex dimensions, in Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proc. Symposia Pure Math., Amer. Math. Soc, Providence, R. I. 2005.

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M. L. Lapidus and M. van Frankenhuysen (eds.), Dynamical, Spectral, and Arithmetic Zeta Functions, Proc. Special Session (AMS Annual Meeting, San Antonio, Texas, Jan. 1999), Contemporary Mathematics, vol. 290, Amer. Math. Soc, Providence, R. I., 2001.

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M. L. Lapidus and M. van Frankenhuysen, Work in progress.

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N. Levinson, More than one third of the zeros of Riemann's zetafunction are on a = \, Adv. Math. 13 (1974), 383-436.

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A. N. Parshin and I. R. Shafarevich (eds.), Number Theory, vol. II, Algebraic Number Fields, Encyclopedia of Mathematical Sciences, vol. 62, Springer-Verlag, Berlin, 1992. (Written by H. Koch.)

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H. von Mangoldt, Zu Riemann's Abhandlung 'Uber die Anzahl der Primzahlen unter einer gegebenen Grosse', J. Reine Angew. Math. 114 (1895), 255-305.

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E. Witten, Magic, mystery and matrix, Notices Amer. Math. Soc. 45 (1998), No. 9, 1124-1129. (Article based on the Josiah Willard Gibbs Lecture given at the Annual Meeting of the American Mathematical Society held in Baltimore, Maryland, in January 1998.)

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ON THE SOLITON-PARTICLE DUALITIES

Frederic HELEIN CMLA, ENS de Cachan 61, avenue du President Wilson 942S5 Cachan Cedex, France heleinQcmla. ens-cachan.fr Joseph KOUNEIHER Universite Diderot-Paris 7 case 7064 75005 Paris, France & CNRS-URA 2052 (C.E.A.) C.E. Saclay 91191 Gif-sur-Yvette Cedex kouneiherQparis 7. jussieu.fr In some field theories we have the striking feature that we have two different spectra: There is the spectrum of particle-like solitons and the spectrum of the actual particles. From this idea that in certain field theories the two spectra may be inter-related or inter-reliant emerge the particle-soliton duality. This phenomena is surprising and deep and occurs elsewhere in field theory and has had important applications in supersymmetric field theory and in superstring theory. It is similar to, and linked with, the T-duality. In this paper we investigate the foundation and the development of this duality.

1

Introduction

T h e concept of a particle is a n a t u r a l idealization of our everyday observation of m a t t e r . Dust particles or baseballs, under ordinary conditions, are stable objects t h a t move as a whole and obey simple laws of motion. However, neither of these is actually a structureless object. T h a t is, if sufficiently large forces are applied to them, they can readily be broken a p a r t into smaller pieces. 93

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The idea that there must be some set of smallest constituent parts, which are the building blocks of all matter, is a very old one. Democritus 1 is often credited with introducing this idea, though his concept of the building block was quite different from ours today. He introduced the word which in English translates as atom to describe the parts, whatever they might be. History plays tricks with language, however. The word atom has acquired a meaning today that only partly matches Democritus' idea. Certainly we know that matter is indeed composed of the objects we call atoms. Atoms were originally thought to be indivisible, that is, the smallest particle. However we now understand that atoms are built up of smaller parts. Therefore the question concerning the particle nature is reminiscent of the atom's one. So we pull up the level and ask again what are the particles? how are they made? are they just points? or solid balls? or kind of vortices in some invisible fluid as imagined by Descartes for the stars? or something else ... We are still very far from any answer. Indeed, The experiments of the nineteenth century have caused the classical definitions of particles and waves to be blurred 2 . Consequently, we were convinced for some time that we need

1

Born about 460 BC in Abdera, Thrace, Greece From this perspective, the particles are discrete, their energy is concentrated into what appears to be a finite space, which has definite boundaries and its contents we consider to be homogenous (the same at any point within the particle). Particles exist at a specific location. If they are shown on 3D graph, they have x, y, and z coordinates. They can never exist in more than one place at once, and to travel to a different place in space, a particle must move to it under the laws of kinematics, acceleration, velocity and so forth. Interactions between particles have been studied for many centuries, and a few simple laws underpin how particles behave in collisions and interactions. The most primary of these are the conservation of energy and momentum which allow us to simplify calculations between particle interactions on scales of magnitude which vary between planets and quarks. Waves unlike particles cannot be considered a finite entity. Their energy cannot be considered to exist in a single place since a wave by definition varies in both displacement and in time. For example, a sound wave is a deformation in air pressure, and water waves a deformation of the water surface. In an area of space, unlike a particle, a wave can propagate until it exists in all locations and at all times, mathematically we can use a pure sine wave as an example, which has no beginning or end, but repeats every 27r. However, like particles, we can analyze a part or phase of the wave and obtain a value for its velocity within this area. 2

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to exclude for the particle the pointlike 3 and the solid balls conceptions. Moreover when A. Einstein, after the pioneer work of M. Planck, understand one century ago that the light itself — understood as an oscillation of the electromagnetic field since J.C. Maxwell — must to be quantized into photons, the problem became even more puzzling and one was led, through the elaboration of the quantum field theory, to think each particle as a quantized excitation of some field. There is however a second mechanism by which a field analogous to the electromagnetic field could lead to structures which behave like a particle: the merging out of a vortex or a soliton as a solution of a non linear partial differential equation. May be such an idea came in mind of some scientist a long time ago, by observing the persistence of a vortex patch in the water. When two centuries ago we were able to modelize the motion of fluid by the Euler equations or the Navier-Stokes equation, then it was possible to check mathematically that, for an incompressible fluid (like in a good approximation the water) the vorticity was conserved. Eventually this led Sir Thomson 4 to a sophisticated theory of rings and knots of vortices as the ultimate constituents of the matter. Of course the quantum theory revolution forced everybody to abandon such naive descriptions and the success of the atomic model temporarily satisfied people. But still the question remained what are the electrons, neutrons, protons, or, as we should rephrased it now, the quarks ? They all are fermions, i.e. particles which possesses strange properties as imposed by Pauli exclusion's principle: two different fermions cannot be in the same quantum state, there is only one place for one fermion in one quantum stat, in contrast with the bosons. Tony Skyrme (1922-1987), a British physicist, proposed5[21] to go back to the idea of vortices merging out from a kind a fluid dynamics (or from a field dynamics) to modelize fermions. Such an hypothesis could have more success in the end of the twentieth century than before, since beside the examples of the vortices other phenomena 3

For then charged electrons would have infinite energy. Kelvin suggested that molecules are knots in the ether. While we now know that there is no ether and that molecules are not the fundamental constituents of matter, the idea that matter has a topological origin remains beautiful and compelling. 5 This a very speculative non-linear theory of nuclear interactions which has a very rich mathematical structure. The fundamental particles in the Skyrme model are pions and the solitons, which are called Skyrmions, are classical nuclei. 4

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of "particles merging out" from a smooth field were known. There are called "solitons" by physicists and we shall see that, as for vortices, solitons were observed more than 150 years ago in the nature. Now we know a lot of models of fields satisfying a partial differential equation which are solitons equations. It seems at first glance that the idea that particles, and in particular fermions, are solitons is in contradiction with the dogma of the quantum field theory 6 that particles are quantization of the energetic excitation of a field (a dogma which is confirmed by every day physical experiment). Nevertheless G. 't Hooft showed that if we combine both ideas it helps in solving the difficult problem of the renormalisation of the Yang-Mills-Higgs model (and there is no other way known). The idea that fermions could be solitons was actually confirmed in theoretical models in 1975 by S. Coleman[5] in the case when the space-time is twodimensional and with the sine-Gordon model 7 . More precisely S. Coleman showed that two different classical models lead, when the quantum theory is constructed, to describing the same fermion particle. But in one model the fermion is a quantum excitation of the field and in the other model the particle is a soliton 8 . Hence both points of view can be reconciliated! So in those field theories which have the striking feature that their classical dynamical equations have particle-like solutions known as solitons, there will be two different spectra. There is the spectrum of particle-like solitons and the spectrum of the actual particles. Prom this idea that in certain field theories the two spectra may be inter-related or inter-reliant emerge the particle-soliton duality. 6 Physicists use quantum field theories to describe fundamental particles. These quantum field theories are derived by quantizing classical field theories. Whereas classical field theories describe the dynamics of continuum fields, quantum field theories can be interpreted as describing the interactions of individual particles. Unlike the particles introduced by the quantization procedure, solitons are germane to the classical, continuum, theory. They owe their particle-like properties, not to quantization, but to the topology of the field theory itself. 7 T h e sine-Gordon model was invented by Tony Skyrme, the name is a joke because it sounds like Klein-Gordon. 8 T h e basic properties of solitons, like propagation and interaction without change in their velocity and shape, make it possible to treat them as the robust localized objects. Solitons show their duality, having both the properties of particles and waves. A soliton has the wave nature and finite width but it behaves itself as a particle when interacting with other solitons. That is why the solitons are often spoken of as quasiparticles.

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In the quantized sine-Gordon model for example there are two different particle spectra: a soliton spectrum and a spectrum arising out of quantization. What makes this equation so remarkable is that there is a non-local transformation of the field which turns it into another one dimensional equation known as the Thirring model. The transformation maps the soliton particles of the sine-Gordon equation onto the ordinary quantum excitations of the Thirring model[22], so the two types of particle are not so different after all. We say that there is a duality between the two models, the sineGordon and the Thirring. They have different equations but they are really the same. This phenomena is surprising and deep and occurs elsewhere in field theory. This idea has had important applications in supersymmetric field theory and in superstring theory. It is similar to, and linked with, the T-duality. The first of these concerns duality between electric and magnetic monopoles. Maxwell's equations for electromagnetic waves in free space are symmetric between electric and magnetic fields. A changing magnetic field generates an electric field and a changing magnetic field generates an electric one. The equations are the same in each case, apart from a sign change which is irrelevant here. However, it is an experimental fact that there are no magnetic monopole charges in nature which mirror the electric charge of electrons and other particles. Despite some quite careful experiments only dipole magnetic fields which are generated by circulating electric charges have ever been observed. In classical electrodynamics there is no inconsistency in a theory which places both magnetic and electric monopoles together. In quantum electrodynamics this is not so easy. To quantize Maxwell's equations it is necessary to introduce a vector potential field from which the electric and magnetic fields are derived by differentiation. This procedure cannot be done in a way which is symmetric between the electric and magnetic fields. 70 years ago Paul Dirac[6][7] was not convinced that this ruled out the existence of magnetic monopoles. He always professed that he was motivated by mathematical beauty in physics. He tried to formulate a theory in which the gauge potential could be singular along a string joining two magnetic charges in such a way that the singularity could be displaced through gauge transformations and must therefore be considered physically

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inconsequential. The theory was not quite complete but it did have one saving grace. It provided a tidy explanation for why electric charges must be quantized as multiples of a unit of electric charge. In the 1970's it was realized by 't Hooft and Polyakov[ll] that grand unified theories which might unify the strong and electro-weak forces would get around the problem of the singular gauge potential because they had a more general gauge structure. In fact these theories would predict the existence of magnetic monopoles. Even their classical formulation could contain these particles which would form out of the matter fields as topological soli tons. In 1977 C. Montonen and D. 01ive[16] conjectured9 that this kind of duality could exists, but the mathematics of field theories in 3 space dimensions is much more difficult than that of one dimension and it seems beyond hope that such a duality transformation can be constructed. But they made one step further forward. They showed that the duality could only exist in a supersymmetric version of a GUT. Until 1994 most physicists thought that there was no good reason to believe that there was anything to the Olive-Montonen conjecture. Recently Seiberg and Witten[20]obtained exact expressions for the metric on moduli space and the dyon spectrum oi N = 2 supersymmetric S't/(2)-Yang-Mills theory by using a version of the Montonen-Olive duality and holonomy on Ad supersymmetric theories. As a bonus their method can be used to solve many unsolved problems in topology and physics. In the last year physicists have discovered how to apply tests of duality to different string and p-brane theories in various dimensions. A series of conjectures have been made and tested. This does not prove that the duality is correct but each time a test has had the potential to show an inconsistency

it has failed to destroy the conjectures. What makes this discovery so useful is that the dualities are a non-perturbative feature of string theory. Now many physicists see that p-brane theories can be as interesting as string theories in a non-perturbative setting. The latest result in this effort is the discovery that all string theories which are known to be perturbatively

9

Olive and Montonen showed how such dualities could be incorporated to Yang-Mills equation and generalized by this way the electromagnetic duality (exchanging the electric field with the magnetic field) to Yang-Mills in a supersymmetric setting.

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finite are now thought to be derivable from a single theory in 11 dimensions known as M-theory[24]. M-theory is a hypothetical quantum field theory which describes 2-branes and 5-branes related through a duality. Finally we want to stress three interesting points about solitons on the mathematical side: 1 Integrability: When solitons were observed in the nature and reported for the first time in 1844 (see below) their influence on the development of mathematics was relatively limited (beside the work of Boussinesq, Rayleigh and Korteweg and de Vries in fluid mechanics) . In particular very little progresses were obtained for almost one century, maybe because mathematicians were under the impression that we could not solve exactly nonlinear partial differential equations. Then around 1960 solitons were rediscovered and gave rise to vast and deep developments of ideas in mathematics. And because of the miraculous properties which are hidden in the structure of completely integrable systems, their study reveals unexpected links between analysis, Lie group theory, algebraic geometry, Hamiltonian mechanics (without speaking on the relations with classical and quantum physics). Moreover solitons led us to recognize that through a combination of such diverse subjects as quantum physics and algebraic geometry, we can actually solve some nonlinear equations exactly, which gives us a tremendous "window" into what is possible in nonlinearity. Note however that many of the magical properties of integrable systems had been detected during the nineteenth century by differential geometers working on surfaces of constant mean curvature or of constant Gauss curvature for instance. Geometers like O. Bonnet, A. Enneper, Ribeaucour, A.V. Backlund, L. Bianchi, S. Lie, G. Darboux, E. Goursat, J. Clairin... developed an impressive science on the subject, a knowledge which was almost forgotten during one century and which is partly rediscovered today. 2 Nonlinear Superposition: Before the discovery of solitons, there was no analogue of the superposition construction for nonlinear equations, but the way that a 2-soliton solution can be viewed as a combination (though not a simple linear combination) of two 1soliton solutions leads to a recognition that (at least for soliton equations) that there is a nonlinear superposition principle as well.

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(This is now understood algebraically in terms of the symmetries known as Darboux-Backlund transformations (but first discovered by Ribeaucour) and geometrically in terms of the structure of the Sato Grassmannian.) 3 Algebraic-Geometry: Perhaps the most surprising thing about solitons is the way they interface the subjects of differential equations and algebraic geometry. This has been used to the benefit of each subject. For example, there are large classes of solutions to soliton equations that we know only because they are provided by classical results in algebraic geometry. (In particular, solutions can be written in terms of the theta function of complex projective algebraic curves.) In the other direction, the KP hierarchy of soliton equations provides the only known method for determining whether a given Abelian variety is a Jacobian variety (a problem known as the Schottky problem in algebraic geometry). Most interesting is the way that algebraic varieties play a role for solitons analogous to a vector space in the case of linear equations. In particular, just as the linear combinations of a set of known solutions form a vector space, the "nonlinear superpositions" of solutions to the KP hierarchy form an infinite dimensional grassmannian manifold.

1.1

T h e discovery of solitons in n a t u r e

The best introduction to the soliton is that contained in the Scottish naval engineer J. Scott-Russell's seminal 1844 report to the Royal Society. In the less formal scientific writing style of the time, Scott-Russell 10 wrote in his Report on Waves^n}: I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded smooth and well-defined heap of water, which continued its course along the channel apparently without change of 10 Russell (1808-1882) was conducting experiments to determine the most efficient design for canal boats. Following this discovery, Scott Russell built a 30' wave tank in his back garden and made further important observations of the properties of the solitary wave.

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form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon ... It was a further decades until first Boussinesq [4] and later independently two Dutchmen, Korteweg and de Vries[13], developed a nonlinear partial differential equation to model the propagation of shallow water waves applicable to situation that Scott-Russell fortuitously witnessed. This famous equation is known simply as the KdV equation 11 . A key feature of KdV is that the speed of solitary waves12 is proportional to their height. In 1955, research into stable and non-dispersive but localized solutions in non-linear media was taken up again when the equipartition of energy between 64 weakly and non-linearly coupled harmonic oscillators was modelled numerically[28]. Starting with only one oscillator excited the energy distributed itself over the whole mode system but returned almost completely to the first excited one. Thermodynamic equilibrium was not reached and the excitation was stable in that sense 13 . From then onwards solitary waves or solitons 14 have become more and more important. On the other hand one of the obvious question that arises was: what happens when a taller solitary wave overtakes a shorter (and therefore slower) solitary wave; in particular, do the individual pulses survive the collision? For nearly seventy years, an article of faith in the physics community was that a collision involving two solitary waves would destroy the otherwise stable input pulses due to the nonlinear nature of the interaction. 11 Note, the very slowly diminishing height of the water wave that Scott-Russell observed is due to "frictional" losses and is not taken into account in the standard KdV equation. 12 A solitary wave is denned as a spatially confined (localized), non-dispersive and nonsingular solution of a non-linear field theory, i.e. one without superposition principle (possible for example in shallow water but not on a simple string). Therefore they have been thought impossible because a dispersive and non-linear medium has been expected to alter any shape of any wave over time. That the non-linearity can compensate for the dispersion leading to a propagating and stable wave having constant velocity and shape came as a surprise. 13 This is what commonly we call the Fermi-Pasta-Ulam paradox. 14 Remark that their definitions change from author to author.

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It was not until the late 1960s that Norman Zabusky and Martin Kruskal numerically 15 discovered that KdV solitary waves maintained their identity following collisions: here we have a nonlinear physical process in which interacting localized pulses do not scatter irreversibly. Zabusky and Kruskal coined the term soliton to reflect the particle-like nature of these robust travelling solitary waves. For quite some time only classical solutions (i.e. not quantized) in low dimensional spaces have been considered. Actually, not having a superposition principle renders the quantization of solitary objects complicated. One cannot think of the shape of the solitary wave as being the shape of the wave function for the reason alone that a quantum soliton cannot be localized in space all the time. The uncertainty principle will cause a spreading. Thus the definition of the quantum soliton is bound to the corresponding classical one. There are a lot of ways of qunatising the solitary waves (e.g. functional, canonical etc) and both, solitons and instantons are quantized semi classically, notably among the important methods is the WKB. The quantization is only possible if the non-linear coupling constants are small although the maybe most promising feature of solitons is that they are approached in non-perturbative ways. Only the quantum corrections are perturbative. For any non-linear theory the soliton is at least as fundamental a solution as the sine wave. Recently, there have been profound advances in finding solitons in higher dimensional theories and in quantizing them. Doing quantum mechanics one finds relations between solitons that go very deep and are entirely unexpected from a classical viewpoint. Their importance was recognized in quite different areas of physics. For particle physicists a localized and stable wave might be a good model for elementary particles opening up in a non-linear field theory 16 the possibility of what would have to be a wave packet in a linear one. Notably among others is the Skyrme model that aims to describe nucleons and 15 In 1967, this numerical curiosity was placed on a firm mathematical basis with Gardner and al.'s discovery of the inverse-scattering-transform method. 16 Newer fundamental gauge theories are non-Abelian and therefore non-linear.

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nucleon-nucleon interactions. Topological solitons give rise to pre-quantum mechanical quantization of charges. To conclude this introduction we want to mention that since their discovery a mere thirty years ago, solitons have been invoked to explain such diverse phenomena as the long lived 'giant red spot' in the highly turbulent Jovian atmosphere. Ion-acoustic waves in a plasma. Energy storage and transfer in proteins via the Davydov soliton. The propagation of short laser pulses in optical fibers over long distances with negligible shape change. For instance nonlinear pulse propagation in optical fibers has been studied over 30 years. The idea of using optical solitons 17 as information bits in highspeed telecommunication systems was first proposed in 1973[10], and then demonstrated experimentally in 1980[15]. In the following years, as fiber technology advanced, interest in optical soliton transmission started to increase. Multigigabits transmission over more than lOOOOfcm was reported regularly starting in 1991. To further increase the bit rate and transmission distance, novel techniques such as wavelength division multiplexing (WDM), polarization division multiplexing (PDM), and dispersion management have been proposed and utilized. Various optical ultrafast soliton switches have also been designed and demonstrated. These techniques have spurred fresh interest in the theoretical modelling of pulse propagation and collision in the underlying communication systems. In an ideal fiber, optical solitons can be modelled approximately by the nonlinear Schrodinger (NLS) equation, whose solution behaviors are completely known [25] [1]. But in reality, optical fibers are birefringent. Pulses travel at

17 A key observation was that an intense cylindrical beam of laser light propagating though certain dielectric media had a natural tendency to self-focus or pinch, the result being that in the focal region the medium would be literally torn apart (usually with an accompanying bang) due to the exceptionally high electric field strengths reached. This result is all the more surprising because at lower laser powers, the beam spreads apart due to diffraction. Given that the refractive index depends on frequency, it is not too much of a j u m p to imagine that it is also dependent on the intensity of the light passing through the medium. That is, at every point in the material, the refractive index depends on the intensity of the light at that point. Since a beam is most intense at its center, the refractive index would be highest at the middle and decrease as one moved outwards towards the beam edge. We can now explain the self-focusing of intense cylindrical beams - light rays travelling at a given radius have a natural tendency to bend inwards towards areas of increased index of refraction.

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slightly different speeds along the two orthogonal polarization axes. This effect has been analyzed in [14], where two coupled nonlinear Schrodinger (CNLS) equations were derived for the pulse propagation along the two polarization axes. Birefringent fibers also support optical solitons they are called vector solitons [2]. The collision of vector solitons is critical in many optical switching devices and nonlinear optical telecommunication networks. In all optical soliton switching, various ultrafast digital logic gates have been pro posed and experimentally demonstrated [8]. These logic gates utilize the phase or frequency shift created by the collision between orthogonally polarized solitons. They can operate at bit rates up to 0.2THz.

2 The Soliton To explain the nature of solitons, we will consider the behavior of water waves on shallow water. The water wave soliton is a result of a dynamic balance between dispersion, i.e. the wave's tendency to spread out, and nonlinear effects. In order to substantiate this statement, we have to pass to some mathematics. The dynamics of water waves in shallow water is described mathematically by the Korteveg-de Vries (KdV) equation: ut + uxxx + uux = 0,

(1)

where u = u(x, t) measures the elevation at time t and position x, i.e. the height of the water above the equilibrium level. The second and the third term 18 in the equation is the dispersive and the nonlinear term, respectively. Let us first investigate the effect of the dispersive term. Thus we neglect the nonlinear term in the KdV equation. This leaves us with the following: ut + uxxx = 0.

The subscripts denote partial differentiation.

(2)

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We know from the wave dynamics that waves with different wavelengths travel with different velocities. Since the initial wave is composed by many small waves with different wavelengths, it will soon spread out in the many components and can no longer be described as an entity or object 19 . Now let us see the effect of the nonlinear term. We neglect the dispersive term in the KdV equation, which leaves us with the following: ut + uux = 0.

(3)

The top of the wave moves faster than the low sides and this causes the wave to shock in the same way as the waves we see on the beach. When both the dispersive and the nonlinear term are present in the equation the two effects can neutralize each other. If the water wave has a special shape the effects are exactly counterbalanced and the wave rolls along undistorted. The soliton shape can be found by direct integration of the KdV equation: u(x, t) = asech2[b(x — vt)],

(4)

with b = (a/12) 2 and v = 3a. The constant a is the only free parameter in the solution. It defines the amplitude and the width in such a way that a large (tall) soliton will be narrow, while a low soliton will be broad. The constant v defines the velocity of the soliton Since « = 3 a a tall soliton will move faster than a low one. The concept of a soliton provides a new line in research in systems which are wave-like in nature, e.g. systems involving water or light. The soliton is regarded as an entity, a quasi-particle which conserves its character and interacts with the surroundings and other solitons as a particle. Instead of trying to describe the complete field of water or light, one can concentrate on the evolution of solitons which will stay around in the system (in principle) forever and will constitute the most important part of the dynamics.

19

In

more

mathematical

7/fc J-oo e%^x+i

terms:

any

solution

of

(2)

writes

u(t, x)

=

' M O d £ > so it is a (Fourier) superposition of sinusoidal waves of ve-

locity (-£•) , where L is its spacial wavelength.

106

F. Helein & J. Kouneiher

Lastly we want to pinpoint the supersymmetric extension 20 of a nonlinear evolution equation (KdV for instance) where the integrable (in the sense of Lax pair) variant of (N — l ) 2 1 supersymmetric-KdV is $ t + £>6$ + 3£>2($£>$) = 0, In order to have nontrivial extension for KdV we choose $ to be fermionic, having the expansion $(£, x, 9) = £(£, x)+9u(t, x), which on the components has the form 22 ut = uxxx - 6uux + 3££xx £t = -£xxx - 3£xU - 3£u x .

(5)

If we choose $(x, t) to be a bosonic field i.e. to have the expansion $(*, x, 0) = u(t, x) + 0£(t, x). then we have the following supersymmetric extension of the KdV equation: $t + D6$ -f 6 $ D 2 $ = 0, or equivalently ut = uxxx — 6uux£t = -£xxx - 6(£u)x.

(6)

This system is trivial in the sense that is linear in the fermionic field and, as a result, one of the two equation is simply the KdV equation itself.

3 Quanta Versus Solitons 3.1 Quantum particles In 1900 M. Planck explained the black body spectrum by assuming that the energy of the light at a given frequency is quantized and can only be equal to an integer multiple of its frequency, by means of a proportionality factor, now known as the Planck structure h. Later A. Einstein explained the photoelectric effect by postulating that the light is made of particles called photons. This went against the conception that light is made of waves, an hypothesis which was largely confirmed in the nineteenth century by 20

We have to extend the classical space (x, t) to a larger space (superspace) (t, x, 6) where 0 is a Grassmann variable and also to extend the pair of fields (u, £) to a larger fermionic or bosonic superfield $(t, x, 0).The derivative on the superspace will be defined in the form D = dg + 0dx • 21 W i t h one supersymmetry generator. 22 Notice that D2 = dx.

On the Soliton-Particle Dualities

107

experiments with interferences of the light. The physicists were then faced to the puzzling question of understanding the light as made of particles and made of waves, summarized in the wave-particle duality. Then L. de Broglie proposed in 1923[3] to assume that conversely, to each particle of matter (electrons, atoms...) it corresponds a wave. This led E. Schrodinger to its equation dij) rfi

rfi

h2

ft^

where A = -^ + -g-? + -^ and ip is a function which depends on time (t) and space (a;, y, z) variables with values in C And a satisfactory model for the atom was built. The hypothesis of de Broglie was confirmed experimentally by Davisson and Germer in 1927, who were able to realize interference experiments with particles of matter. But Schrodinger equation was not relativistic, but rather an undulatory (quantum) version of the Newton law ^ = - W ( x ) . So one needed to find analogous equations which would be at least invariant by the Poincare group, the group of symmetries of special relativity. The simplest one is called the Klein-Gordon equation which is -^-A<{>

+ m2(t> = 0 .

The field is here just a function on the Minkowski space-time. We could assume that <j> takes its values in K, C or any vector space. This equation was actually considered by Schrodinger even before he came with its equation. However at this time people did not see how to associate in a simple way a particle to such a field. Indeed on the one hand Schrodinger equation was relatively simple to interpret: the square of the modulus ofij), \"4>\2, was the spatial density of probability of presence for "finding" the particle. In particular a simple mathematical consequence of the Schrodinger equation is that the integral / R 3 \ip{t,x,y, z)\2dxdydz is constant in time. And this is consistent since, if we follow our probabilistic interpretation of ip, this quantity should be equal to 1, i.e. the probability of finding the particle somewhere in the space! On the other hand such an interpretation cannot work for the Klein-Gordon equation since, if <j> is a solution of the KleinGordon equation, the integral JR3 \4>(t,x,y,z)\2dxdydz is not constant in general, because the equation involves a second derivative with respect to time instead of a first derivative.

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F. Helein & J. Kouneiher

Note that, beside the Klein-Gordon equation, another relativistic equation was known, namely the Maxwell equations for the magnetic field B, the electric field E: 1 f)R curLE + —— = 0, c at

divB = 0

curLB

divE = p,

IdE KT=J, c at

where p is the electric charge density and j is the electric current density. In the absence of charges (i.e. if p and j vanish) we remark that by elimination of one the variables £ or B , we find that E and B are solutions of the wave equation l d 2 * AX = 0, c2 dt2 a version of the Klein-Gordon equation with a mass coefficient m equal to 0. But Maxwell equation presents the same difficulty as the Klein-Gordon equation in order to interpret how particles could be extracted from such a wave equation.

The more beautiful attempt to find a relativistic equation which could be interpreted from a quantum point of view was the Dirac equation for electrons Vip + mi; = 0, where the Dirac operator V is chosen in such a way that it involves first order derivatives in time (for the purpose of "conserving" the probability density) and in space (for relativistic symmetries). It did work at the cost of allowing that tp is now a vector with 4 complex components (one should say rather a spinor) and that V = 7 0 J^ + 7 1 ^ + 7 2 •§- + •y3 ^ , where the coefficients 7M are now 4 x 4 complex matrices. Then in a fashion analogous to the Schrodinger equation, there is way to produce a real density function \ip\2 out of tjj, in such a way that, if tp is a solution of the Dirac equation, then JR3 \ip(t, x, y, z)\2dxdydz is constant. But again the physical interpretation ran into new difficulties: one would like to interpret l^l 2 as a probability density for a particle, but unfortunately \ip\2 is not nonnegative in general, it is equal to | ^ i | 2 + l ^ l 2 - | ^ 3 | 2 - \4>4\2, a difference of two positive quantities! Dirac was then obliged to imagine an infinite see (in the space of quantum states) filled by electrons. The holes in this see would

On the Soliton-Particle Dualities

109

give a negative contribution ot the total / R 3 \tp(t, x,y, z)\2dxdydz and electrons above the see a positive one. This could make sense since electrons are fermions and thus, because of Pauli's exclusion principle, each of these occupies a whole quantum state. But this interpretation suffered from the hypothesis of this infinite see: an infinite set of electrons in the universe! Fortunately three years after Dirac's invention of its equation, antimatter was discovered experimentally and one then realized that Dirac had found the right equation for it: there was no more need for holes, but for antielectrons! In fact a correct interpretation of all these relativisitc equations needed the introduction of the quantum theory of fields: the fields in these equation are not amplitude for density probabilities but operators acting on an auxiliary Hilbert space called Fock space, where all particles existing are registered. This allows hence a variable number of quantum particles, an assumption which is necessary if one want to study scattering of particles. Let us explain briefly the principle of the quantum field theory on a simple equation: the Klein-Gordon equation on Minkowski space-time

Solutions to this equation can be found by means of the space-time Fourier transform: 4>{t,S) = - ^ / 3 (a{£)e** V 2-K" "'R y/2n J®3 V

+a(£)e-**)2 w ' £

where x := (t,x), £ := (w?, £), CUJ?:= \lm2 + |£| 2 and £ • x := c2u>A — £ • x. It means that is a superposition of monomochromatic waves with spacetime wave vector equal to £ satisfying c2u)2- — |£| 2 = m 2 (the mass-shell constraint). Thus ^£z ls a Lorentz invariant measure on the mass shell hypersurface. Next we will imagine that each wave vector £ satisfying the mass shell constraint is associated with a quantum particle. The quantum field interpretation of this equation requires to consider an auxiliary Hilbert space T called Fock space. The point is the construction of a particular Hilbertian basis of this space and the way we label each vector of this basis, according to its physical meaning: as in buddhism the

110

F. Helein & J. Kouneiher

more important one is |0), the vacuum, i.e. the quantum state without any particle. The other ones are denoted by |£i • • • £,- ), for an arbitrary positive integer p and an arbitrary collection of wave numbers £1,..., £,- which satisfy the mass shell constraint. This vector represents a state were p particles of wave number respectively equal to £1, •••,£?,, fill the space. Note that we must assume that f 1 < • • • < £j in order to avoid repetition of two linearly dependant vectors. Moreover if the particles are bosons we allow arbitrary repetitions of a given wave number, but if they are fermions a wave number can occur at most one time. The Fock space is the Hilbert completion of the vector space spanned by all these vectors 23 . Lastly to each wave number £ we associate a creation operator a | and an annihilation operator aj. The creation operators act by a||fi • • • f;jp) = |££i • • -£jp) and the annihilation operator acts by a^|^i • • • £Jp) = ±|£i • • • ^ _ i ^ + i • • • £,p) if there exists some j such that £ = £,-; otherwise a^\ •••£}„) = 0. Back to the Klein-Gordon equation, in the quantum interpretation we are not interested anymore in solving this equation, but in finding a kind of "universal" solution to it, in the sense that this single solution potentially describe all quantum evolutions: we replace 0 by a function with values into a set of self-adjoint operators 24 . And the Fock space T has been built as a representation space for the family of operators ((f>(t,x))(ttzy Hence the meaning of ( ^I§^I — A + m2) (f> = 0 is just that, for all |state) £ T, (c? ST7 — ^ + TTj2) (Estate) J = 0. This is the Heisenberg picture: dynamics is encoded in the evolution of operators and a given vector in T is in correspondence with a full space-time scenario. This representation problem

23

In fact the collection |£i •• • £.jp) is not exactly a basis: one should replace these vectors by smeared vectors using smooth test functions cj>j:

i ••• I MM J«

• • • ^ P f e P ) l € i • • • & , ) < ? ! • • • dfc p .

./R

But the notation |£i • • • £,jp), although it is not mathematically correct, is more intuitive. T h i s process, often called second quantization, is quite analogous to the route that in ordinary quantum mechanics leads from a set of classical coordinates qi to a set of quantum operators qi. There is one important technical difference, though, since ip(x,t) is a field, i.e., an object which depends on the coordinate x. The latter plays the role of a continuous-valued index, in contrast to the discrete index i, which labels the set qi. Field theory therefore is concerned with systems having an infinite number of degrees of freedom. 24

On the Soliton-Particle Dualities

111

is solved here by posing V27T -/K3

iU)

i

and d(j)(t,x) _ ot

1 \/2TT

/• , , . . w _ . t f . x

„*_-i£.^ dg

3

JU

This method works for all standard linear equations without additional difficulties (excepted for the Maxwell equations because of the gauge symmetries). However when we deal with non linear equations like the instance of the non linear Klein-Gordon equation

where n is some integer larger or equal than 3, then the situation becomes very bad. Even in this simple situation there is no satisfactory mathematical theory. Physicists however developed a powerful machinery in order to compute correlation functions, the relevant quantities for physical predictions. But it is only a perturbative expansion in powers of the dimensionless coupling parameter A and its accuracy depends on the smallness of A. For instance the simplest physically interesting model is the Dirac-Maxwell system for electrodynamics with a coupling constant of order -^: this constant is very small, and so the model leads thus to very good predictions. But other models like the Yang-Mills-Higgs one used for chromodynamics involve larger coupling constant, a source a limitation for their application. Note that, although the perturbative theory of physicist is not well founded mathematically, it involves rich and subtle concepts like the use of Feynman diagrams and renormalisation theory which are sources of fruitful mathematical ideas.

3.2

T h e kinks

As a first example of a nonlinear field equation, let us consider 1 d2(j) d2cj) t ojL/ 2 2 + 24>(4> - l) = o. c 2 dt2 dx2 The solutions of this equation are critical points of the functional

F. Helein & J. Kouneiher

112

Let us try to find a solution of the type 4>{t, x) = f(x — vt): it leads to the equation

/ " = 2p2f(f2 - 1),

(7)

where j3 := ( 1 — ^j-j . We shall assume that \v\ < c in the following. A trick is to look first at solutions of the first order partial differential equation / ' + / ? ( / 2 - l ) = 0,

(8)

since — as the Reader could check himself — any solution of (8) is automatically a solution of (7). And then one sees easily that solutions to the first order equation (8) are of the form f(s) = tanh/?s. We can pause to stress out the fact that this trick rests on finding Bogomol'nyi solutions to the second order equation (7). It is based on the fact that the functional A is part of a supersymmetric action functional and that the first order Bogomol'nyi condition is just asking the solution to be invariant by one supersymmetry generator (the Bogomol'nyi-PrasadSommerfeld states) 25 . Let us look now at the solution
25

A way t o characterize the Bogomol'nyi solution is t h e following. A consequence of the supersymmetry involved is that the action has the form A[] : = JKXR ( 2 ? I ^ l 2 " \ l ^ l 2 ~ 2(w'())2) plies in particular

that a function

is a critical point of the functional can write down this functional

dtdx

>

where

here

w s

i)

= h (ss3

~s)-

lt

im

-

/ of x is a solution of (7) if and only if it £[f] := / R ((f)2

+ 4f32 {W(f))2\

as £[f] = Jm ( ( / ' + 2/3W'(f))2

dx. Now we

- 4 / 3 / ' W ' ( / ) ) dx =

JR ( ( / ' + 2f3W'(f))2 - 4 g (W(f))) dx. If we assume that l i m ^ i o o f(x) = / ± , then the last term in the right hand side is just C := 4 (W(f-) - W ( / + ) ) So £ [ / ] - C is the integral of the square of / ' + 2/3W'(f) and a trivial solution is t o set / ' + 2f3W'(f) = 0: this is exactly (8) (see [9]).

113

On the Soliton-Particle Dualities

• the difference Q := \ (<j>(t, oo) - <j)(t, -oo)) does not depend on t, it is just equal to 1. We could interpret this number as a charge 26 . A way to see that is to consider the 1-form d = 7$ dt + ^dx: it is obviously closed and its integral over a constant time slice is equal to 2Q as soon as the field is assumed to be asymptotically constant in time at infinity. • another solution is — tanh(/3(a; - vt)). Its charge is -L It is also a Bogomoln'nyi solution corresponding to the equation / ' — (3(f2 — 1) = 0. One could think this solution as an antiparticle. Moreover we could imagine one kink and one antikink (in such a way that the total charge is 0) travelling each to the other one. When they meet they cancel together, like the disintegration of a pair electronantielectron. All that suggest strongly that the kinks behave like fermions. Note that here the total charge could only be equal to - 1 , 0 or 1, because of the conditions at infinity imposed by the finiteness of the energy JR f ^

|£

2

+ i M

2 +

i (02 _ X )2\ dx

We

recover here the Pauli's ex-

clusion principle: there is no place for two kinks on the same line!

3.3 The sine—Gordon equation A more refined model is the following 1 d2

d2

a . ,

1

„

equation are critical points of the o f this t The solutions 0 : R x R -—> IR of functional d<j>

A[}: JMxM 26

\2c

dt

2

1 2 dx

2

\ + —j (cosA(?!> — 1) I dtdx.

In 1917 German mathematician Emilie Emmy Noether had shown that the mass, charge and other attributes of elementary particles are generally conserved because of symmetries. For instance, conservation of energy follows if one assumes that the laws of physics remain unchanged with time, or are symmetric as time passes. And conservation of electrical charge follows from a symmetry of a particle's wave function. Sometimes, however as in our case, attributes may be maintained because of deformations in fields. Such conservation laws are called topological, because topology is that branch of mathematics that concerns itself with the shape of things.

114

We

F. Helein & J. Kouneiher

are

interested

JR ( 2c7 ~5t + 1 &

in

solutions

such

that

their

energy

+ 3*(1 ~~ cos\) J dx is finite. This forces that for

any time limx_»±0o (t, x) = ^-n±, for n± integer. Thus we could define a kind of topological charge Q := n+ — n_ as before, the only difference being that this charge could be any number in Z in principle (instead of being in { — 1,0, +1}). Beside the constant functions the finite energy solutions of the type (t, x) = f(x — vt) has the form 27 2 2TT 4>{t,x) = -r a r c cos (± tanh(3y/a(x — v£)) + —n, A

A

for \v\ < c,

where again (3 := (1 — ^j) and n G N. Again it turns out that this solution behaves like a soliton and possesses the same properties as the kinks of the previous equation. So we can think on these kinks as being fermions of charge ± 1 . Note that there is no translating solution with a charge different from — 1, 0 or 1. This reflects the exclusion principle for fermions. Superposition of several fermions would show up repulsion of fermions with the same charge and attraction of fermions with opposite charge, with the possibility of annihilation.

3.4

T h e massive Thirring model

The surprising result— already suggested by Skyrme 28 — was that these solitons behave really like quantum fermions! This was proved in 1975 by S. Coleman. For that purpose we introduce the massive Thirring model: the fields are spinors fields ip from the 2-dimensional space-time with two complex components ip\ and ip2- We consider the Lagrangian L(i}>) := V'( i 7 /i £^7 - m)i/j - -g ( t / W ) 27

(^7M^)

,

<j> is also a Bogomol'nyi solution of the form <j>(t, x) = f(x — vt), where f' + 2/3W'(f)

= 0

and W(s) : = - ^ c o s ^ t . 28 Skyrme's explanation was that, in the full quantum theory, it is possible to construct a new quantum field whose fluctuations are the solitons. The new field operator is obtained by an exponential expression in the originalfieldcf> if>±(x) = e

vv

J

-°°

Bt

'

(5)

with two spin components (and a normal ordering understood). The construction (5) is an example of the vertex operator construction later to be so important in string theory and in the representation theory of infinite dimensional algebras (resembling quantum field theories).

On the Soliton-Particle Dualities

115

where o

/O

1\

7 =oi=(1 J ,

!

(0

7 =-^=L

-1 Q

and 7 P = 1)^1" for ??oo = -Vu = *> ^oi = »7io = 0. Using a perturbative approach, S. Coleman compared the quantum theories of the sine-Gordon equation with the quantum massive Thirring model. He found that both models are equivalent if we assume that

And then, in the sense of quantum operators,

V^V dt + ^7
m

*(l i ) ^ = - ^

and m

(10)

KS S)* = -£ e ~**

where we have omitted a renormalisation constant which depends on the regularization. Unfortunately there is no rigorous mathematical proof of that, since both equations here are non linear and so the quantum theory of such equations has no solid mathematical base. So the result of S. Coleman is a verification that n-point Green functions (which are coefficients of the perturbative expansion) of both theories agree. This result has been confirmed by other methods by S. Mandelstam and since this time by many other authors. Note that equation (10) relies two forms which, for classical solutions, are closed, but each one for different reasons. The left hand side of this equation is the Noether current associated with a U(l) invariance of the Thirring equation: it is divergence free only if ip is a solution of the equation; it described the electric current density due to the motion of the fermions in space-time. The right hand side of (10) is always closed, even if 4> is not a solution of the sine-Gordon equation; for that reason it is called a topological current and integration of this current on a constant time hypersurface produces the topological charge Q that we described earlier. An important remark also is to observe how relation (9) relates the coupling constants A and g: if A is close to 0, then the perturbative theory of the sineGordon is relatively accurate, but then g is very large and the perturbative theory of the Thirring model is almost meaningless. On the other hand if A < \/4TT is close to %/47r, then g is close to 0 and the situation is inversed: the perturbative theory for the sine-Gordon is not very good whereas it is good for the Thirring equation.

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We are hence in a situation of duality between two models, particularly interesting because we could for instance use the perturbative quantum theory of the sine-Gordon model to obtain results on the Thirring model in the case where g is very large.

3.5

R e m a r k s on t h e mass mechanisms

This problem has two possible solutions, at first sight different, but related, at least in four dimension. The first is the idea that mass arises from the vacuum spontaneously breaking some of the gauge symmetry via a "Higgs" scalar field. The second insight into the origin of mass comes from another area of physics. Yet, it seems connected to the first, in four dimensions at least and it is a principle due to Zamolodchikov. Zamolodchikov made the observation which has rationalized the theory of solitons and more. In two dimensions, the local conservation laws characteristic of conformal symmetry are chiral. This means that the densities are either left moving or right moving (at the speed of light) and so can be added, multiplied or differentiated. If conformal symmetry is judiciously broken, a certain (infinite) subset of the chiral densities remain conserved, although no longer chirally so. Their conserved charges, that is their space integrals, generate an infinite dimensional extension of the Poincare algebra in which the charges carry integer spins. The charges with spin plus or minus 1 are the conventional light cone components of momentum. The sinh-Gordon equation illustrates this nicely. It can be written

The last term, proportional to e _ A ^, can be multiplied by a variable coefficient k so that k = 1 yields (3), while k = 0 yields the hyperbolic Liouville equation. It is interesting to investigate the behavior as k varies from 1 to 0. As long as k > 0, a simple redefinition of the field 0 by a displacement restores the sinh-Gordon form (3) but with fi replaced by jik1^. As /ih is the mass of the particle which is the quantum excitation of <j>, we see that it is singular as k approaches zero with critical exponent 1/4. So we see how mass arises from the breaking of conformal symmetry. This short discussion was classical but it extends to the quantum regime as envisaged by Zamolodchikov[26].

On the Soliton-Particle Dualities

117

The sine-Gordon equation is obtained from the sinh-Gordon equation by replacing A by iX. It then exhibits the symmetry

and, consequently, as we saw it possesses an infinite number of vacuum solutions n = ^ p , n 6 Z, all with the same minimum energy, zero. The particle of mass /j,h describes fluctuations about any of these vacua. But there also exist classical solutions which interpolate two successive vacua and which are stable with respect to fluctuations. Therefore as we saw this sort of integrable field theory has two "sorts" of particle, the quanta of the fluctuation of the field (obtained by second quantization) and the solitons which are classical solutions. There are multicomponent generalizations of the sine-Gordon equations called the affine Toda theories, likewise illustrating Zamolodchikov's principle in a nice way, and revealing the role of algebraic structures such as affine Kac-Moody algebras. Again there are particles created by each field component, now possessing interesting mass and coupling patterns (related to group theory). Remarkably, there are an equal number of soliton species and these display very similar properties.

4 On the Road to the Dualities Motivated by the previous dualities and the electromagnetic-duality, Claus Montonen and David Olive made, in 1977, a bold conjecture 29 . Might there exist an alternative formulation of physics in which the roles of 29

OHve and Montonon want to generalize the electromagnetic duality rotation symmetry to include matter. They also consider generalizations to non-abelian gauge theories of the type which seem to unify the fundamental interactions. In either case they meet the same difficulty that the gauge potentials enter the equations of motion. And so they have to include them in the transformation. The second difficulty is related to the equations of motion in non-abelian gauge theories; that they are conformaly invariant (in 3 + 1 dimensions). As a consequence, the gauge particles which are the quanta of the gauge potentials, should be massless. This problem seems to be rather general and deep: unified theories chosen according to geometric principles tend to exhibit unwelcome conformal symmetry. This occurs in string theory too, at least as a world sheet symmetry. As a consequence, there is a general problem of understanding the origin of mass through a geometrical mechanism for breaking conformal symmetry (see the section "Remarks on the mass mechanisms").

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E Helein & J. Kouneiher

Noether charges (like electrical charge) and topological charges (like magnetic charge) are reversed? In such a "dual" picture, the magnetic monopoles would be the elementary objects, whereas the familiar particles-quarks, electrons and so on-would arise as solitons. More precisely, a fundamental particle with charge e would be equivalent to a solitonic particle with charge 1/e. Because its charge is a measure of how strongly a particle interacts, a monopole would interact weakly when the original particle interacts strongly (that is, when e is large), and vice versa. The conjecture, if true, would lead to a profound mathematical simplification. In the theory of quarks, for instance, physicists can make hardly any calculations when the quarks interact strongly. But any monopoles in the theory must then interact weakly. One could imagine doing calculations with a dual theory based on monopoles and automatically getting all the answers for quarks, because the dual theory would yield the same final results.

4.1 Electromagnetic duality From a historical point of view we can say that many of the fundamental concepts of twentieth century physics have Maxwell's equations at its origin. In particular some of the symmetries that have led to our understanding of the fundamental interactions in terms of relativistic quantum field theories have their roots in the equations describing electromagnetism. As we will now describe, the most basic form of the duality symmetry also appears in the source free Maxwell equations rewritten as: div (E + i B\ = 0, — (E + iB) + icurl (E + iB\

= 0.

(11)

These equations are invariant under Lorentz transformations, conformal and gauge invariance, which have later played important roles in our understanding of phase transitions and critical phenomena, and in the formulation of the fundamental interactions in terms of gauge theories. The simplest form of duality is the invariance of (11) under the interchange of electric and magnetic fields: B^E, E ^ - B .

(12)

On the Soliton- Particle Dualities

119

In fact, the vacuum Maxwell equations (11) admit a continuous 50(2) transformation symmetry 30 (E + iB)-^ei't'(E

+ iB).

(13)

If we include ordinary electric sources the equations (11) become: div(E + iB) = q, ^-t{E + iB) + icm\(E + iB)=je.

(14)

In presence of matter, the duality symmetry is not valid. To keep it, magnetic sources have to be introduced: div(£ + iB) = (q + ig), — (E + iB) + icui\(E + iB) = (je +ijm).

(15)

Now the duality symmetry is restored if at the same time we also rotate the electric and magnetic charges (q + ig)-+ei+(q + ig).

(16)

The complete physical meaning of the duality symmetry is still not clear, but a lot of work has been dedicated in recent years to understand the implications of this type of symmetry.

4.2 Dirac's charge quantization From the classical point of view the inclusion of magnetic charges is not particularly problematic. Since the Maxwell equations, and the Lorentz equations of motion for electric and magnetic charges only involve the electric and magnetic field, the classical theory can accommodate any values for the electric and magnetic charges. Turning from the classical to the quantum theory, we immediately find a difficulty, namely that the electromagnetic coupling of the matter wave functions require the introduction of gauge potential, a procedure which is not straightforward in the presence of magnetic charge. Nevertheless, Dirac in 1931 overcame this difficulty and showed that the introduction of magnetic charge could be consistent with the quantum theory, provided its Notice that the duality transformations are not a symmetry of the electromagnetic action.

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F. Helein & J. Kouneiher

allowed values were constrained. We derive Dirac quantization condition here by the quantization of the angular momentum. Our approach will be (semiclassical) heuristic. Consider a non-relativistic charge q in the vicinity of a magnetic monopole of strength g, situated at the origin. The charge q experiences a force31 mr — qf x B, where B is the monopole field given by B = gf/4nr3. The change in the orbital angular momentum of the electric charge under the effect of this force is given by d ( ^ L\ _ ^ — mr x r ) = mr x r dt V J

-i^'M-!(£?)•

(17)

Hence, the total conserved angular momentum of the system is f=fxmf-

— -. (18) 4TT r A result dating to 1896 and due to Poincare. The second term on the right hand side (henceforth denoted by J e m ) is the contribution coming from the electromagnetic field. This term can be directly computed by using the fact that the momentum density of an electromagnetic field is given by its Poynting vector, E x B, and hence its contribution to the angular momentum is given by Jem = I d3xfx

(E x B) = j - I d3xfx

(E x -^ J .

In components and denoting xl := ^-,

J

™ = £ld3xEJ

(*« -x-9)\ = hS d?xE3d^)

= •$- I xlE-ds-— I d3x(divE)xi. (19) 4?r JS2 4TT J Where divE is the electric charge times the Dirac mass. When the separation between the electric and magnetic charges is negligible compared to their distance from the boundary S 2 , the contribution of the first integral to Jem vanishes by spherical symmetry. We are therefore left with Jem = -Y-f. 4TT 31

Using the Lorentz force law.

(20)

On the Soliton-Particle Dualities

121

Returning to equation (18), if we assume that orbital angular momentum is quantized, then it follows that

<*>

£ = 532

where n is an integer . Equation (21) is the Dirac's charge quantization condition. It implies that if there exists a magnetic monopole of charge g somewhere in the universe, then all electric charges are quantized in units of 2-ir/g.

The Dirac quantization condition has the following physical interpretation. Classically, there is not much of a distinction between a a magnetic monopole and a very long and very thin solenoid. The field inside the solenoid is of course, different, but in the limit in which the solenoid becomes infinitely long (on one end only) and infinitesimally thin, so that the inside of the solenoid lies beyond the probe of a classical experiment, the field at the end of the solenoid is indistinguishable from that of a magnetic monopole. Quantum mechanically, however, one can in principle detect the solenoid through the quantum interference pattern predicted by the BohmAharanov effect. The condition for the absence of the interference is precisely the Dirac quantization condition.

4.3 The Dirac string In the following, we present a more rigorous derivation of the Dirac quantization condition which is based on the notion of a Dirac string. Let Bmon denote the magnetic field around a monopole. Since V • Bmon ^ 0, it is not possible to construct a well-defined Amon such that Bmon = VAmon. To overcome this problem, Dirac introduced a semi-infinite solenoid (or string) running from (0,0, —oo) to the monopole position, (0,0,0). This solenoid carries a magnetic field Bsoi = g8(—z)5(x)5(y)z which also is not divergence free. However, the total magnetic field, B = Bmon + Bsoi = - ^ 32

+ g8(-z)5(x)8{y)z

,

(22)

Here we have assumed the total angular momentum of the charge-monopole system to be quantized in half-integral units. This is a strange assumption considering that we did not have to treat the electrically charged particle or the monopole as fermions. However, it turns out that this actually is the case and that the situation does not contradict the spin-statistics theorem. We will not discuss this issue further but remark that the derivation of the same equation presented in the next subsection does not depend on this assumption.

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satisfies V • B = g6(f) — g5(r) = 0. It is therefore possible to construct a non-singular A = Amon + Asoi corresponding to the monopole-solenoid system. In fact, the singular Asoi associated with the Dirac string is used to cancel the singularity in Amon. The position of the singularity in Amon, and therefore the position of the Dirac string, can be shifted by singular gauge transformations. Since the Dirac string is an artificial construct, it should be unobservable and should not contribute to any physical process. However, in an Aharonov-Bohm experiment, the presence of the string can affect the wave function of an electric charge by contributing to its phase. This resulting Aharonov-Bohm phase along a contour T encircling the string and enclosing an area S can be easily computed to be e j> Asoi • dl = e

Bsoi • ds = eg.

Here, e = q/h is the electromagnetic coupling constant. This phase, and therefore the Dirac string, is unobservable provided eg = 2im, which is again the Dirac charge quantization condition.

5

't Hooft-Polyakov Monopoles

The previous discussion of magnetic monopoles was not compelling, because ordinary electrodynamic does not require that monopoles exist. Electrodynamic without monopoles is perfectly consistent. However, in certain gauge theories the spontaneous symmetry breaking is intimately connected with the existence of monopoles solutions. It can be shown that pure gauge theory does not, by itself, possesses any static nonsingular monopole configurations. However, a more general case, such as gauge theory coupled to scalar fields, does posses singular monopole solutions. We will be interested here by the 't Hooft-Polyakov monopole which is more likely to exist because it is topologically possible if the group space is that of 50(3) which is doubly connected (recall that SU(2) has no solitons in 2 + 1 dimensions because its group space is simply connected) and this group is important for some theories 3 + 1 dimensions. Furthermore, the magnetic charge comes from a theory with only electrical charges at the outset. The magnetical ones arise topologically. More, this monopol solution has been shown by 't Hooft to be energy finite indeed.

On the Soliton-Particle Dualities

123

The Lagrangian one uses and which leads to a non-Abelian theory with an 0(3) symmetry that will be broken is as follows:

c = -l-F;vF^a + i(DM0a)(D"0°) - ~rr - Krrf

(23)

where the gauge field F is F^u=dllAav-dvA^+e^bcAblAcu

(24)

a is the group index and the isovector Higgs field <pa comes with the covariant derivative {Dtir)

= d^a

+ eeabcAb^c.

(25)

The equation of motion following from this is (I>M(£>'>a)) + (m 2 + A\b(j>b)cj)a = 0

(26)

which allow for solutions 33 : limr-+

xb

Aa = Ab = 0

limr-^ •oo
-6m 2 xa

(27)

=

A

r

This is the 'tHooft-Polyakov monopole. To compare this monopole, defined for 0(3) symmetry, with the usual Dirac monopole, we will have to define a new Maxwell tensor F^ that will reduce to the usual one when a(dltb){dl/a) ^

= ^

a

A l

(28)

with this definition, we can now calculate the magnetic and electric charge of the monopole. We find that A^ — 0 and that 1

F0i = 0 , Fij =

fc

3 eijkxk

,Bk = ^

(29)

Here we have made a nontrivial linkage between three-dimensional physical space and three dimensional isospin space.

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so we have a radial magnetic field just like the Dirac monopole only that the magnetic flux is $ = ~. This is twice as much as the Dirac quantization condition gives as the minimum. This follows from the fact that the electromagnetic U{\) is embedded in SO(3) in such a way that the electric charge is the eigenvalue of the T3 isospin generator, which here is in the adjoint representation, which has integral isospin. The minimum electric charge is therefore eTO;„ = | e , relative to which the charge of the 't Hooft-Polyakov monopole is indeed one Dirac charge, again up to a sign. In summary, we see that the 't Hooft-Polyakov solution describes an object of finite size which from far away cannot be distinguished from a Dirac monopole of charge =$£-. In contrast with the Dirac monopole, the 't HooftPolyakov monopole is everywhere smooth—this being due to the massive fields which become relevant as we approach the "core" of the monopole. To reveal the topological nature of these monopole solutions, we remark that the sole contribution to F^v comes from the Higgs sector, since A^ = 0. A magnetic current K^ can easily be defined in such a way that d^K^ = 0 without it being a Noether current[19]. The topological nature of this current can be seen again by expressing the magnetic charge by an integration Ma j K d x. It turns out that this can be rewritten as in integration over the sphere S2 at infinity — that is the boundary of coordinate space. It follows that M = -; (30) e n is an integer, the so-called "Brouwer degree" due to the fact that the single valued vector <j> will be covered an integral number of times while the integration covers S2 once. Again we meet the concept of a mapping of group space — here the vacuum manifold S2 after symmetry breaking describing the orientations of isovector cj>a — onto the boundary of the coordinate space. However the non-Abelian electroweak group is given by the Weinberg-Salam model and is SU(2) x U(l) which does not have the topology of SO(3). 't Hooft-Polyakov monopoles exist in case that iT2(G/H) is non trivial, where G denotes the gauge group and H the unbroken subgroup (here always U(l)). Therefore the Weinberg-Salam model is monopole free. They may exist in GUTs where SU(2) x £/(!) is embedded in SU(5) for

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On the Soliton- Particle Dualities

example. Magnetic monopoles and even dyons which are objects with magnetic and electric charge, have been found in a variety of models.

5.1 The BPS states We discuss the BPS states from 't Hooft and Polyakov point of view using the Hamiltonian picture. As we saw 't Hooft and Polyakov found monopole solutions, corresponding to the maps S2 —> S2 m adjoint SU(2) Higgs theory. The adjoint Higgs breaks the symmetry SU(2) —> U(l). The leftover f/(l) gauge symmetry corresponds to electromagnetism and the monopoles are magnetically charged under this gauge group. The appearance of monopoles is a general feature of symmetry breaking in which factors are left unbroken. The SU(2) Higgs theory, with time independent fields has the following Hamiltonian: J

8

4

2

The ground state solution corresponds to A = 0, and 4> = £0 , where £0 satisfying34 Tr(£ = 2rf. To construct the soliton state at large distances from the origin consider a fixed vacuum solution 35 , £0 = Wz and the map U(6,
— ei

(32)

and construct 36 UtoU* = 7]-. (33) r Now for any finite energy solution when one continues this asymptotic form an r-dependent factor vanishing at r = 0 must be introduced. Thus, the ansatz for the Higgs field should be 4> = V°f(r) and using the transverse gauge, <j> can be chosen as

(34) -^ja(r).

The boundary conditions for the radial wave functions are that a and f(r) vanish at r = 0 while /zm r _,oo/(r) = 1.

Here we chose the vacuum expectation value of the Higgs field to be 17. <7j is an element of the basis in the isovector space. Where C is a function which playing no role in our analysis.

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Of course, we can show that these forms for the fields are consistent with the original field equations 37 . One can substitute the ansatz into the Hamiltonian to get

« = ^/ d r r ' [ ^ + ^ ( 2 a ^ ) 2 + ^ / , 2 + Z ? ( 1 - a ) 2 + 1 ! ( / 2 - 1 ) 2 1 (35) The asymptotic behavior of a is obvious from the last but one term of the Hamiltonian. Unless lim a = 1 the energy is not finite. For a general monopole solution one would have the topological charge, n on the right hand side. The field equations can only be solved numerically, except in the case when X —* 0 . Though the lim f(r) = 1 r—»oo

limit is not required, one can impose it and get the analytic solution of Bogomol'nyi, Prasad, and Sommerfeld (BPS).

6 Duality of Dualities In 1990 several theorists generalized the idea of Montonen-Olive duality to four-dimensional superstrings, in whose realm the idea becomes even more natural. This duality, which nonetheless remained speculative, goes by the name of S-duality. In fact, string theorists had already become used to a totally different kind of duality called T-duality. T-duality relates two kinds of particles that arise when a string loops around a compact dimension. One kind (call them "vibrating" particles) is analogous to those predicted by Kaluza and Klein and comes from vibrations of the loop of string. Such particles are more energetic if the circle is small. In addition, the string can wind many times around the circle, a rubber band on a wrist; its energy becomes higher the more times it wraps around and the larger the circle. Moreover, each energy level represents a new particle (call them "winding" particles). T-duality states that the winding particles for a circle of radius R are the same as the "vibration" particles for a circle of radius 37

Note the particular role of the point x = 0 . At this point the scalar field vanishes. Also, at this point, and only at this point the gauge transformation U becomes singular. The singularity must be present, because the symmetry of the system changes. While the gauge rotation symmetry is U(l) in every other point, at this point the complete St/(2)-symmetry is restored.

On the Soliton-Particle Dualities

127

1/R, and vice versa. To a physicist, the two sets of particles are indistinguishable. This duality has a profound implication. For decades, physicists have been struggling to understand nature at the extremely small scales near the Planck length of 10~ 33 centimeter. We have always supposed that laws of nature, as we know them, break down at smaller distances. What T-duality suggests, however, is that at these scales, the universe looks just the same as it does at large scales. One may even imagine that if the universe were to shrink to less than the Planck length, it would transform into a dual universe that grows bigger as the original one collapses. Duality between strings and fivebranes still remained conjectural, however, because of the problem of quantizing five-branes. The solution of the problem was by sidestepping it. If four of the 10 dimensions curl up and the five-brane wraps around these, the latter ends up as a one-dimensional object — a (solitonic) string in six-dimensional space-time. In addition, a fundamental string in 10 dimensions remains fundamental even in six dimensions. So the concept of duality between strings and five-branes gave way to another conjecture, duality between a solitonic and a fundamental string. The advantage is that we do know how to quantize a string. Hence, the predictions of string-string duality could be put to the test. One can show, for instance, that the strength with which the solitonic strings interact is given by the inverse of the fundamental string's interaction strength, in complete agreement with the conjecture. In 1994 Christopher M. Hull along with Townsend[12], suggested that a weakly interacting heterotic string can even be the dual of a strongly interacting Type IIA string, if both are in six dimensions. The barriers between the different string theories were beginning to crumble. It occurred that string-string duality has another unexpected payoff. If we reduce the six-dimensional spacetime to four dimensions, by curling lip two dimensions, the fundamental string and the solitonic string each acquire a T-duality. But here is the miracle: the T-duality of the solitonic string is just like S-duality of the fundamental string, and vice versa. This phenomenon — in which the interchange of charges in one picture is just the inversion of length in the dual picture — is called the Duality of Dualities. It places the previously speculative S-duality on just as firm a footing as the well established T-duality. In addition, it predicts that the strength with which objects interact is related to the size of the invisible dimensions. What is charge in one universe may be size in another.

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References [1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform SIAM, Philadelphia, (1981). [2] K. H. Ahn, M. Vaziri, B. C. Barnett, G. R. Williams, X. D. Cao, M. N. Islam, B. Malo, K. O. Hill, and D. Q. Chowdhury, J. Lightwave Technol. LT14, 1768 (1996). [3] L. de Broglie: Comp. Rend. 77, 506, 548, 630 (1923). [4] J. Boussinesq, Theorie de I 'intumescence liquide appelee onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus Academie des Sciences Paris 72: 755-759; 73: 256-260; 73: 1210-1212, (1871). [5] Coleman, S., Phys. Rev. D 11, 2088 (1975). [6] Dirac, P. A. M. Proc. Roy. Soc. London A133, 60 (1931). [7] Dirac, P.A.M., Phys. Rev. 74, 817 (1948). [8] S. G. Evangelides, L. F. MoUenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. LT-10, 28 (1992). [9] D. Freed, Five lectures on supersymmetry, A.M.S. (1999). [10] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). [11] G. 't Hooft, Nucl. Phys. B79, 276 (1974) or A.M.Polyakov, JET P Lett. 20, 194 (1974). [12] C. Hull and P. Townsend, Nucl. Phys. B451, 525 (1995); [hep-th/9505073]. [13] D.J.Korteweg and G. de Vries, Phil Mag. 39, 422 (1895). [14] C. R. Menyuk, IEEE J. Quantum Electron. QE23, 174 (1987). [15] L. F. MoUenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980). [16] C. Montonen and D. Olive, Phys. Lett. B72, 117 (1977). [17] H.B. Nielsen and P.Olesen, Nucl. Phys. B 6 1 , 45 (1973). [18] R.Rajaraman:5o/iions and Instantons, North-Holland Publishing Company, Amsterdam (1982). [19] L.H.Ryder: Quantum Field Theory, Cambridge University Press, Chapter 10, Page 391 (1985). [20] N. Seiberg and E. Witten, Nucl. Phys. B426, 19 (1994); [hep-th/9407087]. [21] Skyrme, T. H. R., Proc. Roy. Soc. A 247, 260 (1958). [22] Thirring, W., Ann. Phys. (N.Y.) 3, 91 (1958). [23] P. Townsend, Phys. Lett. 350B, 184 (1995); [hep-th/9501068]. [24] E. Witten, Nucl. Phys. B443, 85 (1995); [hep-th/9503124]. [25] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) k. Sov. Phys. J E T P 34, 62 (1972). [26] A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B379, 602 (1992). [27] Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844 (London 1845), pp 311-390, Plates XLVII-LVII. [28] Studies of Non Linear Problems in: Collected Papers of Enrico Fermi, general editor E. Segre, University of Chicago Press, II, 978 (1965).

Partn

Knots in Mathematics and Nature

"By doing mathematical physics on topological structures, all sorts of new invariants and a new way of thinking about topology has emerged. (...) The knot or link as an embedding in three dimensional space is a whole topological form, and it has long been the aim of topology to treat this form all at once." Louis H. Kauffman, Knots and Physics, 1997.

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KNOTS LOUIS H. KAUFFMAN Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045 U10451 @ UICVM.BITNET

Abstract: This paper is an introduction to the landscape of knot theory and its relationships with statistical mechanics, quantum theory and quantum field theory. The paper is a self-contained introduction to these topics. CONTENTS I. Preface II. Knot Tying and the Reidemeister Moves IE. Invariants of Knots and Links - A First Pass IV. The Jones Polynomial V. The Bracket State Sum VI. Vassiliev Invariants VII. Vassiliev Invariants and Lie Algebras VIII. A Quick Review of Quantum Mechanics IX. Knot Amplitudes X. Topological Quantum Field Theory - First Steps

131 132 138 150 155 166 171 178 184 192

I. Preface This essay constitutes an introduction to the theory of knots as it has been influenced by developments concurrent with the discovery of the Jones polynomial in 1984 and the subsequent explosion of research that followed this signal event in the mathematics of the twentieth century. I hope to give the flavor of these extraordinary events in this exposition. Even the act of tying a shoelace can become an adventure. The familiar world of string, rope and the third dimension becomes an inexhaustible source of ideas and phenomena. As indicated by the table of contents, Sections 2 and 3 constitute a start on the subject of knots. Later sections introduce more technical topics. The theme of a relationship of knots with physics begins already with the Jones polynomial and the bracket model for the Jones polynomial as discussed in Section 5. Sections 6 and 7 provide an introduction to Vassiliev invariants and the remarkable relationship between Lie algebras and knot theory. The idea for the bracket model and its generalizations is to regard the knot itself as a discrete physical system - obtaining information about its topology by averaging over the states of the system. In the case of the bracket model this summation is finite and purely combinatorial. Transpositions of this idea occur throughout, involving ideas from quantum mechanics (Sections 8 131

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and 9) and quantum field theory (Section 10). In this way knots have become a testing ground not only for topological ideas but also for the methods of modern theoretical physics. This essay concentrates on the construction of invariants of knots and the relationships of these invariants to other mathematics (such as Lie algebras) and to physical ideas (quantum mechanics and quantum field theory). There is also a rich vein of knot theory that considers a knot as a physical object in three-dimensional space. Then one can put electrical charge on the knot and watch (in a computer) the knot repel itself to form beautiful shapes in three dimensions. Or one can think of the knot as made of thick rope and ask for an "ideal" form of the knot with minimal length to diameter ratio. There are many aspects to this idea of physical knots. It is a current topic of my own research and the research of many others. I wish that there had been space in this essay to cover these matters. That will have to wait for the next time! In the meantime, it gives me great pleasure to thank Vaughan Jones, Edward Witten, Nicolai Reshetikhin, Mario Rasetti, Sostenes Lins, Massimo Ferri, Lee Smolin, Louis Crane, David Yetter, Ray Lickorish, DeWitt Sumners, Hugh Morton, Joan Birman, John Conway, John Simon and Dennis Roseman for many conversations related to the topics of this paper. This research was partially supported by the National Science Foundation Grant DMS-2528707. II. Knot Tying and the Reidemeister Moves For this section it is recommended that the reader obtain a length of soft rope for the sake of direct experimentation. Let's begin by making some knots. In particular, we shall take a look at the bowline, a most useful knot. The bowline is widely used by persons who need to tie a horse to a post or their boat to a dock. It is easy and quick to make, holds exceedingly well and can be undone in a jiffy. Figure 1 gives instructions for making the bowline. In showing the bowline we have drawn it loosely. To use it, grab the lower loop and pull it tight by the upper line shown in the drawing. You will find that it tightens while maintaining the given size of the loop. Nevertheless, the knot is easily undone, as some experimentation will show. The utility of a schema for drawing a knot is that the schema does not have to indicate all the physical properties of the knot. It is sufficient that the schema should contain the information needed to build the knot. Here is a remarkable use of language. The language of the diagrams for knots implicitly contains all their topological and physical properties, but this information may not be easily available unless the "word is made flesh" in the sense of actually building the knot from rope or cord. Our aim is to get topological information about knots from their diagrams. Topological information is information about a knot that does not depend upon the material from which it is made and is not changed by stretching or bending that material so long as it is not torn in the process. We do not want the knot to disappear in the course of such a stretching process by slipping over one of the ends of the rope.

Knots

133

The knot theorist's usual convention for preventing this is to assume that the knot is formed in a closed loop of string. The trefoil knot shown in Figure 2 is an example of such a closed knotted loop.

Figure I. The bowline

Figure 2. The Trefoil as closed loop

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L. H. Kauffman

A knot presented in closed loop form is a robust object, capable of being pushed and twisted into many topologically equivalent forms. For example, the knot shown in Figure 3 is topologically equivalent to the trefoil shown in Figure 2. The existence of innumerable versions of a given knot or link gives rise to a mathematical problem. To state that a loop is knotted is to state that nowhere among the infinity of forms that it can take do we find an unknotted loop. Two loops are said to be (topologically) equivalent if it is possible to deform one smoothly into the other so that all the intermediate stages are loops without self intersections. In this sense a loop is knotted if it is not equivalent to a simple flat loop in the plane. The key result that makes it possible to begin a (combinatorial) theory of knots is the Theorem of Reidemeister [REI] that states that two diagrams represent equivalent loops if and only if one diagram can be obtained from the other by a finite sequence of special deformations called the Reidemeister moves. I shall illustrate these moves in a moment. The upshot of Reidemeister's Theorem is that the topological problems about knots can all be formulated in terms of knot diagrams.

Figure 3. Deformed Trefoil There is a famous philosophy of mathematics called "formalism", in which mathematics is considered to be a game played with symbols according to specific rules. Knot theory, done with diagrams, illustrates the formalist idea very well. In the formalist point of view a specific mathematical game (formal system) can itself be an

Knots

135

object of study for the mathematician. Each particular game may act as a coordinate system, illuminating key aspects of the subject. One can think about knots through the model of the diagrams. Other models (such as regarding the knots as specific kind of embeddings in three dimensional space) are equally useful in other contexts. As we shall see, the diagrams are amazingly useful, allowing us to pivot from knots to other ideas and fields and then back to topology again. The Reidemeister moves are illustrated in Figure 4.

Figure 4. Reidemeister moves The moves shown in Figure 4 are intended to indicate changes that are made in a larger diagram. These changes modify the diagram only locally as shown in the Figure. Figure 5 shows a sequence of Reidemeister moves from one diagram for a trefoil knot to another. In this illustration we have performed two instances of the second Reidemeister move in the first step, a combination of the second move and the third move in the second step and we have used "move zero" (a topological rearrangement that does not change any of the crossing patterns) in the last step. Move zero is as important as the other Reidemeister moves, but since it does not change any essential diagrammatic relationships it is left in the background of the discussion.

36

L. H. Kauffman

Figure 5. A sequence of Reidemeister moves from one diagram for a trefoil knot to another

Knots

137

Knots as Analog Computers We end this section with one more illustration. This time we take the bowline and close it into a loop. A deformation then reveals that the closed loop form of the bowline is topologically equivalent to two trefoils clasping one another, as shown in Figure 6.

Figure 6. The closed loop form of the bowline is topologically equivalent to two trefoils clasping one another

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This deformation was discovered by making a bowline in a length of rope, closing it into a loop and fooling about with the rope until the nice pair of clasped trefoils appeared. Note that there is more than one way to close the bowline into a loop. Figure 6 illustrates one choice. After discovering them, it took some time to find a clear pictorial pathway from the closed loop bowline to the clasped trefoils. The pictorial pathway shown in Figure 6 can be easily expanded to a full sequence of Reidemeister moves. In this way the model of the the knot in real rope is an analog computer that can help to find sequences of deformations that would otherwise be overlooked. It is a curious reversal of roles that the original physical object of study becomes a computational aid for getting insight into the mathematics. Of course this is really a two way street. The very close fit between the mathematical model for knots and the topological properties of actual knotted rope is the key ingredient. Knots are analogous to integers. Just as we believe that objects follow the laws of arithmetic, we believe that the topological properties of knotted rope follow the laws of knot topology. III. Invariants of Knots and Links - A First Pass We want to be able to calculate numbers (or bits of algebra such as polynomials) from given link diagrams in such a way that these numbers do not change when the diagrams are changed by Reidemeister moves. Numbers or polynomials of this kind are called invariants of the knot or link represented by the diagram. If we produce such invariants, then we are finding topological information about the knot or link. The easiest example of such an invariant is the linking number of two curves, which measures how many times one curve winds around another. In order to calculate the linking number we orient the curves. This means that each curve is equipped with a directional arrow, and we keep track of the direction of the arrow when the curve is deformed by the Reidemeister moves. If the curves A and B are represented by an oriented link diagram with two components, attach a sign (+1 or -1) to each crossing as in Figure 7. Then the linking number, Lk (A,B), is the sum of these signs over all the crossings of A with B. Of course, two singly linked rings receive linking number equal to +1 or -1 as shown in Figure 8.

Figure 7. Crossing signs

Knots

Lk = - 1

139

Lk = +1

Figure 8. The linking number of two linked rings

It can be shown that the linking number is invariant under the Reidemeister moves. That is, if we take a given diagram D (representing the curves A and B) and change it to a new diagram E by applying one of the Reidemeister moves, then the linking number calculation for D will be the same as the calculation for E. The calculation is unaffected by the first Reidemeister move because self-crossings of a single curve do not figure in the calculation of the linking number. The second Reidemeister move either creates or destroys two crossings of opposite sign, and the third move rearranges a configuration of crossing without changing their signs. With these observations we have in fact proved that the singly linked rings are indeed linked! There is no possible sequence of Reidemeister moves from these rings to two separated rings because the linking number of separated rings is equal to zero, not to plus or minus one. It may seem a minor accomplishment to prove something as obvious as the inseparability of this simple configuration, but it is the first step in the successful application of algebraic topology to the study of knots and links. The linking number has a long and interesting history, and there are a number of ways to define it, many considerably more complicated than the sum of diagrammatic signs. We shall discuss some of these alternative definitions at the end of this section. One of the most fascinating aspects of the linking number is its limitations as an invariant. Figure 9 shows the Whitehead link, a link of two components with linking number equal to zero. The Whitehead link is indeed linked, but it requires methods more powerful than the linking number to demonstrate this fact. Another example of this sort is the Borromean (or Ballantine) rings as shown in Figure 10. These three rings are topologically inseparable, but if any one of them is ignored, then the other two are not linked.

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Figure 9. The Whitehead link

Figure 10. The Borromean rings

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Just in case these last few examples leave you pessimistic about the prospects of the linking number, here is a positive application. We shall use the linking number to show that the Mobius strip is not topologically equivalent to its mirror image. The Mobius strip is a circular band with a half twist in it as illustrated in Figure 11. The Mobius is a justly famous example of a surface with only one side and one edge. An observer walking along the surface goes through the half-twist and arrives back where she started only to discover that she is on the other local side of the band! It requires another trip around the band to return to the original local side. As a result there is only one side to the surface in the global sense. It is as though the opposite side of the world were infinitesimally close to us by drilling into the ground, but a full circumnavigation of the globe away by external travel. To make matters even more surprising, there are actually two Mobius bands depending on the sense of the half twist. Call them M and M* as illustrated in Figure 11.

Figure 11. Mobius and mirror Mobius

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If you make these two Mobius bands from strips of paper and try to deform one into the other without tearing the paper, you will fail (Try it!). How can we understand the topological nature of the handedness of the Mobius band M? Draw a curve C down the center of the band M as shown in Figure 11. Compare this curve with the space curve formed by the boundary of the band. Orient these curves in parallel and compute the linking number. It is +1. The very same calculation for the mirror image band M* yields the linking number o f - 1 . If it were possible topologically to deform M to M* then the corresponding links (formed by the core curve and the boundary curve of the band) would be topologically equivalent, and hence they would have the same linking number. Since this is not the case, we conclude that M cannot be deformed to M*. We have shown that there are two topologically distinct Mobius bands. The two bands are mirror images of one another in the sense that each looks like the image of the other in a reflecting mirror. When an object is topologically inequivalent to its mirror image, it is said to be chiral. We have demonstrated the chirality of the Mobius band. Three Coloring a Knot There is a remarkable proof that the trefoil knot is knotted. This proof goes as follows: Color the three arcs of the trefoil diagram with three distinct colors. Let's say these colors are red, blue and purple. Note that in the standard trefoil diagram three distinct colors occur at each crossing.

Figure 12. The three colored Trefoil Now adopt the following coloring rule: Either three colors or exactly one color occur at any crossing in the colored diagram.

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Call a diagram colored if its arcs are colored and they satisfy this rule. Note that the standard unknot diagram is colored by simply assigning one color to its circle. A coloring does not necessarily have three colors on a given diagram. Call a diagram 3-colored if it is colored and three colors actually appear on the diagram. Theorem. Every diagram that is obtained from the standard trefoil diagram by Reidemeister moves can be 3-colored. Hence the trefoil diagram represents a knot.

Figure 13. Inheriting coloring under the type two move

Rather than write a formal proof of this Theorem, we illustrate the coloring process in Figures 13 and 14. Each time a Reidemeister move is performed, it is possible to extend the coloring from the original diagram to the diagram that is obtained from the move. These extensions of colorings involve only local changes in the colorings of the original diagrams. The best way to see that this proof works is to do a few experiments yourself. The Figures 13 and 14 should get you started!

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Figure 14. Coloring under type two and three moves

Note that in the case of the second move performed in the simplifying direction, although a color is lost in the arc that disappears under the move, this color must appear elsewhere in the diagram or else it is not possible for the two arcs in the move to have different colors (since there is a path along the knot from one local arc to the other). Thus 3-coloration is preserved under Reidemeister moves, whether they make the diagram simpler or more complicated. As a result, every diagram for the trefoil knot can be colored with three colors according to our rules. This proves that the trefoil is knotted, since an unknotted trefoil would have a simple circle among its diagrams, and the simple circle can be colored with only one color.

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The Quandle and the Determinant of a Knot There is a wide generalization of this coloring argument. We shall replace the colors by arbitrary labels for the arcs in the diagram and replace the coloring rule by a method for combining these labels. It turns out that a good way to articulate such a rule of combination is to make the label on one of the undercrossing arcs at a crossing a product (in the sense of this new mode of combination) of the labels of the other two arcs. In fact, we shall assume that this product operation depends upon the orientation of the arcs as shown in Figure 15.

Figure 15. The Quandle operation In Figure 15 we show how a label a on an undercrossing arc combines with a label b on an overcrossing arc to form c=a*b or c=a#b depending upon whether the overcrossing arc is oriented to the left or to the right for an observer facing the overcrossing line and standing on the arc labelled a. This operation depends upon the orientation of the line labelled b so that a*b corresponds to b pointing to the right for an observer approaching the crossing along a, and a#b corresponds to b pointing to the left for the same observer. All of this is illustrated in Figure 15. The binary operations * and # are not necessarily associative. For example, our original color assignments of R (red), B (blue) and P (purple) for the trefoil knot correspond to products R*R=R, B*B=B, P*P=P, R*B=P, B*P=R, P*R=B. Then R*(B*P)=R*R=R while (R*B)*P=P*P=P. We shall insist that these operations satisfy a number of identities so that the labeling is compatible with the Reidemeister moves. In Figure 16 I have illustrated the diagrammatic justification for the following algebraic rules about * and #. An algebraic system satisfying these rules is called a quandle [JOY]. 1. a* a = a and a#a = a for any label a. 2. (a*b)#b = a and (a#b)*b = a for any labels a and b. 3. (a*b)*c = (a*c)*(b*c) and (a#b)#c = (a#c)#(b#c) for any labels a,b,c.

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These rules correspond, respectively to the Reidemeister moves 1, 2 and 3. Labelings that obey these rules can be handled just like the 3-coloring that we have already studied. In particular a given labeling of a knot diagram means that it is possible to label (satisfying the rules given above for the labels) any diagram that is related to it by a sequence of Reidemeister moves. However, not all the labels will necessarily appear on every related diagram, and for a given coloring scheme and a given knot, certain special restrictions can arise.

A*A=A

and

A*A=A

(A*B)*B

(A*B)*B = A

and

(A*B)*B = A

(A*B)*

O* (B*C) and

[A*B) *C = (A*C) *(B*C)

Figure 16. Quandle identities

To illustrate this, consider the color rule for numbers: a*b = a#b = 2b-a. This satisfies the axioms as is easy to see. Figure 17 shows how, on the trefoil, such a coloring must obey the equations a*b=c, c*a=b,b*c=a. Hence 2b-a=c, 2a-c=b, 2c-b = a. For example, if a=0 and b=l, then c=2b-a = 2 and a = 2c-b = 4-1 = 3. We need 3 = 0. Hence this system of equations will be satisfied for appropriate labelings in Z/3Z, the integers modulo three, a modular number system.

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For the reader unfamiliar with the concept of modular number system, consider a standard clock whose dial is labeled with the hours 1,2,3,..., 11,12. We ask what time is it 4 hours past the hour of 10? The answer is 2, and one can say that in the arithmetic of this clock 10+4=2. In fact 12=0 in this arithmetic because adding 12 hours to the time does not change the time indicated on the clock. We work in clock arithmetic by remembering to set blocks of 12 hours to zero. One can multiply in this arithmetic as well. The square of the present time is 1 o'clock, what time is it? The answer is 7 since 7 squared is 49 and 49 is equal to 1 on the clock. We say that the clock represents a modular number system Z/12Z with modulus 12. It is convenient in mathematics to think of the elements of Z/12Z as the set {0,1,2,...,11}. Since 0=12 this takes care of all the hours. In general we can consider Z/nZ where n is any positive integer modulus. The resulting modular number system has elements {0,1,2,...,n-l} and is handled just as though there were a clock with n hours rather than 12. In such a system one says that x = y (mod n) if the difference between x and y is divisible by n. For example 49 =1 (mod 12) since 49-1=48 is divisible by 12.

Figure J 7. Equations for the Trefoil knot

The modular number system, Z/3Z, reproduces exactly the three coloring of the trefoil, and we see that the number 3 emerges as a characteristic of the equations associated with the knot. In fact, 3 is the value of a determinant that is associated with these equations, and its absolute value is an invariant of the knot. For more about this construction, see [K10, Part 1, Chapter 13].

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Here is another example: For the figure eight knot E, we have that the modulus is 5. This shows that E is indeed knotted and that it is distinct from the trefoil. We can color (label) the figure eight knot with five "colors" 0,1,2,3,4 with the rules: a*b = 2b-a (mod 5). See Figure 18.

1 =4

Labelling the Figure Eight Knot from the Integers Modulo Five Figure 18. Five colors for the figure eight knot

Note that in coloring the figure eight knot we have only used four out of the five available "colors" from the set {0, 1, 2, 3, 4}. Figure 18 uses the colors 0, 1, 2 and 4. In [KH] we define the coloring number of a knot or link K to be the least number of colors (greater than 1) needed to color it in the 2b-a fashion for any diagram of K. It is a nice exercise to verify that the coloring number of the figure eight knot is indeed four. In general the coloring number of knot or link is not easy to determine. This is an example of a topological invariant that has subtle combinatorial properties. Other knots and links that we have mentioned in this section can be shown to be knotted and linked by the modular method. The reader should try it for the Borommean rings and the Whitehead link. The coloring (labeling) rules as we have formalized them can be described as axioms for an algebra associated with the knot. This is called the quandle [JOY]. It has been generalized to the crystal [K10], the interlock algebra [K15], and the rack [FR]. The quandle is itself a generalization of the fundamental group of the knot complement [CF].

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The Alexander Polynomial The modular labeling method has a marvelous generalization to the Alexander polynomial of the knot. This comes about through generalized coloring rules a*b =ta + (l-t)b and a#b=t"^a + (l-t'l)b, where t is an indeterminate. It is a nice exercise to verify that these rules satisfy the axioms for the quandle. This algebraic structure is called the Alexander Module. The case t=-l gives the rule 2b-a that we have already considered. By coloring diagrams with arbitrary t, we obtain a polynomial that generalizes the modulus. This polynomial is the Alexander polynomial. Alexander [AL] described it differently in his original paper, and there is a remarkable history to the development of this invariant. See [CF], [FOX], [CON], [Kl], [K2], [K4] for more information. The flavor of this relationship can be seen by doing a little experiment in labeling the trefoil diagram shown in Figure 19. The circularity inherent in the knot diagram results in relations that must be satisfied by the module action. In Figure 19 we see directly by labeling the diagram that if arc 1 is labeled 0 and arc 2 is labeled a, then (t + (1-t) 2 a = 0. In fact, t + (1 - 1 ) 2 = t 2 - 1 +1 is the Alexander polynomial of the trefoil knot. The Alexander polynomial is an algebraic modulus for the knot.

Figure 19. Alexander polynomial of the Trefoil knot

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IV. The Jones Polynomial Our next topic describes an invariant of knots and links of quite a different character than the modulus or the Alexander polynomial of the knot. It is a "polynomial" invariant of knots and links discovered by Vaughan Jones in 1984 [J02]. Jones' invariant, usually denoted VK(t), is a polynomial in the variable t " 2 and its inverse t~l' 2 . One says that VK(t) is a Laurent polynomial in tl/2. Superficially, the Jones polynomial appears to be just another polynomial invariant of knots and links, somewhat similar to the Alexander polynomial. When I say that the Jones polynomial is of a different character, I mean something deeper, and it will take a little while to explain this difference. A little history will help. The Alexander polynomial was discovered in the 1920's and until 1984 no one had found another polynomial invariant of knots and links that was not a simple generalization of the Alexander polynomial. Vaughan Jones discovered a new polynomial invariant of knots and links that had some very remarkable properties. The Alexander polynomial cannot detect the difference between any knot and its mirror image. What made the Jones polynomial such an exciting discovery for knot theorists was the fact that it could detect the difference between many knots and their mirror images. Later other properties began to emerge. It became a key tool in proving properties of alternating links (and generalizations) that had been conjectured since the last century [K3], [MUR1], [MUR2], [TH], [MT]. It turns out the the Jones polynomial is intimately related to a number of topics in mathematical physics. Curiously, it is actually easier to define and verify the properties of the Jones polynomial than for any other invariant in the theory of knots (except of course the linking number). We shall devote this section to the defining properties of the Jones polynomial, and later sections to the relationships with physics. Here are a set of axioms for the Jones polynomial. The polynomial was not discovered in the form of these axioms. The axioms are in a format analogous to the framework that John H. Conway [CON], [Kl], [K2], discovered for the Alexander polynomial. I am starting with these axioms because they give a quick access to the polynomial and to sample computations. Axioms for the Jones Polynomial 1. If two oriented links K and K' are ambient isotopic, then VK(t)=VK'(t). 2. If U is an unknotted loop, then VU(t) = 1. 3. If K+, K-, and K0 are three links with diagrams that differ only as shown in the neighborhood of a single crossing site for K+ and K- (see Figure 20), then t"1 VK+(t) - tVK-(t) = ( t 1 / 2 - r 1 / 2 ) V K 0 ( t ) .

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KO Figure 20. Links with diagrams

The axioms for VK(t) are a consequence of Jones' original definition of his invariant. He was led to this invariant by a trail that began with the study of von Neumann algebras [JOl] (a branch of algebra directly related to quantum theory and to statistical mechanics) and ended in braids, knots and links. The Jones polynomial has a distinctly different flavor from the Conway-Alexander polynomial even though it can be axiomatised in a very similar way. In fact, this similarity of axiomatics points to a common generalization (the Homfly (Pt) polynomial) [F], [PT] and to another generalization (the Kauffman polynomial) [K9], and then to further generalizations in the connection with statistical mechanics [K8], [J05], [AW]. To this date no one has found a knotted loop that the Jones polynomial does not declare to be knotted. Thus one can make the Conjecture: If a single component loop K is knotted, then VK(t) is not equal to one. While it is possible that the Jones polynomial is able to detect the property of being knotted, it is not a complete classifier for knots. There are inequivalent pairs of knots that have the same Jones polynomial. Such a pair is shown in Figure 21. These two knots, the Kinoshita-Terasaka knot and the Conway knot, both have the same Jones polynomial but are different topologically. Incidentally these two knots are examples whose knottedness cannot be detected by the Alexander polynomial. Lets get to work and use the axioms to compute the Jones polynomial for the trefoil knot. To this end, there is a useful device called the skein tree. A skein tree is obtained from a given knot or link diagram by recording the knots and links obtained from this diagram by smoothing or switching crossings. Each node of the tree is a knot or link. The nodes farthest from the original knot or link are unknotted or unlinked. Such a tree can be produced from a given knot or link by using the fact that any knot or link diagram can be transformed into an unknotted (unlinked) diagram by a sequence of crossing switches. See Figure 22.

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KINOSHITA-TERASAKA KNOT

Figure 21. Conway knot and Kinoshita-Terasaka knot

Figure 22. A standard unknot

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In Figure 22 I have illustrated a "standard unknot diagram". This diagram is drawn by starting at the arrowhead in the Figure and tracing the diagram in such a way that one always draws an over crossing before drawing an undercrossing. This is the easiest possible knot diagram to draw since one never has to make any corrections just pass under when you want to cross an an already created line in the diagram. Standard unknot diagrams are always unknotted. Try your hand at the one in Figure 22 and you will see why this is so. Using the fact that standard unknot diagrams are available, we can use the difference between a given diagram K and a standard unknot with the same plane projection to give a procedure for switching crossings to unknot the diagram K. This switching procedure can be used to produce a skein tree for calculating the Jones polynomial of K. We have illustrated in Figure 23 a skein tree for the computation of the Jones polynomial of the trefoil knot. The tree reduces the calculation of the Jones polynomial of the trefoil diagram to the calculation of certain unknots and unlinks. In order to see how to calculate an unlink it is useful to observe the behaviour of the axioms in this case: t-i v u + - tVU- = (t 1/2 - r 1 / 2 )VU0. Here U+ and U- denote unknots with single positive and negative twists in them. UO, obtained by smoothing the crossing of U+ or U-, is an unlinked pair of circles with no twists. See Figure 24. Therefore ( r 1 - t)l

=(t1/2-r1/2)VU0.

Hence

d = vuo = ( r 1 - t)/(t 1/2 -r 1/2 ) = - ( t l / 2 + r 1 / 2 ). Thus we see that an extra unknotted component of the link multiplies the invariant by d =-(t1/2 + Vl/2). Here T denotes the trefoil knot, U denotes the unknot and L denotes the link of two unknotted circles as shown in Figure 23. With this fact in place, we find that t-lVT ]

t- VL

- tVU = ( t 1 / 2 - t " 1 / 2 ) V L - td =(t1/2-r1/2)VU

Thus VL = t( td + (tl/2 _ t-1/2)) = _t5/2 . t l/2. Hence VT = t(t + (t 1 / 2 - r 1 / 2 )VL) = t(t + (t 1 / 2 - r 1 / 2 ) ( - t 5 / 2 -1 1 / 2 )) = t (t -t 3 -t+t 2 +l) = t(-t 3 +t 2 +l) = -t4+t3+t.

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Figure 23. Trefoil skein tree The same calculation applied to the mirror image T* (obtained by reversing all the crossings of T) of the trefoil yields the invariant VT* = - t -4 + t-3 + H . This shows how the Jones polynomial discriminates between the trefoil and its mirror image, thereby proving that there is no ambient isotopy from T to T*. This method of calculating the Jones polynomial from its axioms does not tell us why the invariant works. It is possible to analyze this method of calculation and show that it does not depend upon the choices that one makes in the process and that it gives topological information about the knot or link in question. There is a different way to proceed that leads to a very nice formula for the Jones polynomial as a sum over "states" of the diagram. In this formulation, the polynomial is well defined from the beginning, and we can see the topological invariance arise in the course of adjusting certain parameters of a well-defined function. Our next topic is this state summation model for the Jones polynomial.

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u-

155

uo

Figure 24. Unknots with single positive and negative twists, and an unlinked pair of circle with no twists

V. The Bracket State Sum In the last section we gave axioms for the Jones polynomial and showed how to compute it by skein calculations from these axioms. In this section we shall show one way to prove that the Jones polynomial is well-defined by these axioms, and that it is an invariant of ambient isotopy of links in three dimensional space. In order to accomplish this aim, we shall give a different definition of the polynomial as a certain summation over combinatorial configurations associated with the given link diagram. This summation will be called a state summation model for the Jones polynomial. In fact, we shall first construct a state summation called the bracket polynomial [K3], and then explain how to modify the bracket polynomial to obtain the Jones polynomial. The bracket polynomial has a rather natural development, and is defined for unoriented link diagrams. We work with diagrams for unoriented knots and links. To each crossing in the diagram assign two local states with labels A and B, as shown in Figure 25. (In the A-state the regions swept out by a counterclockwise turn of the overcrossing line are joined. In the B-state the regions swept out by a clockwise turn of the overcrossing line are joined.) A state S of a diagram K consists in a choice of local state for each crossing of K. Thus a diagram with N crossings will have 2N states. Two states S and S' of the trefoil diagram are indicated in Figure 26. States are evaluated in two ways. These ways are denoted by and by HSU. The norm of the state S, IISII, is defined to be one less than the number of closed curves in the plane described by S. In the example in Figure 26, we have IISII=1 and IIS'll=0. The evaluation is defined to be the product of all the state labels (A and B) in the state. Thus in Figure 26 we have =A3 and =A2B.

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Figure 25. Local states of crossings

Figure 26. Two states of the trefoil diagram

Taking variables A,B and d, we define the state summation associated to a given diagram K by the formula = S dIIS11. In other words, for each state we take the product of the labels for that state multiplied by d raised to the number of loops in the state. is the summation of this state evaluation over all the states in the diagram for K. (The notation S means "sum over all instances of S".) We will show that the state summation is

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invariant under the second and third Reidemeister moves if we take B = 1/A and d = (A^ + B^). A normalization then enables us to obtain invariance under all three Reidemeister moves, and hence topological information about knots and links. (See [K3] for more information about the bracket and its relationship with the Jones polynomial.) There is a great deal of topological information in the calculations that ensue from the bracket polynomial. In particular, one can distinguish many knots from their mirror images, and it is possible that the bracket calculation can detect whether a given diagram is actually knotted. First Steps in Bracketology The first constructions related to the bracket polynomial are quite elementary. There are two basic formulas that are reminiscent of the exchange relations we have already seen for the Jones polynomial. These formulas are as shown in Figure 27.

Figure 27. Bracket equations

Here the small diagrams indicate parts of larger diagrams that are otherwise identical. Formula 1. Just says that the state summation breaks up into two sums with respect to a given crossing in the diagram. In one sum, we have made a smooothing of type A at the crossing, while in the other sum we have made a smoothing of type B. The factors of A and B indicated in the formula are the contributions to the product of vertex weights from this crossing. All the rest of the two partial sums can be interpreted as bracket evaluations of the smoothed diagrams. Formula 2. Just states that an extra simple closed curve in a diagram multiplies its bracket evaluation by the loop value d. Note that a single loop receives the value 1.

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With the help of these two formulas, we can compute some basic bracket evaluations. Note that we have not yet specialized the variables A,B and d. We shall analyze just what specialization will produce an invariant of knots and links. The advantage to having set up the definition of the bracket polynomial in this way is exactly that we have a method of labeling link diagrams with algebra, and it is possible to then adjust the evaluation so that it is invariant under Reidemeister moves. To this end, the next Lemma tells us how the general bracket behaves under a Reidemeister move of type two. Essential diagrams for this Lemma are in Figure 28. Lemma. Let K be a given link diagram, and let K' denote a diagram that is obtained from K by performing a type 2 Reidemeister move in the simplifying direction (eliminating two crossings from K). Let K " be the diagram obtained from K by replacing the site of the type two move by two arcs in the opposite pattern to the form of the simplified site in K'. (The diagrams in Figure 28 illustrate this construction.) Then = AB + (ABd + A 2 + B 2 ). Proof. Consider the four local state configurations that are obtained from the diagram K on the left hand side of the equation, as illustrated in Figure 28. The formula follows from the fact that one of these states has coefficient AB and the other three have the same underlying diagram and respective coefficients ABd (after converting the loop to a value d), A^ and B^. This completes the proof of the Lemma.// With the help of this Lemma it is now obvious that if we choose B=l/A and d + A2 + B2 = 0, then is invariant under the second Reidemeister move. Once this choice is made, the resulting specialized bracket is invariant under the third Reidemeister move, as illustrated in Figure 29. Finally, we can investigate bracket behaviour under the first Reidemeister move. Lemma. Let denote the bracket state sum with B=A~' and d = - A 2 -A - 2 - Then is invariant under the Reidemeister moves II and III and on move I behaves as shown below. < K(+) > = (-A3)< K > < K(-) > = (-A"3)< K >. Here K(+) denotes a diagram with a simplifying move of type I available where the crossing that is to be removed has type +1. K is the diagram obtained from K(+) by doing the type I move. Similarly, K(-) denotes a diagram with a simplifying move of type I available where the crossing that is to be removed has type - 1 . Figure 30 illustrates the diagrams for K(+) and K(-). Proof. See Figure 30 for the behaviour under type I moves. We have already verified the other statements in this Lemma.//

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Figure 28. Type two changes Framing Philosophy - Twist and Writhe Is it unfortunate that the bracket is not invariant under the first Reidemeister move? No, it is fortunate! First of all, the matter is easy to fix by a little adjustment: Let K be an oriented knot or link, and define the writhe of K, denoted w(K), to be the sum of the signs of all the crossings in K. Thus the writhe of the right-handed trefoil knot is three.

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Figure 29. Type III invariance of the bracket

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>

Figure 30. Bracket under type I move

The writhe has the following behaviour under Reidemeister moves: (i) w(K) is invariant under the second and third Reidemeister moves. (ii)w(K) changes by plus or minus one under the first Reidemeister move: w( K(+) ) = w(K) + 1 w(K(-)) = w(K) - 1. (Here we use the notation of the previous Lemma as shown in Figure 30.) Thus the writhe behaves in a parallel way to the bracket on the type I moves, and we can combine writhe and bracket to make a new calculation that is invariant under all three Reidemeister moves. We call the fully invariant calculation the "fpolynomial" and define it by the equation fK(A) = (-A 3 )" W ( K )(A). Up to this normalization, the bracket gives a model for the original Jones polynomial. The precise relationship is that VK(t) = fK(t"^4) w here w(K) is the sum of the crossing signs of the oriented link K, and is the bracket polynomial obtained by ignoring the orientation of K. We shall return to this relationship with the Jones polynomial in a moment, but first a little extra mathematical philosophy: Another way to view the fact of the bracket's lack of invariance under the first Reidemeister move is to see that the

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bracket is an invariant of knotted and linked bands embedded in three dimensional space. Regard a link diagram as shorthand for an embedding of bands as shown in Figure 31. In Figure 31 we have illustrated a link diagram ;for the trefoil knot in a thick dark mode of drawing. This diagram is juxtaposed with a drawing of a knotted band that parallels that knot diagram . The band has two boundary components that proceed (mostly in the plane) parallel to one another. The curl in the knot diagram becomes a

Figure 31. Bands and twists

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flat curl in the band that is ambient isotopic to a full twist ( two half twists) in the band. This isotopy is indicated in Figure 31. The top of Figure 31 shows a full twist in a band and two flat curls that both give rise to this same full twist by ambient isotopy that leaves their ends fixed. Each component of a link diagram is replaced by a paralleled version — the analog of a ribbbon-like strip of paper attached to itself with an even number of half-twists. The first Reidemeister move no longer applies to this shorthand since we can, at best, replace a curl by a twist as shown in Figure 31. In fact, as Figure 31 shows, there are two distinct curls corresponding to a single full twist of a band. The bracket (and the writhe) behave the same way on both of these twists. This means that we can re-interpret the bracket as an invariant of the topological embeddings of knotted, linked and twisted bands in three-dimensional space. This means that the bracket has a fully three-dimensional interpretation, although its definition depends upon the use of planar projections. Calculating the Bracket In Figure 32 we show a tree for calculating the bracket polynomial of the trefoil knotT. It follows at once from the behaviour of the bracket on curls that the contributions of the three (farthest from the trefoil itself) branches of this tree add to give the bracket polynomial of the trefoil: = A 2 (-A 3 ) + AA-!(-A- 3 ) + A ' ^ - A " 3 ) 2 = -A 5 -A^+A" 7 . Hence fT = (-A 3 )- W ( T ) = A" 4 + A" 1 2 - A" 16 . Note that we managed only three branches in the tree for this calculation rather than the full expansion into the eight states. A savings like this is always possible, because we know how the bracket behaves on curls. The resulting expansion gives a sum of monomials and is useful for thinking about the properties of the invariant. Mirror Mirror The knot K* obtained by reversing all the crossings of K is called the mirror image of K. K* is the mirror image of the knot that would ensue if the plane on which the knot is drawn were a mirror. It is easy to see that (A) = (A"1) and that fK*(A) = fK(A"l). Thus, if K is ambient isotopic to K* (all three Reidemeister moves allowed), then fK(A) = fK*(A) = fK(A-1). Returning to the evaluation of the f-invariant for the trefoil, note that fT(A"l) is not equal to fT(A). Therefore, the trefoil knot T and its mirror image T* are topologically distinct. The proof that we have given for it is the simplest proof known to this author. Note that we have given a complete proof of this fact, starting with the Reidemeister moves, constructing and applying the bracket invariant.

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Figure 32. Tree for bracket of Trefoil

A knot is said to be chiral if it is not ambient isotopic to its mirror image. The words chiral and chirality come from physical chemistry and natural science. A knot that is equivalent to its mirror image is said to be achiral. (Or amphicheiral in the speech of knot theorists). Many knots are achiral. The reader may enjoy verifying that the figure eight knot shown in Figure 33 is ambient isotopic to its mirror image.

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E

E*

Figure 33. Figure eight knot and its mirror image

A complete understanding of the problem of determining whether a knot is chiral remains in the far distance. The new invariants of knots and links have enhanced our understanding of this difficult question. Return to the Jones Polynomial Now lets verify that the bracket does indeed give a model for the Jones polynomial. To see this, consider fK(A) = (-A 3 )" W ( K )(A). Since the writhe, w(K), is obtained by summing signs over all the crossings of K, we can interpret the factor (-A->)"W(K) a s the product of contributions of (-A-*) or (-A 3 )-! one from each crossing and depending upon the sign of the crossing. Thus we can write an oriented state expansion formula for fK as shown below where K+ and K- denote links with corresponding sites with oriented crossings, KO is the result of smoothing the crossing in an oriented fashion and K& is the result of smoothing the crossing against the orientation. f K+ = (-A 3 )-!Af KO + ( - A ^ - U ^ f K& . Hence, fK+ =-A- 2 fK0

-A"4fK&

and similarly, for a negative crossing f K- = -A2fKO

-A4fK&

Letting VK(t) = fK(r 1 / 4 ) we have VK+ =-t 1 / 2 VK0

-tVK&

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and VK- =-r 1 / 2 VK0

-t-!VK&

Therefore, r1VK+

-tVK-

=(t1/2-r1/2)VK0

We leave the rest of the verification that VK(t) is the Jones polynomial (see section 4) to the reader (You should check that it has the right behaviour on unknotted loops).

VI. Vassiliev Invariants We have seen how it is fundamental to take the difference of an invariant at a positive crossing and at a negative crossing, leaving the rest of the diagram alone. The earliest instance of this is the Conway polynomial [CON], CK(z) with its exchange identity C K+ - C K- = z CKO Vassiliev [V] gave new meaning to this sort of identity by thinking of the structure of the entire space of all mappings of a circle into three dimensional space. This space of mappings includes mappings with singularities where two points on a curve touch. He interpreted the equation ZK+ -ZK- =ZK# as describing the difference of values across a singular embedding K# where K# has a transverse singularity in the knot space as illustrated in Figure 34. (In a transverse singularity the curve touches itself along two different directions.)

Figure 34. Difference equation

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The Vassiliev formula serves to define the value of the invariant on a singular embedding in terms of the the values on two knots "on either side" on this embedding. This Vassiliev formula serves to describe a method of extending a given invariant of knots to a corresponding invariant of embedded graphs with controlled singularities of this transverse type. This idea had been considered before Vassiliev. Vassiliev carried out his program of analyzing the singular knot space using techniques of algebraic topology, and in the course of this investigation he discovered a key concept that had been completely overlooked in the context of graph invariants. That concept is the idea of an invariant of finite type. Definition. We shall say that ZG is an invariant of finite type i if ZG vanishes for all graphs with greater than i nodes. This concept was extracted from Vassiliev's work by Birman and Lin [BIL]. A (rigid vertex) invariant of knotted graphs is a Vassiliev invariant of finite type i if it satisfies the identity ZK+ - ZK- = ZK# and it is of finite type i. In rigid vertex isotopy the cyclic order at the vertex is preserved, so that the vertex behaves like a rigid disk with flexible strings attached to it at specific points. Vassiliev invariants form an extraordinary class of knot invariants. It is an open problem whether the Vassiliev invariants are sufficient to distinguish knots that are topologically distinct. Vassiliev began an analysis of the combinatorial conditions on graph evaluations that could support such invariants. The key observation is the Lemma. If ZG is a Vassiliev invariant of finite type i, then ZG is independent of the embedding of the graph G when G has i vertices. Proof. Suppose that G is an embedded graph G with i nodes. If we switch a crossing in G to form G' then the exchange relation for the Vassiliev invariant says that ZG ZG' = ZG" where G" has one more node than G or G'. But then G" has i+1 nodes and hence ZG" = 0. Therefore ZG = ZG'. This shows that we can switch crossings in any embedding of G without changing the value of ZG. It follows from this that ZG is independent of the embedding and depends only on the graph G. This completes the proof of the Lemma. // For a Vassiliev invariant of type i, there is important information in the values it takes on graphs with exactly i nodes. These evaluations do not depend upon the embedding type of the graph. However, not just any such graphical evaluation will extend to give a topological invariant of knots and graphs. There are necessary conditions. Vassiliev found a version of these conditions through his analysis of the knot space and Ted Stanford [ST] , a student of Joan Birman, discovered the beautiful topological meaning of these conditions in relation to the switching identity. Stanford's argument goes as follows: Consider a singular crossing that has an arc from the diagram passing underneath it as shown in Figure 35.

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Figure 35. Embedded four term relation

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Four crossing switches will take that arc above the singular crossing and return the diagram to a position that is topologically equivalent to its original position. Each crossing switch gives an equation. There are four equations. Add them up and you get an identity among the values of the invariant on four diagrams. Call this the four-term relation. This identity is illustrated in the second box in Figure 35. Now recall from the Lemma we proved above that for a Vassiliev invariant of type i, the graphs with i nodes have values that are independent of their embeddings in three dimensional space. This means that at the top level (the i noded graphs for a Vassiliev invariant of type i will be called the top level.) the four term relations will be relations among the evaluations of abstract graphs. At the top level the four term relations will be purely combinatorial conditions related to the topology. How shall we think of abstract 4-valent graphs corresponding to singular embeddings of a knot? An abstract knot is just a circle. An abstract singular knot is a circle with pairs of points marked that become the singular points in the embedding. Indicate these paired points by arcs between them. Call the resulting structure a chord diagram. See the example at the beginning of Figure 36. In the language of the chord diagrams the 4-term relation at the top level (see the discussion of the top level in the paragraph above) becomes the equation shown in Figure 36. This can be seen by translating the relation in Figure 35 into the language of chord diagrams. In Figure 36 we have indicated parts of the chord diagram that are neighbors by showing an outer bracket connecting them. Those sites that are neighbors can have no other chords between them. Otherwise there can be many chords in these diagrams that are not indicated, just so long as the diagrams in the equation for the four-term relation differ only as shown in the Figure. If you can write down a top level evaluation of chord diagrams that satisfies the 4-term relation, then you have the raw data for a Vassiliev invariant. Such an evaluation of chord diagrams is called a weight system for a Vassiliev invariant. By the Theorems of Kontsevich and Bar-Natan [BAR95], this raw data guarantees the existence of at least one invariant that satisfies the top level evaluation. The world is rife with Vassiliev invariants. Birman and Lin [BIL] showed directly that the Jones polynomial and its generalizations give rise to Vassiliev invariants. In the case of the Jones polynomial here is an easy proof of their result: Theorem. Let VG(t) denote the Jones polynomial extended to rigid vertex 4-valent graphs by the formula VK+ - VK- = VK# . Let vi(G) denote the coefficient of y} in the expansion of VG(exp(x)). Then vi(G) is a Vassiliev invariant of type i. Proof. Use the identities from the end of section 5. VK+ =-t 1/2 VKO -tVK& VK- =-r 1 / 2 VK0 -t^VKcfc. Substitute t =exp(x).

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Embedded Graph

Corresponding Chord Diagram

Abstract FourTeim Relation Figure 36. Abstract four term relation via chord diagrams

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It follows at once that VK# = VK+ - VK- is divisible by x. Hence VG is divisible by x1 when G has i nodes. This implies that the coefficients vi(G)=0 if G has more than i nodes. Hence the coefficients vi(G) are of finite type, proving the Theorem.// With the help of theorems of this type it is possible to study Vassiliev invariants by studying the structure of known invariants of knots and links. In particular it is possible to justify the structure of many weight systems in terms of known invariants. We shall not go into these sorts of investigations in this exposition. The next section shows how the algebraic study of Lie algebras is directly related to the construction of Vassiliev invariants. This is one beginning of a whole world of relationships between knot theory and algebra. VII. Vassiliev Invariants and Lie Algebras The subject of Lie algebras is an algebraic study with a remarkable connection with the topology of knots and links. The purpose of this section is to first give a brief introduction to the concept of a Lie algebra and then to show the deep connection between these algebras and the structure of Vassiliev invariants for knots and links, as described in the previous section. In order to understand the idea behind a Lie algebra it is helpful to first consider the concept of a group. A set G is said to be a group if it has a single binary operation * such that 1. Given a and b in G then a*b is also in G. 2. If a,b,c are in G then (a*b)*c = a*(b*c). 3. There is an element E in G such that E*a = a*E = a for all a in G. 4. Given a in G there exists an element a"l in G such that a*a"' = a"'*a=E. One of the most fertile sources of groups is matrix algebra. Recall that an n x n matrix A is an array of numbers Aij (real or complex), A=(Aij), where i and j range in value from 1 to n. One defines the product of two matrices by the formula (AB)ij = £k AikBkj where k runs from 1 to n in this summation. For our purposes it is essential to have a diagrammatic representation for matrix multiplication. This representation is illustrated in Figure 37. Each matrix is represented by a labeled box with one arrow that enters the box and one arrow that leaves the box. The entering arrow corresponds to the left index i in Aij, while the right arrow corresponds to the right index j . In multiplying two matrices A and B together to form AB we tie the outgoing arrow of A to the ingoing arrow of B. By convention, an arrow that has no free ends connotes the summation over all possible choices of index for that arrow. Many facts about matrices become quite transparent in this notation. For example, the trace of A, denoted tr(A), is the sum of the diagonal entries Aii where i ranges from one to n. The diagrammatic proof of the basic formula tr(AB) = tr(BA) is illustrated in Figure 38.

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Sum on k. Figure 37. Diagrammatic matrix multiplication

-*-

->

= HAJ-IjAjj

tr(AB) =

= tr(BA)

Figure 38. Diagrammatic proof of the basic formula tr(AB) = tr(BA) For a given value of n, we let Mn(R) denote the set of all n x n matrices with coefficients in the real numbers R. We let A*B = AB denote the product of matrices and we let E denote the matrix whose entries are given by the formula: Eii=l for all i, and Eij=0 if i is not equal to j . With this choice of multiplication and identity element E, Mn(R) satisfies the first three axioms for a group. However there are matrices A that have no inverse (A~l so that AA"' = E). For example the matrix 0, all

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of whose entries are zero, is a matrix without an inverse. Thus Mn(R) is not itself a group. There is a criterion for a matrix to have an inverse. This is simply that the determinant, Det (A), should be non-zero. Thus the largest group of matrices of size n x n that we can devise is the set of all matrices A such that Det(A) is non-zero. This is called the general linear group and is denoted by GLn(R). There are many interesting subgroups of this large group of matrices. One example is the group Sl(n) of all matrices with determinant equal to one. We may also restrict to orthogonal matrices A over R. These are invertible matrices A such that A1 = A~l where A1 denotes the transpose of the matrix A: Alij = Aji. The group of orthogonal matrices is denoted by O(n). The intersection of O(n) and Sl(n) is denoted SO(n). SO(n) consists in the orthogonal matrices of determinant equal to one. In the case n=2, SO(2) consists in rotations of the plane that fix the origin, and in the case of n=3, SO(3) consists in rotations of three dimensional space about specified axes. SO(3) has a fascinating collection of finite subgroups including the symmetries of the classical regular solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Ultimately, the matrix groups become a language for the precise expression of symmetry. We now ask when a matrix A can be written in the form A = e B = E + (1/1 !)B + (1/2!)B 2 + (1/3!)B 3 + ... for some other matrix B. Since e ° = limit (E+B/m) m where the limit is taken as m approaches infinity, we can regard (E+B/m), for m large, as an "infinitesimal" version of the matrix A, and one refers to B as an "infinitesimal generator" for A. It is interesting and mathematically significant to compare the algebraic properties of A and B. The key property for this comparison is the determinant equation Det (e B ) = e tr (B) where tr(B) denotes the trace of B. (One way to prove this identity is to use the Jordan canonical form for the matrix and the fact that similar matrices have the same trace and determinant.) For example, if Det(e°) = 1 then we need that tr(B)=0. This means that elements of Sl(n) are the exponentials of matrices with trace equal to zero. Let sl(n) denote the set of n x n matrices with trace equal to zero. The set sl(n) is not closed under matrix multiplication, but it is closed under the Lie bracket (or commutator) operation [B,C]=BC-CB. Iftr(B) = tr(C)=0,then tr[B,C] = tr(BC-CB) = tr(BC) - tr(CB) = tr(BC)-tr(BC) = 0 (since tr(BC) = tr(CB) for any matrices B and C).

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Thus, if B and C belong to sl(n), then [B,C] also belongs to sl(n). This closure under the bracket operation leads directly to the notion of a Lie algebra. Definition. A Lie algebra is a vector space L over a field F that is closed under a binary operation, called the Lie bracket and denoted by [B,C] for B and C in L. The bracket is assumed to satisfy the following axioms. 1. [X,Y] = -[Y,X] for all X and Y in L. 2. [aX+bY.Z] = a[X,Z] + b[Y,Z] for all a and b in F and X,Y,Z in L. 3. [X,[Y,Z]] + [Z,[X,Y]] + [Y,[Z,X]] = 0. This last identity is called the Jacobi identity. It is easy to verify that the bracket operation [B,C]=BC - CB on the vector space of all n x n matrices over F (e.g. F=R) satisfies the axioms given above. Thus, we have so far seen that sl(n) is a Lie algebra that is naturally associated with the group of matrices Sl(n). In fact, sl(n) generates Sl(n) by exponentiation. There is a general pattern. Each matrix group has its corresponding Lie algebra. The classification of matrix groups is simplified by a corresponding classification of Lie algebras. As a result, the Lie algebras are a subject in their own right. It has often happened that Lie algebras are connected mathematically with subjects different from their original roots in group theory. In our context the Lie algebras turn out to be related to the formation of weight systems for Vassiliev invariants. One way to see this is to just take the case of matrix Lie algebras with commutator brackets and interpret diagrammatically the formula that states that the Lie algebra is closed under the bracket operation. This formula states that there is a basis {Tl,T2,...,Tm} for the Lie algebra as a vector space over F such that each Ta is an n x n matrix and such that TaTb - TbTa = I fabcTc where fabc is a set of constants in F depending on the three indices a,b,c (running from 1 to n). The right hand side of this equation connotes a summation over all values of the index c=l,...,n. The left hand side is the commutator of Ta and Tb for any given choice of a and b. In Figure 39 we have diagrammed this equation using the conventions for diagrammatic matrix multiplication explained in this section. (Thus we do not have to write the summation sign on the right of the above equation!) The structure constants fabc are represented by a graphical vertex with three lines attached to it, one for a, one for b and one for c. For the purpose of discussion, we shall assume that fabc is dependent only on the cyclic order of abc. It is convenient to regard the graphical vertex as representing a "tensor" that has this cyclic invariance since this means that we can slide the diagram for the structure constant tensor around in the plane so long as we keep the cyclic order of its legs unchanged. Such bases can be obtained in many cases of matrix Lie algebras, and the results that we outline can be generalized in any case.

175

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a

?

T

3

—£

a^

T

T

^—

a—

X

T

^-b

T

--!»

T

•pQ -rb _

j b -pa

=

fab

jC

Figure 39. Diagram for the equation above Now view Figure 40. You will see, first, a formal version of the commutator relation of Figure 39, except that the labels and indices have been removed and the boxes for matrix elements have been replaced by graphical vertices. Imagine that the terms in this formal version of the commutator relation are parts of chord diagrams as illustrated with examples in this figure. In other words, recall the method of chord diagrams from the last section and imagine that along with the chords there are also trivalent graphical vertices among the chords, and that these vertices are related to commutators as shown in the Figure. Finally, view Figure 41 and you will see a formal derivation of the four term relation for chord diagrams from the diagrammatic commutator identity. This means that the four-term relation, that we derived from topological considerations in the last section, is intimately related to the basic structure of a Lie algebra. This is the essence of the relationship of Vassiliev invariants with Lie algebras and their generalizations. Concretely, the relationship we have just described means that it is possible to construct weight systems for Vassiliev invariants by using matrix Lie algebras. To see how this works view Figure 42. Here we have indicated a chord diagram D and a corresponding diagram involving matrices Ta from a Lie algebra basis. The second diagram represents the sum of traces wt(D) = £ tr(TaTbTcTaTbTc) where we are summing over all values for the indices a, b and c. This second diagram represents the weight, wt(D), that is assigned to the first diagram. It follows from our considerations that this weight system satisfies the four term relation and hence, by the Theorem of Kontsevich [BAR95], is the top row evaluation for a Vassiliev invariant.

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^-X--XFigure 40. A formal version of the commutator relation of figure 39 This section has been a sketch of the amazing and deep connection between Lie algebras and invariants of knots and links. The territory is even more surprising as one explores it further. First of all, it should be clear from what we have said that what is really needed here is an appropriate generalization of Lie algebras. In fact, prior to the discovery of the Vassiliev invariants, a very remarkable such generalization called "quantum groups" was discovered through work in statistical mechanics and was applied to knot theory. It was already known that quantum groups provided a strong connection between Lie algebras and their generalizations and invariants of knots and links. Now the matter of finding all weight systems challenges the resources of quantum groups and it is not known if all Vassiliev invariants can be built through the quantum groups.

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Figure 41. A formal derivation of the four-term relation for chord diagrams

Figure 42. Weight systems for Vassiliev invariants by using matrix Lie algebra

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In the next few sections we shall discuss the physical background behind many of the mathematical ideas discussed so far in this introduction to knot invariants.

VIII. A Quick Review of Quantum Mechanics To recall principles of quantum mechanics it is useful to have a quick historical recapitulation. Quantum mechanics really got started when De Broglie introduced the fantastic notion that matter (such as an electron) is accompanied by a wave that guides its motion and produces interference phenomena just like the waves on the surface of the ocean or the diffraction effects of light going through a small aperture. De Broglie's idea was successful in explaining the properties of atomic spectra. In this domain, his wave hypothesis led to the correct orbits and spectra of atoms, formally solving a puzzle that had been only described in ad hoc terms by the preceding theory of Niels Bohr. In Bohr's theory of the atom, the electrons are restricted to move only in certain elliptical orbits. These restrictions are placed in the theory to get agreement with the known atomic spectra, and to avoid a paradox! The paradox arises if one thinks of the electron as a classical particle orbiting the nucleus of the atom. Such a particle is undergoing acceleration in order to move in its orbit. Accelerated charged particles emit radiation. Therefore the electron should radiate away its energy and spiral into the nucleus! Bohr commanded the electron to only occupy certain orbits and thereby avoided the spiral death of the atom - at the expense of logical consistency. De Broglie hypothesized a wave associated with the electron and he said that an integral multiple of the length of this wave must match the circumference of the electron orbit. Thus, not all orbits are possible, only those where the wave pattern can "bite its own tail". The mathematics works out, providing an alternative to Bohr's picture. De Broglie had waves, but he did not have an equation describing the spatial distribution and temporal evolution of these waves. Such an equation was discovered by Erwin Schrodinger. Schrodinger relied on inspired guesswork, based on DeBroglie's hypothesis and produced a wave equation, known ever since as the Schrodinger equation. Schrodinger's equation was enormously successful, predicting fine structure of the spectrum of hydrogen and many other aspects of physics. Suddenly a new physics, quantum mechanics, was born from this musical hypothesis of De Broglie. Along with the successes of quantum mechanics came a host of extraordinary problems of interpretation. What is the status of this wave function of Schrodinger and De Broglie. Does it connote a new element of physical reality? Is matter "nothing but" the patterning of waves in a continuum? How can the electron be a wave and still have the capacity to instantiate a very specific event at one place and one time (such as causing a bit of phosphor to glow there on your television screen)? It came to pass that Max Born developed a statistical interpretation of the wavefunction wherein the wave determines a probability for the appearance of the localized particulate phenomenon that one wanted to call an "electron". In this story the

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wavefunction y takes values in the complex numbers and the associated probability is y*y, where y* denotes the complex conjugate of y. Mathematically, this is a satisfactory recipe for dealing with the theory, but it leads to further questions about the exact character of the statistics. If quantum theory is inherently statistical, then it can give no complete information about the motion of the electron. In fact, there may be no such complete information available even in principle. Electrons manifest as particles when they are observed in a certain manner and as waves when they are observed in another, complementary manner. This is a capsule summary of the view taken by Bohr,Heisenberg and Born. Others, including De Broglie, Einstein and Schrodinger, hoped for a more direct and deterministic theory of nature. As we shall see, in the course of this essay, the statistical nature of quantum theory has a formal side that can be exploited to understand the topological properties of such mundane objects as knotted ropes in space and spaces constructed by identifying the sides of polyhedra. These topological applications of quantum mechanical ideas are exciting in their own right. They may shed light on the nature of quantum theory itself. In this section we review a bit of the mathematics of quantum theory. Recall the equation for a wave: f(x,t) = sin((2p/l)(x-ct)). With x interpreted as the position and t and as the time, this function describes a sinusoidal wave traveling with velocity c. We define the wave number k= 2p/l and the frequency w = (2pc/l) where 1 is the wavelength. Thus we can write f(x,t) = sin(kx - wt). Note that the velocity, c, of the wave is given by the ratio of frequency to wave number, c=w/k. De Broglie hypothesized two fundamental relationships: between energy and frequency, and between momentum and wave number. These relationships are summarized in the equations E = hw, p=fek, where E denotes the energy associated with a wave and p denotes the momentum associated with the wave. Here h = h/2p where h is Planck's constant. For De Broglie the discrete energy levels of the orbits of electrons in an atom of hydrogen could be explained by restrictions on the vibrational modes of waves associated with the motion of the electron. His choices for the energy and the momentum in relation to a wave are not arbitrary. They are designed to be consistent with the notion that the wave or wave packet moves along with the electron. That is, the velocity of the wavepacket is designed to be the velocity of the "corresponding" material particle. It is worth illustrating how DeBroglie's idea works. Consider two waves whose frequencies are very nearly the same. If we superimpose them (as a piano tuner superimposes his tuning fork with the vibration of the piano string), then there will be a new wave produced by the interference of the original waves. This new wave pattern will move at its own velocity, different (and generally smaller) than the velocity of the original waves. To be specific, let f(x,t) = sin(kx - wt) and g(x,t) = sin(k'x - w't).

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L. H. Kauffman

Let h(x,t) = sin(kx - wt) + sin(k'x - w't) = = f(x,t) + g(x,t). A little trigonometry shows that h(x,t) = cos(((k-k')/2)x -((w-w')/2)t)sin(((k+k')/2)x -((w+w')/2)t). If we assume that k and k' are very close and that w and w' are very close, then (k+k')/2 is approximately k, and (w+w')/2 is approximately w. Thus h(x.t) can be represented by H(x.t) = cos((dk/2)x -(dw/2)t) f(x,t) where dk=(k-k')/2 and dw=(w-w')/2. This means that the superposition, H(x,t), behaves as the waveform f(x,t) carrying a slower-moving "wave-packet" G(x,t)=cos((dk/2)x -(dw/2)t). See Figure 43.

2 sln(kx-wt)

h(x,tl = ainfkx - wt) * sintk'K - w't) Waves and Wave Packets Figure 43. Waves and wave packets Since the wave packet (seen as the clumped oscillations in Figure 43) has the equation G(x,t)=cos((dk/2)x -(dw/2)t), we see that that the velocity of this wave packet is vg = dw/dk. Recall that wave velocity is the ratio of frequency to wave number. Now according to DeBroglie, E = few and p=fek, where E and p are the energy and

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momentum associated with this wave packet. Thus we get the formula vg = dE/dp. In other words, the velocity of the wave-packet is the rate of change of its energy with respect to its momentum. Now this is exactly in accord with the well-known classical laws for a material particle! For such a particle, E = mv 2 /2 and p=mv. Thus E=p 2 /2m and dE/dp = p/m = v. It is this astonishing concordance between the simple wave model and the classical notions of energy and momentum that initiated the beginnings of quantum theory. Schrodinger's Equation Schrodinger answered the question: Where is the wave equation for DeBroglie's waves? Writing an elementary wave in complex form y = y(x,t) = exp(i(kx - wt)), we see that we can extract De Broglie's energy and momentum by differentiating: ihd v /dt = Ey and - i h ^ x = py. This led Schrodinger to postulate the identification of dynamical variables with operators so that the first equation, itH5v/<5t = E v , is promoted to the status of an equation of motion while the second equation becomes the definition of momentum as an operator: p = -ihd/dx. Once p is identified as an operator, the numerical value of momentum is associated with an eigenvalue of this operator, just as in the example above. In our example py = hky. In this formulation, the position operator is just multiplication by x itself. Once we have fixed specific operators for position and momentum, the operators for other physical quantities can be expressed in terms of them. We obtain the energy operator by substitution of the momentum operator in the classical formula for the energy: E = (l/2)mv 2 + V E = p 2 /2m + V E = -(hi/2m)82/Sx2 + V. Here V is the potential energy, and its corresponding operator depends upon the details of the application. With this operator identification for E, Schrodinger's equation ih^y/^t = -{\^l2m)d2yldx2

+ Vy

is an equation in the first derivatives of time and in second derivatives of space. In this form of the theory one considers general solutions to the differential equation and this in turn leads to excellent results in a myriad of applications. In quantum theory, observation is modelled by the concept of eigenvalues for corresponding operators. The quantum model of an observation is a projection of the wave function into an eigenstate.

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An energy spectrum {Ek} corresponds to wave functions y satisfying the Schrodinger equation, such that there are constants Ek with Ey = Eky. An observable (such as energy) E is a Hermitian operator on a Hilbert space of wavefunctions. Since Hermitian operators have real eigenvalues, this provides the link with measurement for the quantum theory. It is important to notice that there is no mechanism postulated in this theory for how a wave function is "sent" into an eigenstate by an observable. Just as mathematical logic need not demand causality behind an implication between propositions, the logic of quantum mechanics does not demand a specified cause behind an observation. This absence of an assumption of causality in logic does not obviate the possibility of causality in the world. Similarly, the absence of causality in quantum observation does not obviate causality in the physical world. Nevertheless, the debate over the interpretation of quantum theory has often led its participants into asserting that causality has been demolished in physics. Note that the operators for position and momentum satisfy the equation xp - px = hi. This corresponds directly to the equation obtained by Heisenberg, on other grounds, that dynamical variables can no longer necessarily commute with one another. In this way, the points of view of DeBroglie, Schrodinger and Heisenberg came together, and quantum mechanics was born. In the course of this development, interpretations varied widely. Eventually, physicists came to regard the wave function not as a generalised wave packet, but as a carrier of information about possible observations. In this way of thinking y*y (y* denotes the complex conjugate of y.) represents the probability of finding the "particle" (a particle is an observable with local spatial characteristics) at a given point in spacetime. Dirac Brackets We now discuss Dirac's notation, , [D]. In this notation are vectors and covectors respectively. is the evaluation of , hence it is a scalar, and in ordinary quantum mechanics it is a complex number. One can think of this as the amplitude for the state to begin in " a " and end in "b". That is, there is a process that can mediate a transition from state a to state b. Except for the fact that amplitudes are complex valued, they obey the usual laws of probability. This means that if the process can be factored into a set of all possible intermediate states cl, c2, ..., en , then the amplitude for a—>b is the sum of the amplitudes for a—>ci—>b. Meanwhile, the amplitude for a—>ci—>b is the product of the amplitudes of the two subconfigurations a—>ci and ci—>b. Formally we have = S where the summation is over all the intermediate states i=l,..., n. In general, the amplitude for mutually disjoint processes is the sum of the amplitudes of the individual processes. The amplitude for a configuration of disjoint processes is the product of their individual amplitudes.

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Dirac's division of the amplitudes into bras is done mathematically by taking a vector space V (a Hilbert space, but it can be finite dimensional) for the bras: belongs to V* so that lb> is a linear mapping lb>:V > C where C denotes the complex numbers. We restore symmetry to the definition by realizing that an element of a vector space V can be regarded as a mapping from the complex numbers to V. Given V, the corresponding element of V is the image of 1 (in C) under this mapping. In other words, V and lb> : V > C. The composition = : C >C is regarded as an element of C by taking the specific value (l). The complex numbers are regarded as the "vacuum", and the entire amplitude is a "vacuum to vacuum" amplitude for a process that includes the creation of the state a, its transition to b, and the annihilation of b to the vacuum once more. Dirac notation has a life of its own. Let P = lyxxl. Let <xl ly> = <xly>. Then PP = lyxxl lyxxl = ly> <xly> <xl = <xly> P. Up to a scalar multiple, P i s a projection operator. That is, if we let Q= P/<xly>, then QQ = PP/<xlyxxly> = <xly>P/<xlyxxly> = P/<xly> = Q. Thus QQ=Q. In this language, the completeness of intermediate states becomes the statement that a certain sum of projections is equal to the identity: Suppose that £i Icixcil = 1 (summing over i) with =l for each i. Then = = £ = S = S . Iterating this principle of expansion over a complete set of states leads to the most primitive form of the Feynman integral [FH]. Imagine that the initial and final states a and b are points on the vertical lines x=0 and x=n+l respectively in the x-y plane, and that (c(k)i(k) , k) is a given point on the line x=k for 0: = S ...

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where the sum is taken over all i(k) ranging between 1 and m, and k ranging between 1 and n. Each term in this sum can be construed as a combinatorial path from a to b in the two dimensional space of the x-y plane. Thus the amplitude for going from a to b is seen as a summation of contributions from all the "paths" connecting a to b. Feynman used this description to produce his famous path integral expression for amplitudes in quantum mechanics. His path integral takes the form SdP exp(iS) where i is the square root of minus one, the integral is taken over all paths from point a to point b, and S is the action for a particle to travel from a to b along a given path. For the quantum mechanics associated with a classical (Newtonian) particle the action S is given by the integral along the given path from a to b of the difference T-V where T is the classical kinetic energy and V is the classical potential energy of the particle. The beauty of Feynman's approach to quantum mechanics is that it shows the relationship between the classical and the quantum in a particularly transparent manner. Classical motion corresponds to those regions where all nearby paths contribute constructively to the summation. This classical path occurs when the variation of the action is null. To ask for those paths where the variation of the action is zero is a problem in the calculus of variations, and it leads directly to Newton's equations of motion. Thus with the appropriate choice of action, classical and quantum points of view are unified. The drawback of this approach lies in the unavailability at the present time of an appropriate measure theory to support all cases of the Feynman integral. To summarise, Dirac notation shows at once how the probabilistic interpretation for amplitudes is tied with the vector space structure of the space of states of the quantum mechanical system. Our strategy for bringing forth relations between quantum theory and topology is to pivot on the Dirac bracket. The Dirac bracket intermediates between notation and linear algebra. In a very real sense, the connection of quantum mechanics with topology is an amplification of Dirac notation. The next two sections discuss how topological invariants in low dimensional topology are related to amplitudes in quantum mechanics. In these cases the relationship with quantum mechanics is primarily mathematical. Ideas and techniques are borrowed. It is not yet clear what the effect of this interaction will be on the physics itself.

IX. Knot Amplitudes At the end of the last section we said: the connection of quantum mechanics with topology is an amplification of Dirac notation. Consider first a circle in a spacetime plane with time represented vertically and space horizontally. See Figure 44.

Knots

/\

->*

Figure 44. A circle in a spacetime plane

t V© V /N

CUP

Figure 45. Quantum representation of the circle with creation of two "particles" The circle represents a vacuum to vacuum process that includes the creation of two "particles", (Figure 45) and their subsequent annihilation (Figure 46). In accord with our previous description, we could divide the circle into these two parts (creation(a) and annihilation (b)) and consider the amplitude . Since the diagram for the creation of the two particles ends in two separate points, it is natural to take a vector space of the form VVV (the tensor product of V with V) as the target for the bra and as the domain of the ket. We imagine at least one particle property being catalogued by each dimension of V. For example, a basis of V could enumerate the spins of the created particles. If {ea} is a basis for V then {eaVeb} forms a basis for VVV. The elements of this new basis constitute all possible combinations of the particle properties. Since such combinations are multiplicative, the tensor product is the appropriate construction.

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t

c CAP V® V

Figure 46. Annihilation of the two "particles"

In this language the creation ket is a map cup, cup = VVV, and the annihilation bra is a mapping cap, cap= lb> : VVV - — > C. The first hint of topology comes when we realise that it is possible to draw a much more complicated simple closed curve in the plane that is nevertheless decomposed with respect to the vertical direction into many cups and caps. In fact, any nonselfintersecting differentiable curve can be rigidly rotated until it is in general position with respect to the vertical. It will then be seen to be decomposed into these minima and maxima. Our prescriptions for amplitudes suggest that we regard any such curve as an amplitude via its description as a mapping from C to C. Each simple closed curve gives rise to an amplitude, but any simple closed curve in the plane is isotopic to a circle, by the Jordan Curve Theorem. If these are topological amplitudes, then they should all be equal to the original amplitude for the circle. Thus the question: What condition on creation and annihilation will insure topological amplitudes? The answer derives from the fact that all isotopies of the simple closed curves are generated by the cancellation of adjacent maxima and minima as illustrated in Figure 47. In composing mappings it is necessary to use the identifications (VVV) VV = VV (VVV) and VVk=kVV=V. Thus in Figure 47, the composition on the left is given by V = VVk - l c u p - > VV (VVV) = (VVV) VV - c a p l - > kVV = V. This composition must equal the identity map on V (denoted 1 here) for the amplitudes to have a proper image of the topological cancellation. This condition is said very simply by taking a matrix representation for the corresponding operators. Specifically, let {el, e2, ..., en} be a basis for V. Let eab = ea V eb denote the elements of the tensor basis for VVV. Then there are matrices Mab and Ma" such that cup(l) = S Ma'>eab with the summation taken over all values of a and b from 1

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V

V

t

4\

k® V /\

CAP ®1

V® V® V 1®CUP V®k 4\

V

V

Figure 47. Topological amplitudes by cancellation to n. Similarly, cap is described by cap(eab) = M a j,. Thus the amplitude for the circle is cap[cup(l)] = cap S M a b eab = S MabM ab . In general, the value of the amplitude on a simple closed curve is obtained by translating it into an "abstract tensor expression" in the Mab and M a b , and then summing over these products for all cases of repeated indices. Returning to the topological conditions we see that they are just that the matrices (Mab) and ( M a b ) are inverses in the sense thatSMaiM i b = Ia b and S M ai Mib = I a b are identity matrices. In the Figure 48, we show the diagrammatic representative of the equation S MaiM ib = Ia b . In the simplest case cup and cap are represented by 2 x 2 matrices. The topological condition implies that these matrices are inverses of each other. Thus the problem of the existence of topological amplitudes is very easily solved for simple closed curves in the plane. Now we go to knots and links. Any knot or link can be represented by a picture that is configured with respect to a vertical direction in the plane. The picture will decompose into minima (creations) maxima (annihilations) and crossings of the two types shown below. (Here I consider knots and links that are unoriented. They do not have an intrinsic preferred direction of travel.) See Figure 49. In Figure 49, next to each of the crossings we have indicated mappings of VVV to itself , called R and R ' l respectively. These mappings represent the transitions corresponding to these elementary configurations.

L. H. Kauffman

a

a-

Figure 48. Diagrammatic representative of the equation S MaiMib = lab

V® V si-

R V®V L V© V

V© Y

-M Figure 49. The mappings which represent the transitions corresponding to the configurations

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a

b V® V 4">

R

va v 1® 1 1® 1 V© V

R: V© V

VHV

Figure 50. Isotopy showing that R and R-l must be inverse

That R and R"l really must be inverses follows from the isotopy shown in Figure 50 (This is the second Reidemeister move.) We now have the vocabulary of cup, cap, R and R_1- Any knot or link can be written as a composition of these fragments, and consequently a choice of such mappings determines an amplitude for knots and links. In order for such an amplitude to be topological we want it to be invariant under the list of local moves on the diagrams shown in Figure 51. These moves are an augmented list of the Reidemeister moves (See Figure 4 in section 2), adjusted to take care of the fact that the diagrams are arranged with respect to a given direction in the plane. The equivalence relation generated by these moves is called regular isotopy. It is one move short of the relation known as ambient isotopy. The missing move is the first Reidemeister move shown in Figure 4 of section 2. In the first Reidemeister move, a curl in the diagram is created or destroyed. Ambient isotopy (generated by all the Reidemeister moves) corresponds to the full topology of knots and links embedded in three dimensional space. Two link diagrams are ambient isotopic via the Reidemeister moves if and only if there is a continuous family of embeddings in three dimensions leading from one link to the other. The moves give a combinatorial reformulation of the spatial topology of knots and links.

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Figure 51. Augmented Reidemeister moves for regular isotopy

By ignoring the first Reidemeister move, we allow the possibility that these diagrams can model framed links, that is links with a normal vector field or, equivalently, embeddings of curves that are thickened into bands. It turns out to be fruitful to study invariants of regular isotopy. In fact, one can usually normalize an

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invariant of regular isotopy to obtain an invariant of ambient isotopy. We have already discussed this phenomenon with the bracket polynomial in section 5. As the reader can see, we have already discussed the algebraic meaning of Moves 0. and 2. The other moves translate into very interesting algebra. Move 3., when translated into algebra, is the famous Yang-Baxter equation. The Yang-Baxter equation occurred for the first time in problems related to exactly solved models in statistical mechanics (See [BA]). All the moves taken together are directly related to the axioms for a quasi-triangular Hopf algebra (aka quantum group). We shall not go into this connection here. There is an intimate connection between knot invariants and the structure of generalized amplitudes, as we have described them in terms of vector space mappings associated with link diagrams. This strategy for the construction of invariants is directly motivated by the concept of an amplitude in quantum mechanics. It turns out that the invariants that can actually be produced by this means (that is by assigning finite dimensional matrices to the caps, cups and crossings) are incredibly rich. They encompass, at present, all of the known invariants of polynomial type (Alexander polynomial, Jones polynomial and their generalizations). It is now possible to indicate the construction of the Jones polynomial via the bracket polynomial as an amplitude, by specifying its matrices. The cups and the caps are defined by (M a j,) = ( M a b ) = M where M is the 2 x 2 matrix (with ii = -1). Note that MM = I where I is the identity matrix. Note also that the amplitude for the circle is S MabM a b = S MabMab = S (Mab) 2 = (iA) 2 + (-iA-!)2 = - A 2 - A"2. The matrix R is then defined by the equation R a b cd = AM a b Mcd + AI a cI b d. Since, diagrammatically, we identify R with a (right handed) crossing, this equation can be written diagrammatically as

Taken together with the loop value of A 2 - A" 2 ,

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these equations can be regarded as a recursive algorithm for computing the amplitude. This algorithm is the bracket state model for the (unnormalized) Jones polynomial [KA87]. This model can be studied on its own grounds as we have already done in section 5.

X. Topological Quantum Field Theory - First Steps In order to further justify the idea of topology in relation to the amplification of Dirac notation, consider the following scenario. Let M be a 3-dimensional manifold; that is, a space that is locally homeomorphic to Euclidean three dimensional space. Suppose that F is a closed orientable surface inside M dividing M into two pieces Ml and M2. These pieces are 3-manifolds with boundary. They meet along the surface F. Now consider an amplitude <MllM2> = Z(M). The form of this amplitude generalizes our previous considerations, with the surface F constituting the distinction between the "preparation" Ml and the "detection" M2. This generalization of the Dirac amplitude amplifies the notational distinction consisting in the vertical line of the bracket to a topological distinction in a space M. The amplitude Z(M) will be said to be a topological amplitude for M if it is a topological invariant of the 3-manifold M. Note that a topological amplitude does not depend upon the choice of surface F that divides M. From a physical point of view the independence of the topological amplitude on the particular surface that divides the 3-manifold is the most important property. An amplitude arises in the condition of one part of the distinction carved in the 3manifold acting as "the observed" and the other part of the distinction acting as "the observer". If the amplitude is to reflect physical (read topological) information about the underlying manifold, then it should not depend upon this particular decomposition into observer and observed. The same remarks apply to 4-manifolds and interface with ideas in relativity. We mention 3-manifolds because it is possible to describe many examples of topological amplitudes in three dimensions. The matter of 4dimensional amplitudes is a topic of current research. The notion that an amplitude be independent of the distinction producing it is prior to topology. Topological invariance of the amplitude is a convenient and fundamental way to produce such independence. This sudden jump to topological amplitudes has its counterpart in mathematical physics. In [WIT] Edward Witten proposed a formulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form Z(M) where Z(M) = SdAexp[(ik/4p)S(M,A)]. Here M denotes a 3-manifold without boundary and A is a gauge field (also called a gauge potential or gauge connection) defined on M. The gauge field is a one-form on M with values in a representation of a Lie algebra. The group corresponding to this Lie algebra is said to be the gauge group for this particular field. In this integral the "action" S(M,A) is taken to be the integral over M of the trace of the Chern-Simons

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three-form CS = AdA + (2/3)AAA. (The product is the wedge product of differential forms.) Instead of integrating over paths, the integral Z(M) integrates over all gauge fields modulo gauge equivalence. This generalization from paths to fields is characteristic of quantum field theory. Quantum field theory was designed in order to accomplish the quantization of electromagnetism. In quantum electrodynamics the classical entity is the electromagnetic field. The question posed in this domain is to find the value of an amplitude for starting with one field configuration and ending with another. The analogue of all paths from point a to point b is "all fields from field A to field B". Witten's integral Z(M) is, in its form, a typical integral in quantum field theory. In its content Z(M) is highly unusual. The formalism of the integral, and its internal logic supports the existence of a large class of topological invariants of 3-manifolds and associated invariants of knots and links in these manifolds. Invariants of three-manifolds were initiated by Witten as functional integrals in [WIT] and at the same time defined in a combinatorial way by Reshetikhin and Turaev in [RT2]. The Reshetikhin-Turaev definition proceeds in a way that is quite similar to the definition that we gave for the bracket model for the Jones polynomial in section 2. It is an amazing fact that Witten's definition seems to give the very same invariants. We are not in a position to go into the details of this correspondence here. However, one theme is worth mentioning: For k large, the Witten integral is approximated by those gauge connections A for which S(M,A) has zero variation with respect to change in A. These are the so-called flat connections. It is possible in many examples to calculate this contribution via both the functional integral and by the combinatorial definition of Reshetikhin and Turaev. In all cases, the two methods agree (See e.g. [FG], [LR]). This is one of the pieces of evidence in a puzzle that everyone expects will eventually justify the formalism of the functional integral. In order to obtain invariants of knots and links from Witten's integral, one adds an extra bit of machinery to the brew. The new machinery is the Wilson loop. The Wilson loop is an exponentiated version of integrating the gauge field along a loop K. We take this loop K in three space to be an embedding (a knot) or a curve with transversal self-intersections. It is usually indicated by the symbolism tr(Pexp(S K A)). Here the P denotes path ordered integration—that is we are integrating and exponentiating matrix valued functions, and one must keep track of the order of the operations. The symbol tr denotes the trace of the resulting matrix. With the help of the Wilson loop function on knots and links, Witten [WIT] writes down a functional integral for link invariants in a 3-manifold M: Z(M,K) = SdAexp[(ik/4p)S(M,A)] tr(Pexp(S K A)). Here S(M,A) is the Chern-Simons Lagrangian, as in the previous discussion.

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If one takes the standard representation of the Lie algebra of SU(2) as 2x2 complex matrices then it is a fascinating exercise to see that the formalism of Z(S 3 ,K) (S3 denotes the three-dimensional sphere) yields up the original Jones polynomial with the basic properties as discussed in section 1. See Witten's paper or [WIT] or [KA95] for discussions of this part of the heuristics. This approach to link invariants crosses boundaries between different methods. There are close relations between Z(S 3 'K) and the invariants defined by Vassiliev (See [BAR95], [KA95].), to name one facet of this complex crystal. Links and the Wilson loop We shall now indicate an analysis the formalism of this functional integral that reveals quite a bit about its role in knot theory. This analysis depends upon some key facts relating the curvature of the gauge field to both the Wilson loop and the Chern-Simons Lagrangian. To this end, let us recall the local coordinate structure of the gauge field A(x), where x is a point in three-space. We can write A(x) = A a | c (x)T a dx' c where the index a ranges from 1 to m with the Lie algebra basis {Tl, T2, T3, ..., Tm}. The index k goes from 1 to 3. For each choice of a and k, Aak(x) is a smooth function defined on three-space. In A(x) we sum over the values of repeated indices. The Lie algebra generators Ta are actually matrices corresponding to a given representation of an abstract Lie algebra. Difference Formula One can deduce a difference formula for the Witten invariants from the formal properties of the functional integral. Let K+ and K- denoting knots that differ at a single crossing with + and - signs respectively, and K## the result of replacing the crossing by a transverse singularity ( i.e. with distinct tangent directions for the two local curve segments). We take K# to denote the insertion of a graphical node at the transverse crossing, as we have done in our discussion of the Vassiliev invariant. The notation K## indicates that the curve intersects itself in space at one point. Let K##TaTa denote the result of placing the matrices of the Lie algebra basis into the Wilson line at the singular crossing as shown in Figure 52. These matrices become part of the big matrix product that generates the Wilson line. Then, up to order (1/k) one has the difference relation (See [KA95].) Z(K+) - Z(K-) = (4pi/k) Z(K##TaTa). This formula is the key to unwrapping many properties of the knot invariants. Graph Invariants and Vassiliev Invariants Recall, from section 6, that V(G) is a Vassiliev invariant if VK+ -VK- =VK#. V(G) is said to be of finite type k if V(G) = 0 whenever #(G) >k where #(G) denotes the number of 4-valent nodes in the graph G. See section 6.

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Figure 52. Matrices of the Lie algebra basis into the Wilson line at the singular crossing With this definition in hand, lets return to the invariants derived from the functional integral Z(K). We have that Z(K+) - Z(K-) = (4pi/k) Z(K##TaTa). This formula tells us that for the Vassiliev invariant associated with Z we have Z(K#) = (4pi/k)Z(K##TaTa). Furthermore, if Vj(K) denotes the coefficient of (4pi/k)i in the expansion of Z(K) in powers of (1/k), then the ambient difference formula implies that (l/k)i divides Z(G) when G has j or more nodes. Hence Vj(G) = 0 if G has more than j nodes. Therefore Vj(K) is a Vassiliev invariant of finite type. (This result was proved by Birman and Lin [BIL] by different methods and by Bar-Natan [BAR95] by methods equivalent to ours.) The fascinating thing is that the ambient difference formula, appropriately interpreted, actually tells us how to compute Vk(G) when G has k nodes. This result is equivalent to the description of weight systems derived from Lie algebras that we described in section 7. Thus the approach to link invariants via the functional integral motivates and explains the fundamental structure of Vassiliev invariants. This deep relationship between topological invariants in low dimensional topology and quantum field theory in the sense of Witten's functional integral is really still in its infancy. There will be many surprises in the future as we discover that what has so far been uncovered is only the tip of an iceberg.

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J. Roberts. Skein theory and Turaev-Viro invariants, (preprint 1993).

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K. Reidemeister. Knotentheorie. chelsea Pub. Co. N.Y. (1948). Copyright 1932. Julius Springer, Berlin.

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N.Y.Reshetikhin. Quantized universal enveloping algebras, the YangBaxter equation and invariants of links, I and II. LOMI reprints E-4-87 and E-17-87, Steklov Institute, Leningrad, USSR.

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N.Y. Reshetikhin and V. Turaev. Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127 (1990), 1-26.

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N.Y. Reshetikhin and V. Turaev. Invariants of Three Manifolds via link polynomials and quantum groups. Invent. Math. 103, 547-597 (1991).

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M. Rosso. Groupes quantiques et modeles a vertex de V. Jones en theorie des noeuds. C.R.Scad.Sci.Paris t.307 (1988), 207-210.

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T. Stanford. Finite-type invariants of knots, links and graphs, (preprint 1992).

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D.W. Sumners. Untangling DNA. Math. Intelligencer. Vol. 12. No. 3. (1990), 71-80.

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P.G.Tait.On Knots I, II, III. Scientific Papers Vol.1, Cambridge University Press., London, 1898, 273-347.

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M.B. Thistlethwaite. Kauffman's polynomial and alternating links. Topology 27 (1988), 311 -318.

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W. Thomson (Lord Kelvin). On vortex atoms. Philosophical magazine. 34. July 1867, 15-24. Mathematical and Physical Papers, Vol. 4. Cambridge (1910).

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V.G.Turaev. The Yang-Baxter equations and invariants of links. LOMI preprint E-3-87, Steklov Institute, Leningrad, USSR. Inventiones Math. 92 Fasc.3, 527-553.

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V.G. Turaev and O. Viro. State sum invariants of 3-manifolds and quantum 6j symbols. Topology, Vol. 31, No. 4, 865-902 (1992).

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V. Vassiliev. Cohomology of knot spaces. In Theory of Singularities and Its Applications. (V.I.Arnold, ed.), Amer. Math. Soc. (1990), 23-69.

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K. Walker. On Witten's 3-Manifold Invariants (preprint 1991).

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J. H. White. Self-linking and the Gauss integral in higher dimensionas. Amer. J. Math. Vol. 91. (1969), 693-728.

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S. Winker. Quandles, Knot Invariants and the N-fold Branched Cover. Ph.D. Thesis, University of Illinos at Chicago (1984).

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E.Witten. Quantum field theory and Commun.Math.Phys. 121, 351-399 (1989).

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F.Y. Wu. Knot theory and statistical mechanics. Reviews of Modern Physics.Vol. 64. No. 4. October (1992), 1099-1129.

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D. Yetter. Quantum groups and representations of monoidal categories. Math. Proc. Camb. Phil. Soc. 108 (1990), 197-229.

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A.B. Zamolodchikov. Factorized S matrices and lattice statistical systems. Soviet Sci. Reviews. PartA (1979-1980).

the

Jones

polynomial.

TOPOLOGICAL KNOTS MODELS IN PHYSICS AND BIOLOGY Mathematical ideas for explaining inanimate and living matter LUCIANO BOI Ecole des Hautes Etudes en Sciences Sociales, Centre de Mathematiques 54, BdRaspail, 75006 Paris (France) Abstract: Our macroscopic world abounds in knotted and linked rings. But even at the nanoscopic scale, works in molecular topology have shown that Nature builds DNA molecules, which form actual knots and links between themselves. Qualitative similarities may be observed between forms of the microscopic world, as well as between those of the macroscopic world; and by extracting the structures, it is possible to study their dynamics thanks to topological methods and techniques. This chapter is aimed to show that knots and links are ubiquitous scaleindependent objects carrying a tremendous amount of precious information on the emergence of new forms and structures especially in quantum physics and in living matter. In fact, knotting and unknotting are "universal" principles underlying these forms and structures. This study is focused at showing that differential geometry and topological knots theory can be used notably (a) to find invariants for closed 3-manifolds related to quantum field theory, and (b) to describe 3dimensional structures of DNA and protein-DNA complexes. Key words and phrases: Topology, knots, Seifert surfaces, Dehn surgery, three-dimensional spaces, Artin braid group, quantum field theory, geometrodynamics, string theory, polynomials invariants, non-linear dynamical systems, fluid mechanics, statistical mechanics, Yang-Baxter equations, geometry of the double helix, topological enzymology, supercoils, DNA replication, form and function. CONTENTS 1. Mathematical Introduction to Knot Theory 1.1. Polynomials invariants and Reidemeister moves 1.2. Some remarks on non-linear science and knots theory 2. Knots, Gravity, and Quantum Field Theory 2.1. Braid group and statistical mechanics 2.2. Knots invariants and topological Chern-Simons theory 2.3. Knots theory, geometrodynamics and quantum gravity 3. The Role of Knots in Enzymatic Processes into the Cell 4. Topological Transformations and Biological Processes Explained by Pictures 5. Geometrical Properties of the Double Helix and DNA Replication 6. Conclusions References

203 213 226 228 228 231 239 242 256 264 273 275

1. Mathematical Introduction to Knot Theory Knots have been in the last century one of the most fruitful and unifying mathematical concepts. Not only because they are at the same time of topological, geometrical and algebraic nature, but also because they encompass many striking features of inanimate and living mater. For example, one of the most fertile developments in the last decades in theoretical physics is the work relating quantum field theory and topology in three dimensions. Thus, E. Witten (1988), M. Atiyah (1988) and Vladimir Turaev (1994), among others, where led to construct link invariants for closed 3-manifolds. 203

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Knots invariants were soon found to occur in quantum field theory. Indeed, Witten and others have shown that topological quantum field theory provide a natural way of expressing new ideas about knots. This advance, in turn, has allowed a beautiful generalization about the invariants of knots in more complicated three-dimensional spaces known as three-manifolds, in which space itself may contain holes and loops. More generally, in the very recent years, it appeared clearly that quantum field theory, knot theory and topological and arithmetical invariants theory are intimately linked and involved in the same reality. It should be especially emphasized that physicists think that physical space is of string nature and that it is knots-shaped (See C. Nash and S. Sen, 1983; and A. S. Schwarz, 1993). It is worth of note that knots have formed the basis of fruitful speculations in the realm of matter, and that inorganic and organic physicists and chemists claimed that the molecules they study and observe might form knots or links. For example, the doublehelix structure of DNA is a geometrical entity, or more precisely, a topological configuration. As we shall see, this topological configuration is itself a manifestation of linking and knotting. DNA inside of cells is a very long molecule with a remarkably complex topology. Topological properties of DNA are defined as those that can be changed only by breakage and reunion of the backbone. There are three important topological properties of DNA: (i) the linking number between the strands of the double helix, (ii) the interlocking of separate DNA rings into what are called catenanes, (iii) and knotting. As the number of crossing in a knot or catenane increases, the number of possible isomers grows exponentially. The linking number of DNA in all organisms is less than the energetically most stable value in unconstrained (relaxed) DNA. This puts the DNA under stress which causes it to buckle and coil in a regular way called ()supercoiling. The (-) sign indicates that the linking number is less than in the relaxed state. The name supercoiling arise because it is the coiling of a molecule which is itself formed by the coiling of two strands about each other. Although supercoiling is, strictly speaking, a geometric property, it is a consequence of a topological one, the linking number difference between supercoiled and relaxed DNA. (See J. H. White, 1989; and N. R. Cozzarelli, T. C. Boles, J. H. White, 1990). On the other hand, it should be underlined that the complex topology of DNA is essential for the life of all organisms. In particular, it is needed for the process known as DNA replication, whereby a replica of the DNA is made and one copy is passed on to each daughter cell. The most direct evidence for the vital role played by DNA topology is provided by the results of attempts to change the topology of DNA inside of cells. The topology of DNA in vivo is set by a remarkable group of enzymes called topoisomerases. These enzymes promote the passage of DNA segments through each other - that appear to make DNA incorporeal - until a stable state is achieved. Knotted and linked structures are ubiquitous in nature and in fluid flow in particular. Their scale lengths range from lCT^-lCT6 m for superfluid vortices, to 10~2-102 m for fluid eddies, vortex filaments and tornadoes, and many reach 106-1010 m as in the case of magnetic flux tubes, plasma loops and magnetic circles in stellar atmospheres. There is now observational and experimental evidence of complex braided and entangled fluid structures in the context of both classical fluid mechanics and magneto-hydrodynamics (see H. K. Moffat and A. Tsinober, 1990). A large family of situations reveal that a set of vortices or magnetic field lines will intertwine each other, while staying more or less parallel. A topological description of the intertwining involves braid theory. In many

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initial configurations one observes two unlinked rings in parallel planes, placed one in front of the other, and as time passes, they interact and reconnect at two sites, with an exchange of strands. The final result is a linked pair of vortex rings with some dissipation and diffusion of vorticity. Let's now, before of embarking in a mathematical statement of knot theory, make some brief historical remarks on the dawning of knot theory in the 19th century in mathematics and physics. In spite of the extensive use of knots through the millennia, they did not become a subject deemed worthy of mathematical study until the late eighteenth century. The beginnings of topological knot theory can be seen in the interest given to the study of knots by the German mathematician, astronomer and physicist, Carl Friedrich Gauss. In 1794, Gauss prepared sketches of thirteen knots, which were among other sketches of knots found in his papers after his death in 1855; from these it became clear that during his working life he had given much thought to the problem of capturing the essence of knots in mathematical terms. In particular, he invented a notation for regular projections, which was re-invented in a slightly different form by the Scottish physicist Peter Guthrie Tait, and which was used in a further modified form by the topologist Hugh Dowker. To obtain a symbol in this notation, one first chooses a starting point on the curve, and a direction. One then travels round the curve, labelling the different crossing point A, B, C, ... in the order in which they occur, and also recording the sequence, complete with repetitions, of crossing points traversed. For example,

Figure 1.1. Crossing points on a curve according to the Gauss's method.

would be denoted by ABCABC. Gauss mentioned the problem of finding necessary and sufficient conditions for a sequence of 2ra letters, where each letter occurs twice, to be realizable in this way as a closed curve, but apparently he did not address himself further to the matter. The Dowker notation is an extremely simple way to describe a projection of a knot. First let's start with an alternating knot. Suppose we have a projection of an alternating knot that we want to describe. Choose an orientation on the knot, given by placing coherently directed arrows along the knot. Pick any crossing and label it 1. Leaving that crossing along the understand in the direction of the orientation, label the next crossing that you come to with a 2. Continue through that crossing on the same strand of the knot, and label the next crossing with a 3. Continue to label the crossings with the integers in sequence until you have gone all the way around the knot once. When you are done, each crossing will have two labels on it, as the knot passes through each crossing twice (Figure 1.2). Notice that, in fact, each crossing has one even number and one odd number labelling it.

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Figure 1.2. An alternating knot in which each crossing is labelled with two numbers.

The topological research of Johann Benedict Listing, Gauss's student in 1834 who later became professor of physics at Gottingen, led him to publish some of his work on knots in 1847, in an essay entitled Vorstudien ziir Topologie1. He considered the handness of spirals and discerned between their ability to be either left- or right-handed, which he termed respectively dexiotropy and laeotropy. Planar projections of these spirals led him to introduce the concept of handedness for an oriented crossing. To each type he assigned a certain symbol distribution consisting of Ss and As. The orientation becomes insignificant after the regions have been assigned a type. Listing was the first to persist in representing knots as knotted circles, and obtaining diagrams by projecting these onto a plane. By attaching symbols to the crossing in a diagram, to indicate their types, and considering the resulting symbol distribution in each of the diagram's regions, he was able to propose an 'invariant' for a knot (see P. van de Griend, 1996; and M. Epple, 1998). In knot theory, the easiest invariant to visualize, but one which is not very useful for distinguishing between knots, is the number of components in an «-link L. By definition it equals n, and remains so whatever continuous deformations L is subjected to. Invariants are useful aids in the classification of knots for the following reasons. Suppose we compute the value of a particular invariant from two knot's diagrams, and obtain two different values. Then we can conclude that the two knots forms from which the diagrams were obtained are different knots. However, the converse is not true; diagrams having the same invariant value may or may not come from the same knot. Listing concocted his invariant as follow. He called a region of a knot diagram monotypical if all the angles on the region's boundary had been assigned the same typesymbol (that is, were all Sor all A); in which case, the region was itself given the same type-symbol. If a mixture of Ss and As had occurred, he called the region amphitypical. In this way, Listing typified each region, including the unbounded one (which he called the amplexum). He defined a diagram to be in reduced form if it had a minimal number of crossings, over all possible diagrams obtainable from the knot. He knew that if one or more regions were amphitypical in a diagram, the diagram would not necessarily be in reduced form; but he gave no methods for reducing the number of crossing in such diagrams. He proposed an invariant for knots which have a monotypical diagram in reduced form; and he gave it the name complexions-symbol. However; one might give examples which shows that it is not a true invariant. Indeed, a simple knot (for instance the 7-crossing knot used by Listing himself) can give rise to two different complexionssymbols. Briefly, a pair of polynomials are computed from the diagram, one in the

J. B. Listing, "Vorstudien ziir Topologie," Gottingen, Gottinger Studien, 1847, 857-866.

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"variable" 8, and the other in the "variable" k. The exponents on the terms of these polynomials correspond to the numbers of sides surrounding the regions in the diagram: thus, for example, suppose that there are 5 regions, each having 3 sides and bearing the symbol 8; then the term 58 will appear in the 8-polynomial of the complexions-symbol.

Figure 2. A 7-crossing knot with labelled regions. (Original free-hand drawing by Listing).

The middle of nineteenth century saw the dawning of knot theory in the work of Sir William Thomson and Peter Guthrie Tait, studies that in some way owe their origins to the pioneering work of the mathematicians C. F. Gauss and J. B. Listing. In around 1833 Thomson presented a remarkable example of the relationship between topology (or Geometria situs, as he then called topology) and measurable physical quantities such as electric currents. In the course of his research, he discovered a pure number that depends only on the nature of the link, and not on the underlying geometry. This number is in fact a topological invariant now known as the linking number (see bellow for a mathematical definition). The formula, along with some intrinsic features of the topological structure of knot, was first studied by Listing, who obtained many interesting results regarding some common combinatorial properties of what at a first glance seem to be very different classes of knots. These results became known to Thomson and Tait, and also to James Clark Maxwell. Thomson was in search of an ultimate theory of matter. The idea of vortex motion provided him with a wonderful inspiration, and he became a fervent believer in the eternal existence of vortex atoms as fundamental constituents of nature. In his theory, atoms were thought to be tiny vortex filaments or tiny topological twists, or knots, embedded in an elastic-like fluid medium called ether. The infinite variety of possible chemical compounds was given by the endless family of topological combinations of linked or knotted vortices. Even the fluid ether surrounding these vortices could have a complex topology in which empty holes and inaccessible closed channels were present. Lord Kelvin's revolutionary idea of describing fundamental physics through topological properties not only motivated the first studies on the existence and stability of knotted vortices2, but also stimulated the interest of many of his most distinguished colleagues and friends. According to the Thompson picture, the stability of matter might be explained by the stability of knots; their topological nature prevents them from untwisting. He believed that the variety of chemical elements could be accounted for by the variety of different knots. The main feature of Thompson's vortex theory was that its

2

W. Thomson began writing about his model of the atom, his vortex theory, in the mid-1860s in the Philosophical Magazine, Vol. 34, 1867, 15-24.

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indivisible components would be held together by the "forces" of topology3. For example, the ability of atoms to change into other atoms at high energies could be interpreted as the cutting and recombining of knots. Although Tait never succeeded in experimentally reproducing knotted vortex rings, he was motivated by Thomson's ideas of vortex atoms to produce the first knot table, which is similar to the periodic table of the modern atom. His mathematical classification on knots and links became a standard work on the subject. Tait opened his landmark papers "On Knots. Part I, Part II and Part IH"4 with the following interesting remarks: I was led to the consideration of the forms of knots by Sir W. Thomson's Theory of Vortex Atoms, and consequently the point of view which, at least at first, I adopted was that of classifying knots by the number of their crossings; or, what comes to the same thing, the investigation of the essentially different modes ofjoining points in a plane, so as to form single closed plane curves with a given number of double points. The enormous numbers of lines in the spectra of certain elementary substances show that, if Thomson's suggestion be correct, the form of the corresponding vortex atoms cannot be regarded as very simple. For though there is, of course, an infinite number of possible modes of vibration for every vortex, the number of modes whose period is within a few octaves of the fundamental mode is small unless the form of the atom be very complex. Hence the difficulty, which may be stated as follows: 'What has become of all the simpler vortex atoms?' or 'Why have we not a much greater number of elements than those already known to us?' It will be allowed that, from the point of view of the vortex-atom theory, this is almost a vital question. Two considerations help us to an answer. First, however many simpler forms may be geometrically possible, only a very few of these may be forms of kinetic stability, and thus to get the sixty or seventy permanent forms required for the known elements, we may have to go to a very high order of complexity. This leads to a physical question of excessive difficulty. But secondly, stable or not, are there after all very many different forms of knots with any given small number of crossings? This is the main question treated in the following paper, and it seems, so far as I can ascertain, to be an entirely novel one. (p. 274.)

He later added: My investigations commenced with a recognition of the fact that in any knot or linkage whatever the crossings may be taken throughout alternately over and under. It has been pointed out to me that this seems to have been long known, if we may judge from the ornaments on various Celtic sculptured stones. It was probably suggested by the processes of weaving or plaiting. There is also a remarkable engraving by Diirer, exhibiting a very complex but symmetrical linkage, in which this alternation is maintained throughout. This and some associated properties of knots (...) are direct consequences of the obvious fact that two closed plane curves in one plane necessarily intersect one another an even number of times. It follows as an immediate deduction from this that in going continuously round any closed plane curve whatever, an even number of intersections is always passed on the way from any one intersection to the same again. Hence, of course, if we agree to make a knot of it, and take the crossings (which now correspond to the intersections) over and under alternately, when we come back to any particular crossing we shall have to go under if we previously went over, and vice versa. This is virtually the foundation of all that follows. But it is essential to remark that we have thus two alternatives for the crossing with which we start. We may

3

On the relationship between topological concepts and physical theories in the 19th century, see the M. Epple's detailed study, "Topology, Matter, and Space, I: Topological notions in 19th-century Natural Philosophy," Arch. Hist. Exact Sci., 52 (1998), 297-392. 4 P. G. Tait, "On Knots. Part I.," Transactions of the Royal Society of Edinburgh, Vol. XXXIX (1877), 273-317; "Part II.," Vol. XL (1884), 318-334; "Part III.," Vol. XLI (1885), 335-347.

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make the branch we begin with cross under instead of over the other at that crossing. This has the effect of changing any given knot into its own image in a plane mirror-what Listing calls Perversion. Unless the form be an Amphicheiral one, this perversion makes an essential difference in its character-makes it, in fact, a different knot, incapable of being deformed into its original shape, (p. 275.) Influenced by Maxwell's appreciation of Gauss's and Listing's ideas, Tait tried in 1876 to measure by electromagnetic means topological properties such as knot type, which he call "beknottedness," i.e. the unknotting number. The gordian number or unknotting number of a knot K is the minimal number of overcrossing-undercrossing changes required to transform K to the unknot. (Let us not forget that a knot or link invariant does not change its value if we apply one of the elementary knot moves. A very important knot-or link-invariant is the linking number, which is, more precisely, an invariant for oriented links5). Suppose that D is an oriented regular diagram of a 2-component link L = {Kv K2). Further, suppose that the crossing points of D at which the projections of Kx and K2 intersect are c,, c2, ... , cn. (Here we are considering only the crossing points of the projections of Kt and K2, which are not self-intersections of the knot component.) Then l/2{sign(c,) + sign(c2) + ... + sign(c„) is called the linking number of K, and K2, which is usually denoted by lk(K„K2). This number has some very striking properties, the most important of which is that the linking number lk(Kv K2) is an invariant of L, that is, it is the same for two or more diagrams of L. Let us calculate, for example, the linking number of the links L and L in Figures 3.1 and 3.2 respectively. (a) We need only calculate the signs at the four crossing points c,, c2, c3, and c4. Since the sign at each crossing point is - 1 , we obtain that lk(K\, K2) = - 2. (b) Similarly, it is easy to show sign(c,) = sign(c4) = +1, while sign(c2) = sign(c3) = 1. Therefore, lk(K„K2) = 0.

(a)

(b)

Figure 3.1. The linking number of the links L and L. 5

The work of Gauss on electromagnetism had led him to compute inductance in a system of two linked circular wires; he introduced thus the concept of winding or linking numbers, which are now a basic tool in knot theory.

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Figure 3.2. Type III Reidemeisetr moves don't affect linking number.

Tait went on to list knots according to their numbers of crossing when drawn on the plane in the most efficient possible way. The crossing-number c(K) measures the complexity of a knot, and is one of its basic invariants. If D is a diagram of K, let c(D) be the number of crossing points of D. Then c{K) = inf c(D). The advantage of this measure is that there are only a finite number of diagrams, hence there are only a finite number of knots to a given crossing-number. Tait called the minimal number of crossing-points achievable for the diagram of a given knotted structure the degree of knottiness of that structure6. The first tabulation by Tait (1885) of alternating knots of up to ten crossings is shown (in part) below. But it was Maxwell who truly saw the physical implications of topology. The whole Preface of his Treatise on Electricity and Magnetism is permeated by topological ideas. He developed Listing's original ideas of multiply connected regions in order to study the relationship of electricity and magnetism to forces and potentials. Maxwell noticed that if we express a locally conservative force as the gradient of a potential function, then that function will be well defined - that is, single-valued - only inside a simply connected region. An example of a simply connected region in the plane is a circular patch, while a doubly connected region is a patch with a hole in it. In his section devoted to magnetism, Maxwell also provided a remarkable example of a particular case of Gauss's linking formula for linked magnetic tubes. While Thomson's dream of explaining atoms as knotted vortex rings in a fluid ether never came to fruition - though there are some remarkable analogies between his ideas and modern string theory - his work was seminal in the development of a topological approach to fluid dynamics7. When Leon Lichtenstein published his book on hydrodynamics in 1929, two of the eleven chapters were dedicated to topological ideas. But the difficulty of an immediate testing of these ideas limited for many years the use 6

A good historical account of the origins of knot theory can be found in Pieter van de Griend, "A History of Topological Knot Theory," in J. C. Turner and P. van de Griend (eds.), History and Science of Knots (Series on Knots and Everything - Vol. 11), Singapore, World Scientific, 1996, 205-260. See also the interesting overview of knot theory by M. B. Thistlethwaite, "Knot Tabulations and Related Topics," in I. M. James and E. H. Kronheimer (eds.), Aspects of Topology, Cambridge, Cambridge University Press, 1985, 1-76. 7 See R. L. Ricca and M. A. Berger, "Topological Ideas and Fluid Mechanics," Physics Today, December 1996, 28-34.

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of topological concepts. In recent years, the applications of modem results from topology and knot theory have led to new developments in the qualitative study of fluid mechanics. In particular, it has been shown recently8 that the helicity of a magnetic link (to be considered in a perfectly conducting, incompressible, viscous fluid), which is invariant under fluid flow, does reflect the topology and the geometry of the magnetic lines of forces within a magnetic link. In other words, the topology of the magnetic link

Figure 4. Tail's Table of the Ten-crossing Alternating

Knots, plate VIII, 1885.

8 See S. J. Lomonaco, jr., "The modern legacies of Thomson's atomic vortex theory in classical electrodynamics," in L.H. Kauffman (ed.), The Interface of Knot and Physics, Proc. Symp. Appl. Math., Vol. 51, Amer. Math. Soc, 1996, 145-166; and M. H. Freedman, Z.-H. He, Z. Wang, "Mobius Energy of Knots and Unknots", Ann. Math., 139 (1994), 1-50.

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g,L, as expressed in knotting and linking, creates a barrier to the full dissipation of the magnetic link's energy, i.e., E^gJ^) has a positive lower bound that results from the topology of g,L. Thus, the magnetic link will reach a non-trivial stable and invariant energy state, much as Kelvin conjectured his atomic vortices would. Another important point, related to the foregoing question, is the problem of finding the "best" shape, i.e., a shape minimal "energy", for knots and links. More specifically, the new research objective is to find flows on isotopy classes of knots and links that terminate in a "best" shape for each given knot or link type, and then to use this "best" shape to develop new knot invariants. The arguments of Kelvin and Tait in favour of the idea that atoms were knotted vortex tubes of ether may be summarised as follows9. (1) Stability. The stability of matter might be explained by the stability of knots, i.e., their topological nature. (2) Variety. The variety of chemical elements could be accounted for by the variety of different knots. (3) Spectrum. Vibrational oscillations of the vortex tubes might explain the spectral lines of atoms. From a modern twentieth-to twenty-first-century point of view we could, in retrospect, have added a fourth. (4) Transmutation. The ability of atoms to change into other atoms at high energies could be related to cutting and recombination of knots. As we have seen above, Tait undertook an extensive study and classification of knots. He enumerated knots in terms of the crossing number of a plane projection and also made some pragmatic discoveries, which have since been christened "Tait's conjectures". An important class of knots is that of alternating knots. A diagram (of a knot) is alternating if (i) it has at least one crossing; (ii) the curve goes alternately over and under at successive crossing points; (iii) the diagram has no 'nugatory' crossing /^^^ , where the shaded disc represents a portion of knot diagram that may or may not be trivial. Thus, although the definition of an alternating knot is a simple matter, it is nevertheless non-trivial. Furthermore, a reduced alternating diagram can readily be understood intuitively, i.e., it is a regular diagram that we cannot change into one with fewer crossings. However, it is only quite recently that this has been mathematically and rigorously proven10. The study of alternating knots has a long history, and over this period a panoply of its characteristics had been proven, but the Tait's conjectures (concerning the primary local problems) had haunted researchers but proved unyielding. These were the problems/conjectures that concerned Tait in the 19th century and from which a specific type of knot, namely, the alternating knots, emerged. The new invariants, the Jones polynomial and the skein (Kauffman) polynomial have played the

9

Here we follow Michael Atiyah in The Geometry and Physics of Knots, "Lezioni Lincee," Cambridge, Cambridge University Press, 1993, 6. 10 See K. Murasugi, "Jones polynomials and classical conjectures in knot theory," Topology, 26 (1987), 187-194.

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crucial role in solving these problems. The three best known of Tait's conjectures can be stated briefly as follows: First conjecture. A reduced alternating diagram is the minimum diagram of its alternating knot (or link). Moreover, the minimum diagram of a prime alternating knot (or link) can only be an alternating diagram. In other words, a non-alternating diagram can never be the minimum diagram of a prime alternating knot (or link). (Here we need to recall what is meant by a prime knot. Let K be a knot. When a true (non-trivial) decomposition cannot be found for K, we say that K is a prime knot. (In this sense, it is equivalent to the way we define a prime number, i.e., a natural number that cannot be decomposed into the product of two natural numbers, neither of which is 1. Thus, a knot K is either a prime knot or can be decomposed into at least two non-trivial knots.) Second conjecture. Suppose that Dx and Di are two reduced alternating diagrams of an alternating knot (or link) K, then the writhe number of D, is equal to the writhe number of D2, that is: vv(D,) = w(D2). Moreover, when D is a reduced alternating diagram, then this number is an invariant of K. (Recall that if D is an oriented regular diagram of an oriented knot (or link), then the sum w(D) of the signs of all the crossing points or D is said to be the Tait number of D (or the writhe of D). Third conjecture. Suppose that D, and D2 are two reduced alternating diagrams of an alternating knot K. Then we can change D, into D2 by performing a finite number of flips. (This conjecture has very recently been shown to be true.) Despite the great strides made by topologists in the twentieth century the Tait's conjectures resisted all attempts to prove them until the late 1980s. The new Jones invariants turned out to be extremely useful and enabled several of Tait's conjectures to be established. The new polynomial invariant of links discovered by Vaughan Jones in 198411 is capable of distinguishing many knots from their mirror images, whereas the Alexander polynomial (the other most important knot invariant) is unable to distinguish between links and their mirror images. So, for example, the trefoil and its trefoil image can be distinguished. What was remarkable was that this new invariant could be defined, as we shall see, using very simple skein relations. 1.1. Polynomial invariants and Reidemeister moves The Jones polynomial is a polynomial in ;, r 1 assigned to a knot K in IP. It is denoted by V/^t). It is normalised so that V(t) = 1 for the unknot (the standard unknotted circle in R2). Moreover it has the key property VK*(t) = V^r1), where K* is the mirror image of K. The Jones polynomial can be defined generally, for any oriented link L (i.e. each component of L is oriented). Reversing the orientation of all components leaves the Jones polynomial unchanged. This explains why, for a knot, orientation is irrelevant. Before to going on, we need at this stage to give some preliminary definitions related to the notion of Reidemeister moves, which played a very important role in the development of knot theory. A link is an embedding of a collection of circles into the oriented three-dimensional sphere or, equivalently, into three-dimensional Euclidean space. A link of n circles is

" V. F.R. Jones, "A polynomial invariant for links via von Neumann algebras," Bull. Amer. Math. Soc., 129 (1985), 103-112. See also "Hecke algebra representations of braid groups and link polynomials," Ann. Math., 126(1987), 335-388.

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said to have n components, and a 1-component link is called a knot if it cannot be continuously deformed into a standard (unknotted) circle. A knot (or link) invariant, by its very definition, does not change its value if we apply one of the elementary knot moves. It is often useful to project the knot onto the plane, and then study the knot (or link) via its regular diagram. We must now ask ourselves what happens to, what is the effect on, the regular diagram if we perform a single elementary knot move on it? This question has a beautiful answer. Two diagrams are equivalent (or, mathematically stated, represent the same isotopy class) if and only if one can get from the first to second by a sequence of (local) deformations known as the Reidemeister moves (from the name of the German mathematician who wrote the first book on topological knot theory)12. Each of these moves, usually called I, II, and III, consists in modifying a small portion of a link diagram while keeping the rest fixed. We will call a small portion of a component of a link, as it appears in a diagram, a strand. Move I consists in modifying a diagram in a neighborhood containing only a single strand by putting a "twist" in the strand. Move U consists in "cancelling" a pair of crossings, as shown below. Move III consists in sliding a strand under a crossing as shown below. (Note that we can also interpret this move as sliding a strand over a crossing.)

Figure 5. Reidemeister moves.

In fact, these Reidemeister moves are sufficient to get between any two diagrams representing the same isotopy class of knots (or links). In other words, he proved that if we have two distinct projections of the same knot, we can get from the one projection to the other by a series of Redemeister moves and planar isotopies. Therefore, according to Reidemeister, there is a series of Reidemeister moves that takes us from the first projection to the second (figure 8.2 shows one series of moves that demonstrate this equivalence). The idea of the proof is simple; in performing an isotopy of a link, its 12

K. Reidemeister, Knotentheorie, Berlin. Springer, 1932; reed, by Chelsea, New York, 1948.

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projection onto the plane will occasionally encounter certain "catastrophes" where it does something non-generic, as in figure below; if one looks at the link diagram before and after one of these catastrophes, it will be seen that a Reidemcister move has occurred.

€) ® ( Figure 6. A Reidemeistcr move's calastrophe.

One can now define an equivalence (isotopic deformation) between two regular diagrams D and D' of knots K and K. Definition. If we can change a regular diagram D to another D' by performing, a finite number of times, the operations (moves) I, II, III and/or their inverses, then D and D' are said to he equivalent: D « D'. Suppose that D and D' are regular diagrams of two knots (or links) K and K', respectively. Then K = K' <-> D ~ D'. Therefore, a knot (or link) invariant may be thought of as a property (quality) that remains unchanged when we apply any one of the above Reidemeister moves to a regular diagram. It is interesting to play around with the Reidemeister moves and discover that some knots are isotopic to their mirror images, while other are not. The trefoil is not isotopic to its mirror image, so there are really two forms of trefoil, the right-handed and left-handed trefoil, shown in figure 7. On the other hand, the figureeight knot is isotopic to its mirror image; such knots are said to be amphicheiral.

a

b

Figure 7. Isotopic and non-isolopic knots and their mirror images. Reidemeister moves and Alexander polynomials are among the most important invariants in topological knot theory. The Alexander polynomial makes it possible to distinguish between knots that otherwise look very similar, or, conversely, to detect new invariants which permit to recognise when two or more seemingly different knots are in fact equivalent. The polynomial, written AA(r), is constructed according to the numbers

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of different kinds of crossing in a knot diagram. A simple trefoil knot, for example, has A^f) = t - 1 - \lt. Differently deformed versions of the same knot have the same Alexander polynomial; knots with different polynomials are different. Yet two knots with the same A are not necessarily equivalent. The Alexander does not distinguish, for example, between the square knot and the granny knot. Reidemeister moves simplify the analysis of knots. Two knots are the same if and only if their diagrams can be made identical by some combination of (two-dimensional) moves. A knot is not perceptively changed if we apply only one elementary knot move. However, if we repeat the process at different places, several times, then the resulting knots seem to be completely different.

Figure 8.1. Perko's pair of knots.

Figure 8.2. Reidemeister moves and planar isotopy. The figure-eight knot is equivalent to its mirror image.

Let us take, for example, Perko's pair of knots (see below). In appearance, Perko's two knots look completely different from each other. In fact, for the better part of a hundred years, nobody thought otherwise. However, it is possible to change the one into the other by performing the elementary knot moves a significant number of times, as

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showed in 1970 by the American lawyer K.A. Perko. More precisely, we say that two knots K, and K2 are equivalent if there exists an orientation-preserving homeomorphism of R? that maps K, to K2. Moreover, if two knots Kt and K2 that lie in S3 are equivalent, then their complements P-K, and S3 - K2 are homeomorphic. Knot invariants are mathematical expressions that describe certain properties of knots. They derive their name from the fact that they do not change according to the way the underlying knot is pushed, pulled or twisted. The path from a loop of string to an equation in powers of t is a complex one, however. How do we determine an invariant, for example, the Alexander or Jones polynomial, of a given knot? The simplest method is the skein relation invented by British mathematician John Conway. Another technique, the Kauffman "states model," is notable for its elegance and profound connection to statistical mechanics. A knot is a line drawn in three-dimensional space, which begins and ends at the same point, and which is never broken. These figures (see figure 9) represent all knots including the circle (a), associated with the trivial knot, or

SS'CQ Figure 9. Knols associaled with Ihc Irivial knot (the circle).

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non-knotted. The nature of a knot depends neither on its form nor on its size, nor on the position of its loops: knots (c) and {d) are equivalent, as are knots (e) and (f), and knots (g) and (h). On the other hand, the circle (a) seems to be of a different nature from the knotted curve (£>). The primary task of the knot theory is to prove that such knots are clearly differentiated. However, there are knots which though seems apparently different, they are actually intrinsically equivalent or isotopic, as for example Perko has shown, or as that is the case for Kinoshita's and Conway's non-trivial knots, which have AK(t) = 1 for Alexander's polynomial. The Jones polynomial is an invariant of oriented links. One can first define it for links equipped with both framing and orientation, and then prove that it is independent of the framing. In addition, the Jones polynomial is invariant under Reidemeister move I. If we represent a link by a general plane projection with over/under crossing the Jones polynomial can be characterised and computed by a skein relation. To every link L Jones associated a Laurent polynomial VL(t) in the variable Vf, which is calculated inductively by the number of crossings according to the following scheme: First choose an orientation of the link and project it onto the plane. Any link can be transformed into a simpler link (with less crossing) by replacing a suitably chosen overcrossing by an undercrossing. This procedure is illustrated in Figure 10 for the positive trefoil L =: L+, where the L_ produced in this way is already equivalent to the unknot. We must also consider a third link L0, obtained from L+ by replacing the chosen overcrossing by the uncrossing. The Jones polynomial of L+ is obtained from the polynomials of the simpler links L_ and L0 by the so-called skein relation: Vijt) = t2VL_(t) + ?(Vr- iHov^Ht). For this algorithm to make sense, it must still be shown that different choices of orientation and projection of the link lead to the same polynomial. Even more: equivalent links have identical Jones polynomials.

Figure 10. The Jones polynomial.

One of the early achievements of modern topology was the discovery in 1928 of the Alexander polynomial of a knot or a link, which has become one of the cornerstones of knot theory. It can be shown that the Alexander polynomial was the same as the L,polynomial, even though Kurt Reidemeister's approach is independent of the work of James Alexander. The Alexander polynomial is closely connected with the topological

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properties of the knot13. Also of great significance is that there are various methods by which we may calculate the Alexander polynomial; one of these is that called the Seifert matrix (from the name of the German mathematician Herbert Seifert who discovered it). Roughly speaking, the Seifert matrix of a knot is a very combinatorial, topological and useful tool that make it possible to show how various surfaces can be constructed from the same knot diagram: it is sufficient that the linking numbers for all these surfaces be equal or rearranged in such way that they turn out to be equivalent. He described a method for associating with each knot a polynomial such that if one form of a knot can be topologically transformed into another form, both will have the same associated polynomial. Now, suppose M is the Seifert matrix of knot (or link) K, then det (M + MT) is an invariant of the knot K. This invariant, discovered by Alexander, is called the determinant of K. Furthermore, the determinant of a knot (or link) K is completely independent of the orientation assigned to K. It is here important to underline that the Alexander polynomial arises from the homology of the infinite cyclic cover of the complement of a knot. Equivalently it can be derived from considering cohomology of the knot complement with coefficients in a flat line-bundle. Let us define mathematically, in a summarized way, the Alexander polynomial. Let K a S3 be an oriented knot or link, and F c S 3 a connected oriented spanning surface for K. Let 9 : H^F) x tf,(F) -> Z be the Seifert pairing. Definition 1. Two polynomials fit), g(t) e Z[t] are said balanced (written / = g) if there is a non-negative integer n such that ± ffit) = g(t) or ± fg(t) =f(t). Definition 2. Let K, F, 6 be as above. The Alexander polynomial, Ag(f), is the balance class of the polynomial Ax(t) = D(6 - tQ'). (It follows from S-equivalence (due to Seifert) that this determinant is well defined on isotopy classes of knots and links up to multiplication by factors of the form ± f.) Again few words are in order to clarify the last definition. The topologist Herbert Seifert established an important result in 1927 according to which there can be different surfaces for isotopic knots. Stated differently, a given knot or link can have many various spanning surfaces. For example, two isotopic diagrams will have rather different Seifert surfaces. How are all the different surfaces spanning a knot related to one another? The answer is, in principle, surprisingly simple. Consider the following way to complicate spanning surfaces (see Kauffman (1987)): 1) cut out two discs, Z),, D2; 2) take a tube Sl x / and embed it in S3 disjointedly from the surface, but with the tube boundary attached to 8D\ and 8D2. This is called doing a 1-surgery to the surface.

13 J. W. Alexander: "A Lemma on Systems of Knotted Curves," Proc. Nat. Acad. Sci., Vol. 9, 1923, 9395; "Topological Invariants for Knots and Links," Trans. Amer. Soc, Vol. 30, 1928, 275-306; J. W. Alexander and G. B. Briggs, "On Types of Knotted Curves," Ann. Math., Vol. 28, 1928, 562-586. See, for example, the classic work on knot theory by R.H. Crowell and R. H. Fox, Introduction to Knot Theory, New York, Ginn and Company, 1963.

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F

F after surgery

The reverse operation consists in finding a curve a on F such that a bounds S3-F. Then cut out a x I from F and cap off with two D2's. This is a 0-surgery. It simplifies the surface (i.e., reduces genus). These two different surgery operations give us different surfaces with the same boundary. We have thus the following definition and theorem: Definition. Let F and F be oriented surfaces with boundary that are embedded in S3. We say that F and F are S-equivalent (noted Fs F) if F may be obtained from F by a combinations of 0-surgery, 1-surgery and ambient isotopy. Theorem. Let F and F be connected, oriented spanning surfaces for ambient isotopic links L, L 5 3 . Then F and F are 5-equivalent. Finally, we have the Corollary. Let K and K be ambient isotopic knots or links with connected spanning surfaces F (for K) and F (for K). Let 9 be the Seifert pairing for F and \y be the Seifert pairing for F. Then 6 and i|/ are S-equivalent. In 1984, after nearly half a century in which the main focus in knot theory was the knot invariant derived from the Seifert matrix, Vaughan Jones announced the discovery of a new invariant. Instead of further propagating pure knot theory, this new invariant and its subsequent offshoots unlocked connections to various applicable disciplines. In particular, the Jones polynomial can be constructed using methods from statistical mechanics, quantum groups, etc. Hence, knot theory was once again entwined with fields outside pure mathematics, generating a tremendous amount of interdisciplinary research and virtually spawning a whole new area of research. Before to give the mathematical definition of the Jones polynomial, let us recall some basic facts. It is a straightforward task to show how to transform a braid into a knot and how to create a group from the braids. Therefore we have the following correspondence: knot

=>

braid =>

braid group, Bn.

Suppose we can map the braid group Bn into some sort of algebraic system, say, A, whose structure we understand, for example, the group of invertible matrices, or more generally, an algebra such as a group ring in which the sum and product have been defined. The aim is to be able to represent an arbitrary knot by an element of A. An initial stumbling block is that the correspondence knot

=>

braid

is not in 1-1 correspondence. To be more precise, to a single knot we may assign an infinite number of braids. So it can be, thanks to the Markov's theorem, that each knot

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corresponds to only one M-equivalence class. Therefore, when a braid a corresponds to a certain value, say <(>(a), then if this value <|>(a) is the same for any other-equivalent braid (3, it follows that this <|>(a) is an invariant of the knot Ka formed from the braid a. So, from the first condition of M-equivalence <|> must have the same value for a and yarf1. Now, if we want to represent the braid group by some algebraic system, A, it must have a similar structure to Bn. Further, A should have a simpler or more restricted algebraic structure than Bn; otherwise, we will probably not gain any further insight into Bn. Jones found that one of the (operator) algebras, which he was studying for other purposes, had a structure that resembled that of the braid group. By means of this insight, Jones was able to define a function that was invariant under both the Markov moves, M\ and Mi- This function could, in fact, be written in terms of a complex number q. Following from, it was possible to assign to each knot a complex number. To be more exact, to every knot it became possible to associate a Laurent polynomial in this complex number q (if we replace q by an indeterminate t, then we shall recover the usually defined Laurent polynomial). This polynomial, one of the new invariants, is now called the Jones polynomial. From the point of view of knot theory, the definition of the Jones polynomial runs as follow: Definition. Suppose A" is an oriented knot (or link) and D is a (oriented) regular diagram for K. Then the Jones polynomial of K, Vj^t), can be defined (uniquely) from the following two axioms. The polynomial itself is a Laurent polynomial in \lt, i.e., it may have terms in which V? has a negative exponent. (We assume (•ft)2 = t)). The polynomial Vg(t) is an invariant of K. Axiom 1: If K is the trivial knot, then V^t) = 1. Axiom 2: Suppose that D+, D-, Do are skein diagrams (see figure below), then the following skein relations holds.

lit vD+(t) - tvDst) = (Vr - v4i)vDo(t).

« D+

D_

) D0

Figure 11. Regular skein diagrams of tree knots (or links).

A number of years after the discovery of the Jones polynomial, an unexpected relationship was found between the Alexander polynomial and in a sense a "truncated" form of the Jones polynomial. In fact, the algorithm to calculate the Jones polynomial is completely analogous to the one of the Alexander polynomial; i.e., it is necessary to form a skein tree diagram. Only the coefficients of the two skein relations are different. If one calculate the Jones polynomial of the trivial (i-component links, 0„, VoM), we get

v0ll(t) = (-iy-l(J + vfy-1.

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As in the Alexander polynomial case, let us write down the following equalities: VD+(i) = t2VD_{t) + tzVDo(t)

vDXt) =

r2vD+(t)-rlzVDo
wherez = (\ft - l/yjt). We can now draw the skein tree diagram for the calculation of the Jones polynomial of the right-hand trefoil knot (see figure below). It follows from the skein tree diagram that VJKO = t2V0(t) + thVooit) + t2z2V0(t) = t +

t3-A

vtz

G vtz

Figure 12. The skein tree diagram for the calculation of the Jones polynomial of the right-hand trefoil knot.

It is now interesting to compare the above Jones polynomial with the one of Alexander drawn bellow. D = DA

"SD +1/

D„=D +

—LD.

(S

+iy

D„

3 Figure 13. Skein operations on the regular diagram of the right-hand trefoil knot.

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Here, we suppose that K is the right-hand trefoil knot and D is a regular diagram of K, which in the above figure is the topmost diagram. Within the dotted circle on D, one can perform a skein operation. Since within this circle the crossing point is positive, it is better to rename this regular diagram D+. By performing a skein operation, D+ is transformed into two other regular diagrams: one, D_, is the regular diagram obtained by changing the original positive crossing point to a negative crossing point; the other, DQ, is obtained by removing the positive crossing point (by splicing the regular diagram at this crossing point). Now connect D+ and D_ (and, similarly, connect D+ and Do) by drawing a line segment and assign +1 (respectively z) to the line segment. The appropriate assignment follows directly from the coefficients of VD_(Z) and Va0(z). So for our skein tree diagram we have our first pair of branches, and they correspond to I ^ D J Z ) + ZVD0(Z) in the evaluation of Vo+(z). It is straightforward to see that £>_ is

equivalent to the trivial knot, and hence VD_(Z) = 1. Therefore D_ will not produce any further branches, i.e.; we cannot perform a subsequent skein operation here. However, Do is not equivalent to the trivial knot or links, and so we can again perform a skein operation within another dotted circle. Since within this circle (see the above figure), the crossing point is also positive, let us rename DQ as D+. Then, as before, let us denote the subsequent left-hand diagram by D_ and the right-hand diagram D0. Again, we draw line segments and assign to them coefficients by means of the bellow operations. It is easy to see that D_ and D0 are, respectively, the trivial 2-component link and the trivial knot. Hence no further branches may be formed and our skein tree diagram for K is complete. V^(z) can now be calculated as the sum of the Conway polynomial of the each terminating trivial knot (or link) multiplied by the product of the coefficients (on the line segments) along the uniquely determined branch path that begins with our original regular diagram, D, of K and terminates, by construction, with the regular diagram of this trivial knot (or link). Therefore, in the above we obtain the following sum : V/Kz) = 1 V0(z) + zVooiz) + z2V0(z). Since V0(z) = 1 and V 00 (z) = 0, the calculation collapses down to Vjdz) = 1 + z2. Therefore, by applying the previous proposition we obtain

Vjtfr) = 1 + (VF - 1/Vo2 = r 1 - 1 +1. The skein tree diagram for the Conway polynomial for the figure 8 knot (or Listing knot) is given bellow (figure 14). A first very important result (a theorem) was proven in 1933 by the German mathematician Herbert Seifert14. It states that, given an arbitrary oriented knot (or link) K, then there exists in R} an orientable, connected surface, F, that has as its boundary K. In other words, there exists an orientable connected surface that spans K, called a Seifert surface of K. The orientation of F is induced naturally from the orientation of the knot K that forms its boundary. This result obtained by Seifert is pretty amazing. On first thought, it's hard to imagine how to get any orientable surface with one boundary component such that the boundary component is knotted at all. We are supposed to take

See also H. Seifert and W. Threlafall, Lehrbuch der Topologie, Teubner, Leipzig, 1934.

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a surface like a torus with one boundary component and embed it in space so that the boundary circle is knotted into a trefoil knot. Seifert's algorithm tells us that not only can we embed surfaces in space with a knotted boundary component but we can do so to get any knot whatsoever.

(positive crossings)

o •<§><§)-

(negative crossings)

o o - <§*«. *<e> • o Figure 14. The skein tree diagram for the Conway polynomial for the figure 8 knot.

Suppose we want to construct such a surface for a particular knot K. Starting with a projection of the knot, choose an orientation on K. At each crossing of the projection, two strands come in and two strands go out. Eliminate the crossing by connecting each of the strands coming into the crossing to the adjacent strand leaving the crossing. Now all of the resultant strands will no longer cross. The result will be a set of circles in the plane. These circles are called Seifert circles. Each circle will bound a disk in the plane. Since we do not want the disk to intersect one another, we will choose them to be at different heights rather than having them all in the same plane. Now we would like to connect the disks to one another at the crossings of the knots by twisted bounds. The result is a surface with one boundary component such that the boundary component is the knot. In fact, the surfaces that we are generating are always orientable. Notice that the Seifert's algorithm can be used to generate lots of different surfaces for the same knot. We could alter the projection of the knot and then obtain a surface that at least looks different. Thus, there can be different surfaces for isotopic knots. Stated differently, a given knot or link can have many various spanning surfaces. The surface, F, constructed from K by the above method, has genus g = \ll(d - s + 1), where d is the number of crossing in K and s the number of Seifert circles arising from K. You might expect this number to be an invariant of the knot; however, a

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different projection might give a different surface and hence a different g. To avoid these problems one can make the following definition: Definition: the genus g(K) of a knot K is the minimal genus of all orientable surfaces which span K. The genus thus defined gives an invariant of the knot, but the form of definition makes it difficult to calculate. Since the Seifert construction does not necessarily give a minimal surface, we can only say that g{K) < l/2(d - s + 1). The second aspect concerns the intrinsically theoretical problem of classifying knots (and links). In algebraic geometry, this is the general classification of manifolds. One of the most important, even fundamental, applications of this general problem in knot theory is the following. One can show that it is possible to create from an arbitrary knot (or link) a 3-dimensional manifold. Hence by studying the properties of knots one can gain insight into the properties of 3-manifolds. Here we shall restrict ourselves to mention two fundamental ways, both of which are closely related to knots, of constructing 3-manifolds. One approach is via Dehn surgery15, while the other is via covering space. Throughout all the knots and links shall be considered to lie in S3. Suppose that K is an oriented knot in Si. Now, 'slightly' thicken K, so that we form a 3dimensional manifold called the tubular neighborhood of K, and denoted by V(K). Since AT is a knot, V(K) is a solid torus, and its boundary dV is a torus. The concept of covering space is one of the most important in topology. Intuitively speaking, the covering space M of space M has locally the same structure as M. Globally, however, M be said to be a 'large' space that covers M equally several times. J. W. Alexander (1888-1971) described a method for associating with each knot a polynomial, so that if one form of a knot can be topologically transformed into another form, both will have the same associated polynomial. He then proved that the polynomial was an especially powerful tool in knot theory; for example, it was able to distinguish 76 knots out of the first 84 in the knot tables; they were found to have unique Alexander polynomials. Now, suppose M is the Seifert matrix of knot (or link) K, then ldet(M + MT)\ is an invariant of the knot K. This invariant, discovered by Alexander, is called the determinant of K. Furthermore, the determinant of a knot (or link) K is completely independent of the orientation assigned to K. Alexander's polynomial proved to be a fairly powerful invariant of isotopy in knots. Differently deformed versions of the same knot yield the same polynomial Ag. We need here to recall another important result in knot theory, due to Max Dehn16. In short, it states that a left-hand trefoil knot cannot be deformed into a right-hand trefoil knot. Dehn's proof is based on the observation that a deformation of one trefoil knot into the other would induce an automorphism of the knot group which associates certain geometrically significant elements, namely "latitude" and "longitude" curves on the neighbourhood torus of the knot, which determine an orientation of the ambient space. The trefoil knot is the simplest knot, i.e. the simplest interwoven space curve which cannot be continuously transformed in space into a circle. It is the only knot with a plane projection having only three double points. In space one can distinguish two different kinds of trefoil knot, which result from each other by reflection, say, in a plane. The 15 See M. Dehn, "Uber die Topologie des dreidimensionalen Raumes" ("On the topology of threedimensional space"), Math. Ann., 69 (1910), 137-168; English edition in Max Dehn, Papers on Group Theory and Topology, translated and introduced by J. Stiwell, Springer-Verlag, New York, 1987. 16 M. Dehn, "Die beiden Kleeblattschlingen" ("The two trefoil knots"), Math. Ann., 75 (1914), 402-413.

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problem is then to find a way of distinguishing the two kinds of trefoil knot in space topologically, i.e. to show that the left knot is not convertible into the right by a continuous space transformation. This enables to point out a very remarkable phenomenon. The space with a left trefoil knot and the space with a right trefoil knot are homeomorphic, i.e. so composed that the knots occupy corresponding regions in the two spaces. Despite this the two knots in the same space are not convertible into each other by a continuous transformation of space. Thus we have before us a kind of topological symmetry, called "mirror symmetry", which plays nowadays a fundamental role in the attempts to explain the essential features of space and matter. 1.2. Some remarks on non-linear science and knots theory Generally speaking, according to the topological theory of forms, that is topology, there may be a continuous deformation of a geometrical object into another one that leaves certain characteristic (global) properties of the primitive object unchanged; such is the topological notion of homeomorphism, that is to say of qualitative equivalence between objects or, more generally, spaces whose shape can be thought of as being the same. In other words, topology is the study of the way that lines, surfaces and manifolds connect to themselves in space. Roughly, the mathematical theory of analogy-;', e. the homology and cohomology theory-rests on the knowledge of these classes of equivalence relations. In more rigorous terms, a continuous map / : X —> Y is called a homotopy equivalence if there is a "homotopy inverse" map g: Y —» X, i.e. such that the two composites g°/: X —> Y and f°g: Y —» X are homotopic to the respective identity maps

\x:X^X(lx(x)

= x\\Y--Y^Y(lY(y)

= y).

The spaces X, Y are then said to be homotopy equivalent, X ~ Y, or to have the same homotopy type. Suppose now that the following conditions hold: X is a subspace of Y (or embedded in Y); f: X —> Y is the inclusion map; g: Y -> X restricted to X is the identity map on X; and, finally, throughout the homotopy F deforming f°g: Y —> Y to the identity map 1 y, we have F(x, f) = x for all JC e X c Y. In this situation the subspace X is called a deformation retract of Y. For instance any open region Y of R" has a deformation retract of lower dimension. The whole Euclidean space R" has any point as a deformation retract. Spaces Y with the latter property are said to be contractible (over themselves) or homotopically trivial: Y ~ 0. A retraction of a space Y onto a subspace X c 7 i s a map /: Y —> X with the property that the restriction of / to X is the identity map on X: f\x = \XThe space X is then called a retract of Y. It is by using these powerful concepts that notably the mathematicians H. Whitney, R. Thom, S. Smale, and V. Arnold proceeded in his well-known theory of singularities, by defining the spatial concepts of boundary, but also of the fold, as singularities, qualitative discontinuities (such as broken symmetries, bifurcations, attractors, caustics, etc.), and archetypal schemas-that is as an unveiling of the original dynamic process. They actually did not consider the knot as a generic and generative mathematical form in his theory. But the knot, which yields a rich structure when it is question of building a homology and cohomology between the parts of space or different phenomena unfolding within it, could well be viewed in this manner.

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The concept of a knotted form could be theorised, in a morphodynamic approach, as a qualitative structure emerging from the substantial "interiority" of matter, and able to be apprehended when it manifests itself phenomenologically, through fine mathematical models. For example, our macroscopic world abounds in knotted and linked rings. But even at the nanoscopic scale, works in molecular topology have shown that Nature builds DNA molecules, which form actual knots and links between themselves (see section 3). Qualitative similarities may be observed between forms of the microscopic world as well as those of the macroscopic world, and by extricating the structures, it is possible to study their dynamics. In fact, over the last twenty years, knots and links became an appealing subject notably in "chaos theory" and "non-linear science," and they found broad application to the methods and ideas of dynamical system theory. The basic world of a dynamical system is its state space: a (smooth) manifold, M, which constitutes all possible states of the system, and a mapping or flow defined on M. It is well-known that in the case of systems of first order ordinary differential equations, the vector field generates a flow <j),: M —» M, t e R. The general problem tackled by dynamical systems theorists is to describe <(>, geometrically, via its action on subsets of M. This implies classification of the asymptotic behaviours of all possible solutions, by finding fixed points, periodic orbits and more exotic recurrent sets, as well as the orbits which flow into and out of them. In many applications (|>, also depends on external parameters, and the topological changes or bifurcations that occur in M as these parameters are varied, are also of interest. In studying these and related phenomena, one abandons the fruitless search for closed form solutions in terms of elementary or special functions, and seeks instead qualitative information. Now we may ask: why knot theory and dynamical systems are close together? The answer lie in the following key idea: a closed (periodic) orbit in a three-dimensional flow is an embedding of the circle, S', into the three-manifold that constitutes the state space of the system, hence it is a knot. Similarly, a finite collection of periodic orbits defines a link17. Several natural questions immediately arise, directed at the following goal: given a flow, perhaps generated by the vector field of a specific ordinary differential equation (ODE), describe the knot and link types to be found among its periodic orbits. Do nontrivial knots occur? How many distinct knot types are represented? How many of each type? Do well-known families, such as torus knots, algebraic knots, or rational tangles, appear in particular cases? In any cases: Are there "new" families of knots and links which arise naturally in certain flows? Do Hamiltonian and other systems with conservation laws or symmetries support preferred families of links? Do "chaotic" flows contain inherently richer knotting than simple (Morse-Smale) flows? Indeed, how complicated can things get? - is there a single ODE among whose periodic orbits can be found representatives of all knots and links? Indeed, in 1976 it has been conjectured that nontrivial knotting occurs in a well-known set of ODEs called the Lorenz equations18. From the dynamical point of view, in contrast, one might seek to use knot and link invariants to describe periodic orbits and so help them better understand the underlying 17

J. Franks and M. C. Sullivan, "Flows with knotted closed orbits," in Handbook of Geometric Topology, R. J. Daverman and R. B. Sher eds., Amsterdam, Elsevier, 2002, pp. 471-498. 18 See R. F. William, "The structure of Lorenz attractors", in A. Chorin, J. Marsden, and S. Smale (editors), Turbulence Seminar, Berkeley 1976/77, Springer Lecture Notes in Mathematics, 1977, 94-116.

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ODEs. In a parameterised family of flows, for example, one can observe sequences of bifurcations in which a simple invariant set containing, say, one or two periodic orbits, "grows" into a chaotic set of great complexity, containing a countable infinity of periodic orbits. In many cases, the periodic orbits are dense in the set of interest; sometime that set is a so-called strange attractor. The existence-uniqueness theorem for solutions of ODEs implies that, as periodic orbits deform under parameter variation, they cannot intersect or pass through one another. Knot and link types therefore provide topological invariants which may be attached to families of periodic orbits. Can such invariants be used to identify orbit genealogies - to trace the bifurcation sequences in the two-parameter family of maps introduced by Henon, which provides a model for Smale's famous horseshoe map? Can operations in which new knots are created from old, such as composition and cabling, be associated with specific local bifurcations? Is the complexity of knotting related to other measures of dynamical complexity, such as topological entropy? Does knot theory provide finer invariants than entropy for the classification of flows? Let's only mention that R. W. Ghrist, P. J Holmes and M. C. Sullivan (1997) showed that in a parameterised family of flows, periodic orbits appear and disappear in (often complicated) sequence of bifurcations. But for three-dimensional flows, it is the link of periodic orbits which undergoes bifurcations. Thus, if (1) one "dresses" the periodic orbit set with knotting and linking information; and (2) one computes the topological action of bifurcations on orbits, one produces a set of bifurcations invariants derived from knot theory. For example, the periodic orbits implicated in a saddle-node or pitchfork bifurcation of a three-dimensional flow are isotopic knots and have the same linking number with any other orbit which persists through the bifurcation point. 2. Knots, Gravity, and Quantum Field Theory 2.1. Braid group and statistical mechanics The discovery by Vaughan Jones in the 1980s of a new knot's invariant has been very important not only for mathematics, but also for theoretical developments in many other fields. Instead of further propagating pure knot theory, this new invariant and its subsequent offshoots unlocked connections to various applicable disciplines. In particular, the Jones polynomial can be constructed using methods from statistical mechanics19, quantum groups, etc. Hence, knot theory was once again linked with fields outside pure mathematics, generating a tremendous amount of interdisciplinary study and virtually spawning a whole new area of research. Jones constructed his polynomial via a representation of the Artin braid group (noted B„) into a certain von Neumann algebra. Note at the outset that one can transform a braid into a knot and also create a group from the braids, as we shall explain more in detail below. Consider a set of n vertical strings starting and ending in two rows of n horizontally aligned points. The lines can cross each other an arbitrary number of times, forming a braid. Now arrange the lines in such a way that at each horizontal level there is only one crossing at the rth strand, which we denote g,. One can describe such a braid by a sequence gigjgk-.- . Such an ordered sequence states that if one follows the braid 19 See V. F. R. Jones, '"Knot Theory and Statistical Mechanics," in New Developments in the Theory of Knots, T. Kohno ed., World Scientific, Singapore, 1990.

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from the top to the bottom (or vice-versa) one encounters a twist of the strings at the ith and (i + l)th positions followed by a twist of the strings at the j'th and (/' + l)th positions and so on, as shown in figure 15. Each twist has two possible orientations, denoted by g,and gf1. The twists g,- form a group structure, called the Artin braid group. For n strings Bn has n - 1 generators g,,..., g„_, that satisfy the relations (2.1) (2-2)

gigj = gjgi, gigi+\gi = gi+\gigi+\,

Ii-;'I>1, i
which can be easily checked by drawing n strings and applying the twists. The strings involved in the braid group can be thought of as the space-time trajectory of particles in 2 + 1 dimensions as they orbit around each other. This suggests an immediate connection between the braid group and (2 + l)-dimensional physics. What is the relation with knots? One simply obtains a knot or a link by gluing together the ends of a braid in an order preserving manner (this is called a closure of the braid). Conversely, one can associate to each knot a braid. In fact, according to a theorem of Alexander, given an arbitrary (oriented) knot (or link), then it is equivalent (with orientation) to a knot (or link) that has been formed from a braid. Therefore several properties of knots can be coded in the language of braids. The first question that arises is: given two braids, what are the conditions for their closure to yield the same knot or link up to ambient isotopy? Let's first recall that knots are classified into types in the following way: If K\ and Ki are two knots then K\ is said to be equivalent to Ki if there is a homeomorphism a : R 3 —> R3 under which ^1 is mapped into K^\ we write: aK\ = K^. This clearly defines an equivalence relation on the set of all knots and the elements of each equivalence class are called knots of the same type. A somewhat finer classification of knots is that of isotopy type: Let {at}, f e [ 0 , 1] be a family of homeomorphisms of R 3 where at(x) is continuous in t and x and cxo is the identity; then, if a.\Ki = K2, K\ and K2 are said to be of the same isotopy type. Evidently knots of the same isotopy type are also equivalent; however the converse is false. This is because the continuity in t of {a,} causes all the a, to have the same orientation properties, namely they all preserve orientation since (Xo, being the identity, does so. Hence isotopic knots K\ and K2 satisfy aK\ = K2 where a preserves orientation, but, in general, aK\ = K2 does not imply that a is orientation preserving. For knot diagrams the answer to the last question is given by the Reidemeister moves. In terms of braids they translate themselves into a set of moves called the Markov moves. Reidemeister moves of type (iii) are already included in the braid group relation (2.2). Reidemeister moves of type (ii) are almost included in the relation (2.1), except for the fact that gt f g2g\g2~K whereas both gx and g2£i#2 ' y ' e ^ t n e s a m e n n k under closure. The fact is that to implement fully the second Reidemeister move in terms of braids, one has to identify elements that are conjugate under the adjoint action of the braid group. Two elements of the braid group that are conjugate are said to be related by a Markov move of type 1. Reidemeister moves of type (i) imply that a link diagram associated with the closure of a certain braid b e Bn and the closure of the braid bgn±x e Bn+\ are equivalent. These two elements are said to be related by a Markov move of type 2.

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i

i+1

1 2

3

Figure 15.1. Graphical representation of g, and Sift"'-

M,

ux* iix^1 (a)

(b)

Figure 15.2. Markov moves, (a) shows howjS = cfyOf'cTi, an element of B3) changes when M\ is applied; namely,^ becomes 20 J7', where in this case y- Oicrf'. (b) shows how^J = c^cr'c^oi"1, an element of B3, changes when we apply M2; i.e., we obtain the elementfi
One very interesting fact is that the Jones algebra is formally identical to the Temperley-Lieb algebra, an algebra arising in the study of the Potts model. The Potts model is a generalisation of the Ising model in statistical mechanics. In this way, a fertile interrelationship between statistical mechanics and knot theory suddenly emerged. This relation is not merely Platonic. In fact, it soon became apparent that statistical mechanics could be done on a link diagram and produce new invariants of links in the process. This opened the way for the discovery of a number of new invariants of knots and links that generalised the original Jones polynomial. The parameters involved in doing statistical mechanics on a link diagram are for the most part not physical, but the form of the mathematics is the same as that for the physical theory. In particular, a matrix relationthe Yang-Baxter equation-turns out to be directly related to the knot theory. The YangBaxter equation is a statistical mechanics model, which leads to new invariants of knots. This model corresponds to the (exactly solvable) partition function describing the macroscopic properties of matter with a very high number of elements (atoms and molecules), which has been shown to be closely related via quantum groups to invariants of knots and links. Moreover, the Yang-Baxter equation is essentially nothing other than the Reidemeister move DI. In other words, the partition function is unchanged by the regular moves II and HI. So, in essence, it should provide us with a regular invariant of knot. Therefore, finding solutions to the Yang-Baxter equation seems to imply that we may then transform these solutions into invariants of knots. By doing mathematical

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physics on topological structures, all sorts of new invariants and a new way of thinking about topology has emerged. The renaissance in the interactions between physics and knot theory (recall Gauss's use of properties of knots in the solution of an electromagnetic problem, and Thomson's plans to describe chemistry in terms of knots) was due to statistical mechanics studies. Nowadays there are three other different ways in which physics and knot theory are related. Not only topological quantum field theory, but also the theory of quantum invariants has proved to be closely related to conformal field theories20. In this connection, it is important to note that E. Witten has shown that the Jones polynomial and its generalisations are related to the topological Chern Simons actions. More precisely, he attempted to understand the new link invariants (of Jones) in a systematic way by relating them to the vacuum expectation values of Wilson loops in ChernSimons theory. Using the invariance of the Chern-Simons action under orientationpreserving diffeomorphisms, these vacuum expectations values should be isotopy invariants. One of the most fruitful developments in the last decades in theoretical physics is the work relating quantum field theory and topology in three dimensions. Thus Witten was led to construct link invariants for closed 3-manifolds. The basic definitions of knot theory are the same for links in arbitrary manifolds: a link in a manifold Y is simply a submanifold of Y diffeomorphic to a collection of circles, two links L, L are said to be ambient isotopic if there is a smooth one-parameter family a, of diffeomorphisms of Y mapping L to L'. Now, for the purposes of knot theory, one can replace R3 by Y3 (a compact and oriented 3-manifold) by adding a 'point at infinity.' Any link in R3 can then be regarded as a link in Y3, and two links in R3 are isotopic as links in Y3 if and only if they are isotopic as links in R3. Thus isotopy classes of links in Y3 are in one-to-one correspondence with those in R3. A typical picture is shown below.

Figure 16. Links and Closed Three-Manifolds. Consider a pair (Y, L) where /"is an oriented 3-manifold as before and L c Y is an oriented 1-manifold. If Y has boundary £ then L is assumed transversal to £ and dL a X is an oriented O-manifold, i.e. a collection of signed points. If Y is closed then L is just an oriented link in Y. Our link L, and hence its boundary, is also assumed to carry some further information.

2.2. Knots invariants and topological Chern-Simons theory Let us discuss more in detail the relation between knot theory and Chern-Simons theory, and notably the profound relevance of knot polynomials (knot and link invariants) for the topological quantum field theory. We hope to succeed at least in 20

See T. Kohno (ed.), New Developments in the Theory of Knots, Singapore, World Scientific, 1990; C. N. Yang and M. L. Ge (eds.), Braid Group, Knot Theory, and Statistical Mechanics, Singapore, World Scientific, 1989.

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showing that there is a very intimate connection between both theories and that the one can be expressed and understood in terms of the other, and conversely. First, let us stress that the Jones polynomial is a very powerful knot and link invariant, more powerful than the Alexander polynomial A(r). The Alexander polynomial is not strictly speaking a polynomial, it can contain negative as well positive powers of t; such polynomials are referred to as Laurent polynomials or L-polynomials. Given a knot K its Alexander polynomial A(f) is not unique, rather it is only determined up to a factor ±t" for some positive or negative n. The Jones polynomial is an integer Lpolynomial and it originated in the theory of finite dimensional von Neumann algebras. It is also an invariant of isotopy type and is able, contrary to the Alexander polynomial, to distinguish the trefoil knot from its mirror image. If we denote the Jones polynomial for a general link by V[£f) then, if L is the trefoil and L its mirror image, we have (2.3)

Vdt) = t +fi-14,

VL(t) = Vt + l/fi - 1/t*.

VLO) has some interesting properties extending those already given for the Alexander polynomial A^,. Consider the case where L is just a knot K, then we have (2.4)

VK (e2t./3) = i,

dVK(f)ldt\,=l = 0.

These properties can be easily checked for the case of the trefoil above. The Jones polynomial has been generalised to a homogeneous polynomial in three variables by Freyd et al.2[; this polynomial-known as the Homfly polynomial after the initials of its authors-can reduce, in special cases, to the Alexander polynomial and the Jones polynomial. Also, both for the Jones and Homfly polynomials there exist distinct links on which these polynomials have the same values. The Alexander polynomial for any link can be computed inductively as we now describe. Suppose a link has n crossings, then we can relate its Alexander polynomial to that of another link with only (n - 1) crossings. This is done via the skein relation (see further): Let L+, L_ and LQ three links which are identical except for the interior of a small disc where they are as displayed below

X X )( L+

L_

LQ

Notice that if the L+ have n crossings then L0 has (n - 1) crossings. The Alexander polynomials of these links are related by the simple equation (2.5) 21

A i+ (0 - AL_(0 + (r>« - H'2)ALo(0 = 0.

Freyd P., Yetter D., Hoste J., Lickorish W. B. R., Millet K. C. and A. Ocneanu, "A new polynomial invariant of knots and links," Bull. Amer. Math. Soc, 12 (1985), 239-246.

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Iteration of this relation expresses any Ai(t) in terms of the Alexander polynomial for a finite number of unlinked, unknotted circles. Thus it is only the Alexander polynomial for these latter that we need to calculate. In a similar way, Jones has shown that Vi{t) satisfies (2.6)

tVL+(t) - r ' V L ( 0 + (fin - f^Vi^t)

=0

sometimes written diagrammatically as

'X

-f-> V

+(f" 2 -r" 2 )

I =0.

Hence Vi{t) is also determined by its values on a finite number of unlinked, unknotted circles. If we now take this example so that L consists of p unlinked, unknotted circles then VLO) is given by (2.7)

Vdt) ={-(!-

i-')/(/i* - r" 2 ) J""1.

Witten has shown how to determine Vi(t) from certain correlation functions of the Chern-Simons topological field theory22. Here we discuss Chern-Simons theory and knots, and we deal with the calculation of Vi£t) itself. Witten's work has the great benefit of providing an intrinsically three-dimensional definition of the Jones polynomial, something which is otherwise lacking. In fact many knot calculations are carried out by various projections from three into two dimensions: this can create a technical problem in that differing knots may have the same projection, and such calculations must be shown to be independent of the particular projection used. An interesting feature of the quantum field theory of the Jones polynomial calculation is that, as well as the topological Chern-Simons theory, the machinery of conformal field theory is an indispensable part of the construction. Let's now define the very important concept and the associated formalism of "skein relation", introduced by Louis Kauffman in the 1980th, and other related notions. A knot (or link) polynomial is an assignment of a finite set of numbers to a knot (or link) that is invariant under ambient or regular isotopies. Given a knot K one gets a polynomial P{K)q in an arbitrary variable q such that all the coefficients pi{K) of the polynomial are knot invariants. An important point is that for each knot the polynomial is of finite order, but the order depends on the particular knot. Why are these objects interesting? The reason is they are an ordered way to assigning an unlimited number of invariants to knots according to their complexity. There is therefore the expectation that they could constitute a systematic procedure for classifying knots. Moreover, some of the polynomials are defined by quite succinct recursion relations called skein relations. The price for all this is high: there are only a handful of polynomials explicitly known at

22

E. Witten, 'Topological quantum field theory," Commun. Math. Phys., 117(1988), 353-386; "Quantum field theory and the Jones polynomial," Commun. Math. Phys., 118(1988), 351-400. See also C. Nash and S. Sen, Topology and Geometry for Physicists, Academic Press, London: 1983.

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present. Take for example the Alexander-Conway polynomial C{K)q. It is defined by the skein relations C(U)q=h C(L+)g-C(L_)q = qC(L0)q, where U is the unknot (a knot isotopic to a circle) and L±, L0 refer to the crossings shown in the figure below.

L+

L_

Lo

Figure 17. The crossing L± and LQ.

The way in which the skein relations are to be interpreted is the following. The first one is simply a normalization condition that states that the polynomial evaluated for the unknot is 1. To read the second relation consider a specific knot and focus on a point where the is a line crossing. The previous last relation states that if one evaluates the polynomial for the knot where the crossing we excised is replaced by the crossing L+ and subtracts from it the polynomial evaluated for the same crossing replaced by L_ one gets as a result q (the polynomial variable) times the polynomial evaluated for the crossing replaced by L0. The resulting equation is a relationship between the polynomials associated with three different knots. The strategy is to apply the relationship recursively combined with the Reidemeister moves until one gets a system of equations for the coefficients with a unique solution. The skein relations are relations between projections of the knots and it is quite non-trivial that the polynomial they define is independent of the projection. Another important polynomial is the one due to Jones, V(K)q. The skein relations that define it are V(U)q=l, qV(L+)q - rW(L_)q = (q»i -
W(R, Q = trP exp

[fcA]

In this notation R stands for an irreducible representation of G, tr is the trace in the R representation and P is the familiar path ordering symbol. Clearly C can be a knot and, if

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235

we consider p Wilson lines, we have a p-component link L. The normalised correlation function associated with these p Wilson lines is

(2.9)

< w(/?„ c])...w(R„, cp) > = (i/z(M)) IDA W(RU CX)...W(RP, Cp) exp [-(ikIAiz) fM tr\A A dA + 2/3 A A A A A)]

where Z(Af) is the partition function. Thus the Chern-Simon action is purely a differential topological quantity; introducing a link L in the guise of the Wilson lines does not require any non-topological data. In this way we see that the Wilson correlation functions < W(RU C,)...W(RP, Cp) > have a chance of being purely topological. Now we commit ourselves to the choice M = S> since this will enable us to proceed at once to an explicit calculation. The quadratic action, together with the exponential dependence on A of a Wilson line, allows the entire integrand to be written as a Gaussian after completing the square. The calculation of the functional integral rests just on the calculation of a Green's function which, for S3, is an elementary computation. The result is that (2.10)

<W(nuCl)...W(np,Cp)> = exp [{Mk)zijk X"(m=l nlftJCi dxf J^ dyi ((x-y)Vlx - yl3)) J

with x' and yi local co-ordinates on the knots C, and Cm. The basic integral in (2.10) is the linking number L(Ch Cm) of Gauss. L(Ch Cm) is one of the oldest invariants in knot theory and originated, at least in part, in a study of Ampere's law of electromagnetic theory. Its definition is (2.11)

L(C„ Cm) = &i,J47c)fCi dxi fCm dyi ((x - yfl\x - yl3))

and, from Ampere's laws, where precisely the integral occurs, it follows that L{C,, Cm) is an integer. Let us now describe the Chern-Simons method for calculating the Jones polynomial V[£t). To accomplish this task the attractive topological properties of the Chern-Simons action discussed in the previous section need to be supplemented by something further. The main ingredient of this supplement is the machinery of two-dimensional conformal field theory. Some insight into how the conformal field theory enters can be obtained as follows. Suppose M is a 3-dimensional manifold inside which there are various knots. If we consider some two dimensional (Riemann) surface £ inside M then, near 2, M may look like £ x R and this surface will, in general, be punctured by some of the knots. Thus the presence of the knots suggests that we consider, not just J , but Y. w ith a collection of punctures. But Riemann surfaces with punctures constitute the primary data for a conformal field theory; so we are led to a quest for a relation between the conformal field theory on £ and knots inside M. The quantization of the Chern-Simons action suggested by this picture can now be pursued. For example, if we chose the A0 = 0 gauge on J x R then it becomes natural to regard the phase space as the moduli space Mp of equivalence classes of flat connections on £. This space has a natural symplectic structure and is a compact Kahler manifold; thus quantizing on this phase space should give finite dimensional quantum Hilbert space H^ or Hj_ „• These spaces are the sections of the k'h power of the determinant line

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bundle over Mp. The conformal blocks are also spaces of sections and, in the axiomatic approach to conformal theories, vector bundles on the moduli space Mp also occur. The spaces # £ can now be identified with the VQ]) of the conformal field theory on £. To begin our calculations we need a slightly more compact notation for our Wilson line correlation functions. We introduce the quantity W(M, {/?}) and write (2.12)

< W(/f1,C1)...W(/?p,C,) > = W(M, [R})IZ(M), where {/?} = Ru..., Rp

The symbol {R} serves to record the representation content of the Wilson lines and from now on the group G is taken to be SU(2). If we wish to be more explicit we may also write out {R} in full; for example, if p = 2, we could write (2.13)

W(M,RUR2)

When p = 0 we shall write simply (2.14)

W(M)

which we observe is equal to the partition function Z(M). Finally the normalised correlation function is written as W(M, {R}) where (2.15)

W(M, {/?}) = W(M, {R})IZ{M).

Our first goal will be to derive the skein relation (2.16)

tVL+(t) - t-'VL_(t) + (*•« - r^V^t)

=0

which, in the quantum field theory approach, amounts to the statement that any three vectors in a two dimensional vector space are linearly dependent. To start this derivation we take our manifold M with knots inside and decompose it as a connected sum (2.17)

M = M,#M2

where it is required that none of the knots in M pass through the two sphere joining M, and M2. This requirement means that we have two sets of knots : one in M, and the other in M2; we denote these by [Rm] and {Rm} respectively. Next we wish to show that (2.18)

Z(5') W(M,{R)) = W(M„{fl<'>}) W(M2,{R™}).

Now we consider the state of affairs just before the connected sum operation joins M, to M2. M, has a disc excised from it giving a manifold M, with 3M, = S1; when a manifold has a boundary an associated functional integral is heuristically a wave function on this boundary or, more precisely, an element of a Hilbert space H^ defined on this boundary. This space Hp is that for the conformal field theory on S2. A similar remark applies to M2 except that the orientation of its boundary is opposite to that of M,, which has the consequence that the two Hilbert spaces are dual to one another. If we denote the two vectors in the Hilbert spaces by V, and V2 then we have

Topological Knots

(2.19)

Models

237

W(M,{R}) =

where the inner product is induced by the dual pairing of the Hilbert spaces. We can think of this equation as expressing a factorisation of probability amplitudes similar to that encountered in string theories. This connected sum process can be applied to S3 itself in the case where S3 has no knots inside. This gives the equation (2.20)

Z(y) = < £/„ U2 >

where [7, and U2 are two more vectors in H^. But S2 has no marked points and so corresponds to the trivial representation of G. Since corresponds to M, "evolving" into M ]; and the same applies to M2, the RHS is easily seen to be (2.21)

W(M„ {/?<»}) W(M2, {/?»>})

and so we have accomplished our task and shown that (2.22)

Z(S3) W(M,{R}) = W(M„{Rm}) W(M2,{R™})

which we rewrite in the form (2.23)

(W(M,{R})/Z(&)) = [(W(M„{i?<')})/z(53))] [(W(M,{/?<2>})/Z(S3))].

We can now contemplate decomposing M many times, yielding (2.24)

M = M,#...#MP

where we still insist that no knots pass through any of the joins. Induction then shows that (2.25)

(W(M,{R})/Z(Si)) = (W(M,{/?<»})/Z(S3)) ... (WW,{fl

})/Z(,S3)).

This result can immediately be applied to the case where M = S3 and {/?} consists of p unlinked, unknotted circles; in that case it will certainly be possible to decompose S3 into exactly p components each of which is both a copy of S3 and contains only one circle. This reduces the calculation for p unlinked, unknotted circles to that for a single unknotted circle; we have (2.26)

(W(S\ {R}/Z(S')) = (W(53,fl,)/Z(S3))...(\V(S3,Rp)/Z{S*)) or W(S\ {R}) = W(S\ «,)-W(5 3 , Rp).

Now we must get to the skein relation itself; this is done by abandoning our earlier restriction that no knots may pass through the join in the connected sum Mt#M2. To this end let we decompose M = M,#M2 as before but this time we suppose that the knots have produced 4 punctures in the S2 joining M, to M2; we further require all 4 representations at these punctures to be the 2-dimensional defining representation of SU(2). This means that the Hilbert space H^RRRR = H^4 of the 4-fold punctured S2 has dimension 2-this

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can be seen by nothing that the physical Hilbert space must be 5f/(2)-invariant and the trivial representation occurs precisely twice in the tensor product R ® R ® R ® R. Now suppose that we carry out such a decomposition three times and ordain things so that the only difference between the three decompositions is that, when we look into M2, 2 of the 4 strands are arranged according to the three skein configurations

X L+

L_

LQ

This gives us three equations like 2.19; using +, - and 0 for over, under and zero crossing we have (2.27)

M = Ml#M+2, M = Ml#M~1, M = Ml#M°2 W(M, {R+}) = , W(M, {/?-}) = , W(M,{R°}) = .

But, because H^ 4 is only two dimensional, the three vectors V+2, V~2 and V°2 are linearly dependent, that is (2.28)

a+V+2 + orV-2 + a°V°2 = 0 =>a+ W(M,{R+}) + a- W(M,{R-}) + a° W(M,{R°}) = 0

and this is the skein relation. We are now in a position to calculate W{M,{R}) explicitly for the case when M = S3. By (2.26) we only need to calculate W{S*,{R}) for a single unknotted circle. To do this we use the skein relation (2.29)

tW(M,{R+}) -t'W(M,{R-})

+ (tm -r" 2 ) W(M,{R°}) = 0

with {R+ {representing the single unknotted circle. It is elementary to verify that this choice has the consequence that {/?+} = {/?-} while {R0} consists of two unlinked, unknotted circles; hence, using the normalised correlation functions, we have (2.30)

tW(P,{R+}) -t'W(S\{R-}) + (r"2 - tm)W(S\{R°}) = 0 =>(f - rW(S\{R+}) + {tm- r"2){W(S3, {/?+}) }2 = 0 3 + =>W(5' ,{/? }) = - ((f- f-')/(f"2 - f-"2)) .

It follows from (2.26) that for a link L = {R} consisting of p unlinked, unknotted circles we have (2.31)

W(S\{R})=

{-((r-f-')/(f" 2 -t-" 2 ))}''

Topological Knots Models

239

Thus if we analytically continue t to real values and write (2.32)

Vdf) = ~ ((tm - rmW-

*-')) W(P,{R})

then VL(t) is the Jones polynomial since it satisfies the skein relation (2.16) and agrees with (2.7) when L consists of p unlinked, unknottedjurcles. Finally we must describe the transformation in W(M,{R}) under a change in framing. Let M = M, u z M2 represent a decomposition of M obtained by cutting into two so that the boundaries of M, and M2 are the Riemann surface £. Suppose, then, that M has been cut into these two pieces and a framed Wilson line passes through £ at a point P. If we apply a 2%t Dehn twist around P then the framing of C will change by an amount t. Hence the procedure for changing the framing of C is to cut M as instructed and then to apply the Dehn twist to P e SMj and then rejoin M, and M2. The effect of the Dehn twist is detected as a linear transformation on the space H^ and, from conformal field theory, we can calculate that this transformation is just scalar multiplication by the factor exp[27u7/i/{], where fiR is the conformal weight of the primary field in the R representation. Summarising, the transformation law of W(M,[R}) is (2.33)

W(M,{R})^>exp[2mthR] W(M,[R}).

2.3. Knots theory, geometrodynamics and quantum gravity Another line of development of knot theory in theoretical physics is related with Wheeler's quantum geometrodynamics23, which has taken up again and developed in new and unexpected directions the long-term vision of W. K. Clifford and A. Einstein according to which there is nothing in the world except curved empty space. On the classical level the Einstein-Maxwell equations are the basic set-up of geometrodynamics. They are studies in otherwise empty space-time admitting multiconnected manifolds as spacelike hypersurfaces. Charge may thus be regarded as "flux lines trapped in topology" and mass as manifestation of the energy of the electric and magnetic field. The main extension of this familiar model is to assume that these wormhole initial hypersurfaces W3 = S1 x S1 are in general knotted embedded in spacetime, thus accepting Wheeler's proposal to describe an elementary particle as "a knottedup region of high curvature." The interest in knotting results partly from the possibility of introducing concisely the concept of the "scattering" of embedded manifolds. Thus "wormhole scattering" is a means of changing not only the topology, but also the knot invariants of the scattered hypersurface; i.e., intuitively, the incoming and outgoing manifolds are differently knotted and linked. A characterisation of links according to the number of their basic constituents with respect to topology change can be established as a first result of the "scattering" concept.

J. A. Wheeler, Geometrodynamics, New York, Academic Press, 1962.

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<* _

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m

- higher links of knol wormholes

R" "lime-

Figure 18. "Vacuum Sea" of "Virtual" Knot wormholes. As examples a link h£W2) of a trivial wormhole with a "trefoil" wormhole is drawn in figure 18e, whereas links of unknotted wormholes are shown in figure 18d. Links and knots can exhibit important discrete symmetries, which likewise will appear at least in wormhole links L U (W 2 ). Consider the following inversions of knots: (a) Reversing the string orientation: K - * K; (b) Taking the mirror image: K -> K = {(x\ x2, -x^/x" e K}. Knots obeying the following equivalence relations are called: (i) amphicheiral, K = K; (ii) reversible, K = K; involutory, K = K. As an example, the figure-eight handle (figure 18c) enjoys all these symmetries, whereas the trefoil x is known to be only reversible. Left- and right-handed trefoil wormholes are shown in figures 18b and 18b, respectively.

According to general relativity theory, the framework in which the Universe has to be understood should be viewed as an elastic (both extensible and contractile) spacetime woven by light and curved by matter. In the attempt to construct a theory of quantum gravity, the physicist theorist Abhay Ashtekar introduced in 1990 a formalism in which the spacetime manifold constitutes a valid approximation at large distances, whereas at very short distances the geometry might be viewed as a sort of weaving interwoven by a complex system of threads. More precisely, Ashtekar proposes to replace Riemannian geometry, which successful provides the mathematical framework to formulate general relativity (as well as other modern theories of gravity), with what he called "a nonperturbative quantum geometry." In this new framework, Riemannian geometry emerge only as an approximation on a large scale, upon a suitable coarse graining of the semiclassical states of the full-fledges quantum theory. So then what is the mathematical framework for describing the "atoms" of geometry, its "elementary excitations", its "basic quanta"? Following Ashtekar, one can say that the resulting quantum geometry is strikingly different from the familiar continuum descriptions. In particular, the triad operators-as well as area operators associated with intersecting surfaces-fail to commute giving rise to certain intrinsic uncertainties in the simultaneous measurements of geometric quantities. The basic excitations are 1-dimensional, rather like polymers. And geometry is "quantized" in the old-fashioned sense of the word: geometrical quantities such as length, areas and volumes, which take continuous values classically, are represented in the quantum theory by operators with purely discreta spectra.24 24

A. Ashtekar, "Geometric Issues in Quantum Gravity", in S. A. Huggett et al. (eds), The Geometric Universe. Science, Geometry, and the Work of Roger Penrose, Oxford Univer. Press, 1998, 175-76.

241

Topological Knots Models

The upheaval brought recently by string theory in our conception of space and spacetime is still much more revolutionary. It is well-known that the "elementary"25 particles in terms of which the structure of matter is understood are described as points in space. Forces between particles result from exchanges between them of other elementary particles, and the fundamental interactions occur in a local way, at coincident time at the same point in space. For example electromagnetic interactions forces between charged particles result from the exchange of photons, with local couplings to the charged particles. In string theory, particles are not described as points, but instead as strings: one-dimensional extended objects. If Ms and M (Rx Ms) denote the space and spacetime manifolds respectively, then we can picture strings as follow:

JVI

•

'fixed time)

point particle

\^S

J

open string

closed string

M (space-time picture) Figure 19. A string representation of particles and spacetime.

While the point particle sweeps out a one-dimensional -worldline, the string sweeps out a worldsheet, i.e. a 2-dimensional real surface. For a free string, the topology of the worldsheet is a cylinder (in the case of a closed string) or a sheet (for an open string). Roughly, different elementary particles correspond to different vibration modes of the string just as different minimal notes correspond to different vibrational modes of musical string instruments. In string theory the theory of spacetime is coded in the laws by which the strings propagate. One consequence of replacing world-lines of particles by world-tubes of strings is that Feynman diagrams get smoothed out. (One can think about a Feynman diagram intuitively as representing a history of a spacetime process in which particles interact by the branching and rejoining of their world-lines, although they are used in the perturbative quantum field theory more conventionally to calculate scattering

25

To the present day limit of distance scales up to approximately 10~15 cm, electrons and quarks do not show any substructure, and that is why they are viewed as 'elementary particles'. However, which particles are elementary in this way may change over time, since at smaller distance scales substructure may be found in terms of more elementary particles.

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amplitudes). World-lines join abruptly at interaction events, but world-tubes join smoothly. String theory, if correct, entails a radical change in our concepts of spacetime. That is what one would expect of a theory that reconciles general relativity with quantum mechanics. Thus, once one replaces ordinary Feynman diagrams with strings ones, one does not really need spacetime any more, at least any background fixed spacetime; one just needs a two-dimensional field theory describing the propagation of strings. And perhaps more fatefully still, one does not have spacetime any more, except to the extent that one can extract it from a two-dimensional field theory. 3. The Role of Knots in Enzymatic Processes into the Cell We would now like to offer some observations on the interesting connection between knot theory and biology and, hence, with life. For many years, molecular biologists have known that the spatial conformation of DNA knots are phenomena that make up part of the physical universe and living matter, that is to say that knots and links are macroscopic as well as microscopic objects carrying a tremendous amount of precious information on the emergence of new forms in nature and the functions of organisms, and that knotting and unknotting are "universal" principles underlying these processes. In particular, this spatial conformation and the proteins that act on DNA are vital to their biological functions26. Furthermore, differential geometry and knot theory can be used to describe and explain the 3-dimensional structure of DNA and protein-DNA complexes. Biologists devise experiments on circular DNA which elucidate 3dimensional molecular conformation (helical twist, supercoiling, etc.) and the action of various important life-sustaining enzymes (topoisomerases and recombinases). These experiments are often performed on circular DNA molecules, in which changes in the geometric (supercoiling) or topological (knotting and linking) state of DNA can be directly observed. The information content of DNA molecule is embodied in its sequence of paired nucleotide bases and is independent of how the molecule is twisted, tangled or knotted. In the past decade, it has become clear that the topological form of a DNA molecule has an important influence on how it functions in the cell. Enzymes called DNA topoisomerases, which convert DNA from one topological form to another, appear to have a profound role in the central genetic events of DNA replication, transcription and recombination. It is a long-standing problem in biology to understand the mechanisms responsible for the knotting and unknotting of DNA molecules. Large amounts of DNA are wound up and packed into the average cell. In fact, there is enough DNA in a two metre human body to stretch from the earth to the sun and back fifty times! This of course means that the embedding of the DNA in the cell is exceedingly complicated.

26

See on this subject J. Wang, "DNA topoisomerases," Ann. Rev. Biochem., 54 (1985), 665-697; N. R. Cozzarelli, "Evolution of DNA Topology: Implications for its biological roles," in Scientific Applications of geometry and Topology, De Witt L. Sumners ed., Proc. Sympos. Appl. Math., Vol. 45, Amer. Math. Soc, 1992, 1-16; J. H. White, N. R. Cozzarelli, W. R. Bauer, "Helical Repeat and Linking Number of Surface-Wrapped DNA," Science, 241, July 1988, 323-327; S. A. Wasserman, N. R. Cozzarelli, "Biochemical Topology: Applications to DNA Recombination and Replication," Science, 232, May 1986, 951-960.

Topological Knots Models

243

The DNA in the cell knots and unknots, ties and unties itself according to a definite scheme. Knots and links appear during replication and recombination. Certain enzymes (topoisomerases, which behave like topological entities in living organisms) are responsible for the knotting and unknotting. They are able to cut a strand of DNA at a particular point, grasp another strand, pass it through the opening and then close the opening. In other words, these enzymes replace overerossing by undercrossing (see figure below). A molecule of DNA is usually thought of as two linear strands intertwined to form a double helix with a linear axis, or as a ring consisting either of a single strand or of two strands wound in a double helix; moreover, both kinds of DNA ring can be converted to other topological configurations. Quite different geometrical manipulations (the complete operation of breaking, passage and resealing) of the strands are needed to accomplish such topological conversions. Actually all the transformations are based on the same general mechanism. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking), the enzyme mechanism can be characterised and explained (see further the picture). building blorK* of ONA phosphate sugar

sugar phosphate

base

[Q| lil Q

doubl«-«fat-d«d DNA

.^fe-. Nt

o o o o D D

ss iW IWTTT

K

UJ

niKlaolida ONA double hell*

5'

a o

o n o

so:o

sugar-phoaphate backbone

1M o 3 £

rO o IKX o o ME <) o AJdJJ J k>> 5' L-T—13hydrogen-bonded baaepalra

Figure 20 (a). DNA and its building blocks. DNA is made of four types of nucleotides, which are linked covalcntly into a polynucleotide chain (a DNA strand) with a sugar-phosphate backbone from which the bases (A, C, G, and T) extend. A DNA molecule is composed of two DNA strands held together by hydrogen bonds between the paired bases. The arrowheads at the end of the DNA strands indicate the polarities of the two strands, which run antiparcllel to each other in the DNA molecule. In the diagram at the bottom left of the figure, the DNA molecule is shown straightened out; in reality, it is twisted into a double helix, as shown on the right.

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minor groove

major

",'...

: G

O

hyiifoyfii bond

—"phosphodiosw bond

v

5 on.)

3' e n <|

181

Figure 20 (/>). The DNA double helix. (A) A space-filling model of 1.5 turns of the DNA double helix. Each turn of DNA is made up of 10.4 nucleotide pairs and the ccnler-lo-centcr distance between adjacent nucleotide pairs is 3.4 nm. The coiling of the two strands around each other creates two grooves in the double helix. As indicated in the figure, the wider groove is called the major groove, and the smaller the minor groove. (B) A short section of the double helix viewed from its side, showing four base pairs. The nucleotides are linked together covalently by phosphodiester bonds through the 3'-hydroxyl(-OX) group of one sugar and the 5'-phosphate(P) of the next. Thus, each polynucleotide strand has a chemical polarity; that is, its two ends are chemically different. The 3' end carries an unlinked -OH group attached to the 3' position on the sugar ring; the 5' end carries a free phosphate group attached to the 5' position on the sugar ring.

template S strand gene A

gene B n

I I

sano

r-

C > DNA double Mb

geno oxpr

J protein A

J proiWn B

Sst'.ind 5-07 i ' ( T " 6o _ oQ 0

C6

.

,

3-?S' strand Q Q O

9

9

,

v

•••

/ MM

'- J 3'' 0~^3'

, immi '

5'

30 O O O P now S'slrand

5'1

.

.

O O P

....._...._

9

0 0

.',y

pBtent DNA double helb protein C

3J> P P O P P O O template S' strand

/?T9\9S'

Figure 21 (a). The relationship between genetic information carried in DNA and proteins, (b) DNA as a template for its own duplication. As the nucleotide A successfully pairs only with T, and G with C, each strand of DNA can specify the sequence of nucleotides in its complementary strand. In this way, doublehelical DNA can be copied precisely.

245

Topological Knots Models

3'

5'

6' I

3'

v

<> c: 5'

3'

Right-handed double helix

3' U

^S'

Left-handed double helix

Figure 21 (c). Two possible helical forms of DNA are mirror images of each other. The geometry of the sugar-phosphate backbone of DNA causes natural DNA to be right-handed. The geometry and the chemical composition of the subunits of DNA determine that when hydrogen bonds are formed between bases, the strands assume the form of a double helix with the sugar-phosphate backbones on the outside and the base pairs in the middle. The helix is right-handed: it has the same direction of twist as an ordinary wood screw. The structure of the double helix was first analyzed in crystalline fibers, where it was found that the strands revolve about each other completely once every 10 base pairs; this structure is designated the Watson-Crick B helix. Recent results from Wang's group at Harvard show that in DNA molecules in solution the strands of the double helix make a full turn every 10,5 base pairs.

Genes carry biological information that must be copied accurately for transmission to the next generation each time a cell divides to form two daughter cells. DNA encodes information through the order, or sequence, of the nucleotides along each strand. Each base - A, C, T, or G - can be considered as letter in a four-letter alphabet that spells out biological messages in the chemical structure of the DNA. However, as is known, organisms differ from one another because their respective DNA molecules have different nucleotide sequences and, consequently, carry different biological messages. But how is the nucleotide alphabet used to make messages, and what do they spell out? Now it was known well before the structure of DNA was determined that genes contain the instructions for producing proteins. The DNA messages must therefore somehow encode proteins (figure 21a). This relationship immediately makes the problem easier to understand, because of the chemical character of proteins. Nevertheless, the properties of a protein, which arc responsible for its biological function, are determined in turn by the linear sequence of the amino acids of which it is composed. The linear sequence of nucleotides in a gene must therefore somehow spell out the linear sequence of amino acids in a protein. The exact correspondence between the four-letter nucleotide alphabet of DNA and the twenty-letter amino acid alphabet of proteins-the genetic code-obeys a process, known as gene process, through which a cell translates the nucleotide sequence of a gene into the amino acid sequence of a protein. The complete set of information in an organism's DNA is called its genome, and it carries the information for all the proteins the organism will ever synthesize. The amount of information contained in genomes is staggering: for example, a typical human cell contains 2 meters of DNA. Written in the four-letter nucleotide alphabet, the nucleotide sequence of a very small human gene occupies a quarter of a page of text, while the complete sequence of nucleotides in the human genome would fill more than a thousand books the size of this

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one. In addition to other critical information, it carries the instructions for about 30,000 distinct proteins. At each cell division, the cell must copy its genome to pass it to both daughter cells. The discovery of the structure of DNA also revealed the principle that makes this copying possible: because each strand of DNA contains a sequence of nucleotides that is exactly complementary to the nucleotide sequence of its partner strand, each strand can act as a template, or mould, for the synthesis of a new complementary strand. In other words, we designate the two DNA strands as S and S\ strand S conserve as a template for making a new strand 5', while strand 5' can serve as a template for making a new strand S (figure 21b). Thankfully, DNA is a very malleable, deformable molecule, being able to recombine through a series of phases; otherwise the world would be populated by clones. Moreover, one of the most striking structural properties of DNA molecule is its flexibility or plasticity, which influence in an essential way its biological functions. Thus, the molecule can freely move about, although under certain chemical and geometrical constraints, in the space of the cell's nucleus and convert itself into several sorts of shapes without losing a certain structural stability and a certain energetically (biochemical) favourable condition. This movement is twofold: the three-dimensional two-strands helical structure of DNA molecule can extend and compact. The extended (unfolded) conformation of DNA, which put it under tension as if the molecule was subjected to shear one dynamic force, seems to be especially required for DNA replication. By this process each of the two strands of DNA is used as a template for the formation of a complementary DNA strand. The original strands therefore may remain intact through many cell generations. DNA compaction inside cells occurs by successive orders of coiling. A DNA double helix is compacted in about four successive steps. Only the first step of nucleosome formation is well understood. In this step, DNA coils twice in a left-handed helical fashion around a set of proteins called histones. The nucleosomes are then coiled successively to give the final form, called a chromosome. In the phases of this process (the recombination), the knot type of the DNA molecule is actually changed. The whole process, from the original splicing to the recombination, is the result of the effect of a single enzyme/catalyst called a topoisomerase. The term topoisomerase is relatively easy to explain. Chemically, two molecules with the same chemical composition but different structure are called isomers. It follows that two DNA molecules with the same sequence of base pairs but different linking numbers are also isomers. Do to the difference in linking numbers, "topologically" they are inequivalent. So, these DNA molecules are called topoisomerases, which is, hence, the enzyme that causes the linking number to change. The process of mutation due to a topoisomerase can be in simple terms be described as follows: First a strand of the DNA is cut at one place, then a segment of DNA passes through this cut, and finally the DNA reconnects itself. In figure 22 we give two examples of the action of a topoisomerase on a DNA molecule (for clarity, we have not drawn the helical twist). The place where the strand is cut is denoted by "0". The simple strand, in figure 22(a), has a single cut due to a topoisomerase and the DNA passes through it and recombines; this is called a Type I topoisomerase. While, in figure 22(b), a cut in a double-strand DNA, due again to a topoisomerase, allows a double-strand DNA to pass through it and recombine, this is as expected called a Type II topoisomerase.

247

Topological Knots Models

(a)

(b)

Figure 22 (a), (b). The action of a topoisomerase on a DNA molecule.

A molecule of DNA may also take the form of a ring, and so it can become tangled or knotted. Further, a piece of DNA can break temporarily. While in this broken state the structure of the DNA may undergo a physical change, and the DNA will finally recombine. In fact, in the early 1970s it was discovered that a single enzyme called a topoisomerase can facilitate this whole process, from the original splicing to the recombination. The process of recombination involves some interesting topological changes in the substrate. It is worth noting that knowledge of the topology of the substrate and product(s) can be use to compute the Jones polynomials of other products. For instance, a cut in a double-strand DNA, due to a topoisomerase, allows a doublestrand DNA to pass through it and recombine. Finding such a topoisomerase is relatively straightforward, since these enzymes occur in organisms small and large, from bacteria to the body of the reader of this article. The effect of a certain topoisomerase (called a recombinase) is usually called a site-specific recombination. A site-specific recombination is a local operation. The effect of the recombinase on a DNA molecule is either to move a piece of this DNA molecule to another position within itself or to import a foreign piece of a DNA molecule into it. The result is that the gene transmutes itself. The exact process of a site-specific recombination is fairly easy to understand. Firstly, two points of the same or different DNA molecules are drawn together, either by a recombinase or by random (thermal) motion (or even possibly both). The recombinase then sets to work, causing the DNA molecule to be cut open at two points on the parts that have been drawn together. The loose ends are then recombined bt the recombinase in a different combination that the original DNA molecule. In figure 23(a)~(c), we have shown a simple site-specific recombination that has been carried out in the manner described above.

(a)

(b) Figure 23. The Writhing process of the DNA molecule.

(c)

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8 (a)

(b)

-0 (c)

Figure 24. (a), (b), (c). If the orientation on the DNA molecule and the orientation induced by these local orientations agree, then this arrangement is called a direct repeat, figure 24 (a). On the other hand, if they do not agree, then the arrangement is said to be an inverted repeat, figure 24 (c). For a site-specific recombination one can show the following: (1) If the substrate is a DNA knot and the arrangement is a direct repeat, then the product is a 2-component DNA link, figure 24 (a). If, however, the arrangement is an inverted repeat, then the product is a DNA knot, figure 23 (c). (2) If the substrate is a DNA link (i.e., two DNA molecules entwined), then after recombination the product is a DNA knot, figure 24 (b).

Before the action of the recombinase, the DNA molecule is called a substrate; after the recombination it is called a product. The process of going from the DNA molecule to a state in which two parts of the DNA molecule have been drawn together, is said to be the writhing process. When at this stage the recombinase combines with the substrate, the resultant combined complex is called a synaptic complex, figure 2"(b). Within the synaptic complex, we can assign local orientation to the respective, relatively small part of the DNA molecule (or molecules) on which the recombinase acts (within the circle in figure 23(a), (b), and (c)). The following propositions follows from empirical evidence: Proposition 1. Almost all the products obtained by the site-specific recombination of trivial knot substrates are rational knots (or links), i.e., 2-bridges knots (or links). Proposition 2. The part of the synaptic complex acted on by an enzyme (recombinase), mathematically within the 3-ball, is a (2,2)-tangle (figure 25.1). Therefore, the product is just the replacement of one (2,2)-tangle by another (2,2)tangle. For example, the (2,2)-tangle within the circle Tis figure 25.1 is replaced by the tangle R to form the product shown. Mathematically, it is perfectly reasonable to consider S to be a (2,2)-tangle in T. The numerator of the sum of S and R is then the product, figure 25.2. So the following "equation" holds: N (S + R) = P (the product).

Topological Knots Models

+

249

© R

Figure 25.1. Sum of tangles in a synaptic complex; and replacement of tangles.

Figure 25.2. The numerator of the sum of S and J? is the product.

Further, we may divide the substrate into the external tangle S and the internal tangle E, since the substrate is the numerator of the sum of S and E, figure 25.3. Again, we have a quasi-equation holding: N(S + E) = S (the substrate). We already said that the double-helix structure of DNA is a geometrical entity, or more precisely, a topological configuration. This topological configuration is itself a manifestation of linking or knotting27. Further, it has been shown that when a topoisomerase causes DNA to change its form, the process is very similar to what happens locally in the skein diagrams (see above, section 2.2). Therefore, for the geometrical entity-knotted or linked-the linking number is an important concept, while the action of the toposoimerase is related to the new skein invariants. In fact, the linking number Lk between C, and C2 (where C, and C2 are the two backbone curves that form the boundaries of the ribbon B and that represent the closed DNA strands) is an invariant, and its changes have a very important effect on the structure of the DNA molecule. For example, it is known that if we reduce the linking number of a doublestrand DNA molecule, the DNA molecule will twist and coil, i.e., what is known as supercoiling. In other words, the linking number difference is a measure of supercoiling process (see figure 26). Interestingly enough, in the case of a link formed from C, and C2, the linking number defined on the DNA molecule in biology, and the linking number computed from the mathematical knot theory turns out to be the same. Moreover, the untying mechanisms

27

See N. R. Cozzarelli, op. cit.; and D. W. Sumners, "The Role of Knot Theory in DNA Research," in Geometry and Topology, Manifolds, Varieties, and Knots, C. McCrory and T. Shifrin eds., New York, Marcel Dekker, 1987, 297-318.

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lk(C,,C2)=l Wr(B) = 0, Tw(B) = 1

lk(C,,C 2 ) = - l Wr(B) = - 1 , Tw(B) = 0

(a)

(b) Figure 26. Supercoiling process in the DNA molecule.

used in cells bear an uncanny resemblance to the simplest mathematical method for generating the new polynomial invariants. The number of twists of the ribbon B along the axis C is called the twisting number, and is denoted by Tw(B). The writhe, Wr(B), can be defined as the average value of the sum of the signs of the crossing points, averaged over all projections. These numbers, Wr{B) and Tw(B) are invariants. They are not, however, invariants of the knot (or link) obtained from the DNA molecule, but differential geometrical invariants of the ribbon B as a surface in space. The three invariants mentioned above are related by the following basic formula: Lk (C„ C2) = Tw(B) + Wr(B). The application of this expression to DNA molecules is an explanation of its propensity to supercoil. The remarkable fact about this result is that two geometric quantities that may change under deformations of the curves add up to a topological quantity, which is invariant under such deformations. Moreover, the linking number has a very important topological property: it is unchanged under any continuous deformation of the pair of curves, no matter how the double-strand ring is pulled or twisted, so long as the two strands remain unbroken28. It is worth noticing that one may give a "goal-oriented" interpretation of DNA topology (hence, not reducible to a pure mechanism): i.e., why and how the topological structure of DNA evolved. All biologists are fascinated by evolutionary explanations because they seek to answer the fundamental question: Why are we the way we are? Evolutionary explanations are also heuristic in that they often provide the hunch that is behind the stated rationale for an experiment. It is extremely difficult to determine how something evolved, especially for something so basic to life as the genetic material. 28

See on this very interesting and complex subject J.H. White, An Introduction to the Geometry and Topology of DNA Structure, Boca Raton, CRC Press, 1989.

Topological Knots Models

251

However, some significant steps have been taken towards a better and deeper understanding of the structure from which the genetic material is made. DNA inside cells is a very long molecule with a remarkably complex topology. Topological properties of DNA are defined as those that can be changed only by the breaking and rejoining of the backbone. There are three important topological properties of DNA: 1) the linking number between the strands of the double helix, 2) the interlocking of separate DNA rings into what are called catenanes, 3) and knotting. The linking number of DNA in all organisms is less than the energetically most stable value in unconstrained (relaxed) DNA. This puts the DNA under mechanical stress which causes it to buckle and coil in a regular way called (-) supercoiling. The name supercoiling derives from the fact that it is the coiling of a molecule which is itself formed by the coiling of two strands around each other. Although, strictly speaking, supercoiling is a geometric property, it is a consequence of a topological one, the linking number difference between supercoiled and relaxed DNA.

Figure 27. The linking number of DNA. The winding of the two strands of the DNA double helix about each other, represented by filled and open tubes, can be measured by the linking number between the strands. It is equal to one-half the number of signed crossing of the two strands in any projection of the molecule; the sign convention is given in Fig. 28. The linking number of the small circular DNA in the diagram is equal to 12; but it is orders of magnitude larger for naturally occurring DNA.

Figure 28. DNA catenanes and knots. The duplex DNA is depicted as a tube; the arrows indicate the orientation of the DNA as defined by base sequence. The simplest example of catenanes (A) and knots (B) are shown. There are two crossings in the case of the singly linked catenanes in A and three for the trefoil knots in B. The topological sign of the crossings is given by the convention implicit in the drawings. A(-) crossing is one in which a clockwise rotation (< 180°) of the arrow on top is needed to make it congruent to the underlying one; a (+) crossing is one in which a counterclockwise motion is needed for the operation. The two possible topological isomers of each form are shown. As the number of crossings in a knot or catenane increases, the number of possible isomers grows exponentially.

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Figure 29. Negative supercoiling. Shown are interwound (left) and solenoidal (right) forms of supercoiling of duplex DNA (line) molecules of the same length and linking number. These two forms of supercoiling differ geometrically, not topologically. Free supercoiled DNA in solution adopts the interwound form, but supercoiling around proteins is usually solenoidal.

Supercoiling

o

^^k^-'

C.

Figure 30. Reactions of topoisomerases. Duplex DNA is shown as a tube and the arrows indicate orientation. Illustrated are the change in linking number by introducing (-) supercoils (A), forming a (-) singly interlocked catenane (B), and tying a (+) trefoil knot (C). The reverse reactions of removal of supercoils, decatenation, and unknotting are also carried out by topoisomerases. All the reactions are affected by the same operation of passing a segment of duplex DNA through another at the filled arrows. Because these topological changes are brought about by making duplex DNA transiently permeable to the passage of other DNA segments, the enzymes are called type-2 topoisomerases. Type-1 topoisomerases make single-stranded regions of DNA transiently permeable to duplex or single-stranded DNA.

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Metophaso chromosome

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Figure 31. Model for the packing of chromatin and the chromosome scaffold in metaphase chromosomes. In interphase chromosomes, long stretches of 30-nm chromatin loop out from extended scaffolds. In metaphase chromosomes, the scaffold is folded into a helix and further packed into a highly compacted structure, whose precise geometry has not been determined.

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Figure 31.2. DNA compaction inside cells by successive orders of coiling. A DNA double helix is compacted in aboul four successive steps (a-d). Only the first step of nucleosome formation (a) is well understood. In this step, DNA coils twice in a left-handed helical fashion around a set of proteins called histoncs. The nuclcosomes are then coiled successively to give the final form, called a chromosome (d).

•

Figure 32.1. Topoisomerases of the DNA Molecule. Type I topoisomerases, which cut one strand at a time, can carry out several topological operations. The first topoisomerase to be found, a type I enzyme of E. coli, was discovered by James Wang in 1971. Topoisomerases are now known to exist in many species, including man. By cutting one strand of a supercoiled DNA ring the lype I enzyme can put the ring into the relaxed slate (/). Il can lie a single-strand ring into a knot (2). The knot is lied when Ihe single-strand ring crosses over itself. If the two loops formed in this way are pulled together, the enzyme can cut one loop and pass the other loop through the opening. When the break is sealed, the ring is tied in a knot. Structures 2c and 2d arc topological^ equivalent: each form can be made from the other without cutting the strand. The type I enzyme can also interlock two single-strand rings (3). If the rings have complementary base sequences, a double helix results. Although the operations seem quite different, each requires that a strand be broken, a segment of DNA be passed through the break and the break be resealed.

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Double -stranded helix •

Topo I ) Binding of Topo I

\

(

Nick DNA; form covalont DNAphosphotyrosine bond

Pass cut 3' end under other strand and reseat DNA

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3 negative supercoils

Figure 32.2. Action of E. coti type I lopoisomerase (Topo I). The DNA-enzyme intermediate contains a covalcnt bond between the 5'-phosphorly end of the nicked DNA and a tyrosine residue in the protein (inset). After the free 3'-hydroxyl end of the red cut strand passes under the uncut strand, it attacks the DNA-enzyme phosphocslcr bond, rejoining the DNA strand. During each round of nicking and rcscaling catalyzed by E. coti Topo I, one negative supercoils is removed. (The assignment of sign to supercoils is by convention with the helix stood on its end; in a negative supercoils the "front" strand falls from right to left as it passes over the back strand (as here); in a positive supercoil, the front strand falls from left to right.)

:A

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®. A §, K O, SI ©, §§,

Q ®, @, H Figure 32.3. DNA knots produced by lopoisomerase I. A standard reprcscnlation of each knot produced by lopoisomerase I is shown, nol including cnanliomers of chiral knols. Knots 4, and 63 are amphichiral and knot 6C has Ihrce stereoisomers. All olher knols arc chiral with two enantiomcrs. All possible knol forms with seven or fewer nodes were produced by lopoisomerase I.

It must be emphasized that the complex topology of DNA is essential for the life of all organisms. In particular, it is needed for the process known as DNA replication, whereby a replica of the DNA is made and one copy is passed on to each daughter cell. The most direct evidence for the vital role played by DNA topology is provided by the results of attempts to change the topology of DNA inside cells. Two related questions arise immediately from the recognition that DNA topology is essential for life. How did the complex topology of DNA evolve, and why is it so important for cells? DNA is the only molecule in cells that has a complex topology. The evolution of proteins has taken a contrary course. Proteins also naturally subdivide into domains and thus local knots or links could readily occur, but they do not. In addition, no knots, catenanes, or supercoiling have been found in RNA, polysaccharides, or lipids. With the evolution of

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type-1 topoisomerases, compaction by nucleosomes could occur and the size of DNA could grow to about 105 kb. However, as DNA grew in length, the problems of accidental knotting within domains and catenation of separate domains and segregation of the products of DNA replication became acute. These problems were solved by the evolution of the type-2 topoisomerases, which promote the passage of duplex DNA through transient double strand breaks. A type-2 topoisomerase could have evolved from a type-1 topoisomerase by the development of an interaction between two copies of a type-1 enzyme. Further increases in DNA size required only the evolution of successively higher orders of DNA compaction. So we are faced, once again, with a genuine topological problem. 4. Topological Transformations and Biological Processes Explained by Pictures 4.1. Catenane and knot catalogue with representative examples of linked DNA forms All forms shown are chiral (different from their mirror image). The node sign, displayed for a single node, or crossing, in each form in the top row is determined as follows. Arrows are drawn to indicate an orientation of the DNA primary sequence. If the overlying arrow can be aligned with the underlying one by a clockwise rotation of less than 180°, the node for duplex DNA has a value of - 1. If a counterclockwise rotation produces alignment, the node has a value of + 1. Top row: (a and b) The two forms of the simplest knot, the trefoil. A trefoil is a torus knot, that is, it can be drawn on the surface of a doughnut-shaped object. Other members of the torus family of knots and catenanes are shown in (e to i). (d) A redrawing of the knot (d), illustrating why a trefoil is also a member of the twist knot family. Twist knots are formed by strand passage between two looped ends of a molecule twisted on itself. Another twist knot is shown in (k). Middle row: (e to h) The four types of multiply interwound torus catenanes are represented as doubly linked forms. Drawing of this type makes it possible to distinguish right-handed parallel (e), left-handed parallel (/), right-handed antiparallel (g), and left-handed antiparallel (h) double helical interwindings. (h') Redrawing of

Figure 33. Examples of linked DNA forms. (See S. A. Wasserman and N. R. Cozzarelli, 1986).

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catenane in drawing (h). Bottom row: (() Right-handed torus knot with seven (+) nodes. (/') Right-handed torus catenane with eight nodes, (k) Right-handed twist knot with seven (-) nodes. (/ and /') Compound six-nodded knot composed of two (-) trefoils. 4.2. Topoisomerases of the DN A Molecule The tying of knots in rings of DNA is one of the capabilities of the enzymes known as topoisomerases. The genetic material of many organisms has the form of a ring made up cither of one strand of DNA or of two strands twisted in a double helix. The ring can assume a number of topological configurations. The conversion of the DNA ring from one configuration to another is catalysed by the topoisomerases. The upper electron micrograph shows single-strand DNA rings from a virus known as bacteriphage fd, which infects bacteria. The lower micrograph shows the rings after they were exposed a topoisomerase from the bacterium Escherichia coli. By cutting the DNA strand, passing a segment of the ring through the break and rejoining the cut ends the enzyme has tied a knot in each ring; such knotting has been seen only in the laboratory. The process of breaking, passage and resealing is essential to the action of all topoisomerases. Some of the enzymes, designated type I, cut a single strand of DNA; others, designated type II, cut both strands of a double helix. Type II topoisomerases acts only on a double-strand ring. The first type II enzyme to be found was the one called DNA gyrase; it was discovered in E. coli. DNA gyrase can negatively supercoil a relaxed double-strand ring in the presence of ATP. (7) It can relax a negatively supercoiled ring without ATP, but the reaction proceeds slowly. The type II enzyme can tie and untie knots in a doublestrand ring, as the type I enzyme does in a single-strand ring. (2) Many topoisomerases molecules can work on a ring at once. If the cut ends where free to drift apart during the knot-tying process, the base sequence of the ring might be rearranged when the breaks were sealed. No rearrangement has been seen. When the ring is cut, the enzyme forms a covalent bond to the DNA strand; the bond prevents the cut ends from separating. DNA gyrase can also interlock two double-strand rings and separate interlocked rings (3).

Figure 34. The rings on which acted a topoisomerase from the bacterium E. coli. (See 1. C. Wang. 1985).

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Figure 3S.1. The action of type II lopoisomcrase on a double-strand ring. (See 1. C. Wang, 1985),

(b)

Catenation

Decatenation

Catenane

Figure 35.2. Action of E. coli DNA gyrase, a type II topoisomerase. (a) Introduction of negative supercoils. The initial folding introduces no stable change, but the subsequent activity of gyrase produces a stable structure with two negative supercoils. Eukaryotic Topo II en/.ymes cannot introduce supercoils but can remove negative supercoils from DNA. (b) Catenation and decatenation of two different DNA duplexes. Both prokaryotic and eukaryotic Topo II enzymes can catalyze this reaction. (See N. R. Cozzarelli, 1980, Science, 207.)

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4.3. Supercoiling of the DNA molecule Supercoiling of a double-strand DNA ring deforms the ring into a more twisted and compact shape. The shape of a DNA ring is strongly affected by the number of times one strand goes around the other; the quantity is called the linking number. It is a topological quantity: it cannot be altered while the strands are intact regardless of how the ring is pulled or twisted. If the strands are cut, however (7), and the rotated about each other in the direction opposite to that of the twist of the helix (2), the helix unwinds. When the cut ends are rejoined (J), the linking number is decreased by the number of rotations that have been made. The strands of DNA in a linear molecule revolve once every 10.5 base pairs because that configuration puts the least strain on the double helix. A ring in which the ratio of base pairs to linking number is 10.5 is said to be relaxed. Increasing or decreasing the ratio strains the double helix, which responds by supercoiling (4). Reducing the linking number causes negative supercoiling; raising the linking number leads to positive supercoiling. The upper electron micrograph shows a relaxed DNA ring from a bacterial virus called PM2. The lower micrograph shows a negatively supercoiled DNA ring from the same virus. It is now clear that many DNA are closed molecules, i.e., their axes as well as their backbone curves are closed curves (figure 37b), instead than a linear (open) form (figure 37a). In fact, the axes of these closed DNA can assume almost any path in space. It has been found by direct experiment that most closed DNA have the shape shown in figure 37c. Such DNA are called supercoiled since the axis is seen to coil back on itself and so the backbone helices are supercoiled. The supercoiling of the closed circular molecule into an interwound superhelix can be understood in terms of the relation between three mathematical quantities, which are linking, writhe, and twist. Because molecule is underwound it has a deficit in linking number compared with a relaxed molecule of the same size. It compensates by writhing and by twisting and bending, satisfying equation Wr = Lk - Tw. The remarkable fact about this result is that two geometric quantities (writhing and twisting) that may change under deformations of the curves sum to a topological quantity (linking number) which is invariant under such deformations. Supercoiling is one of the three fundamental aspects of DNA compaction; the others two being flexibility and intrinsic DNA curvature. For example, the problem of DNA compaction in E. coli can be putted in the following words: The DNA must be compacted more than a thousand fold in the cell, yet it still needs to be available to be transcribed. (Recall that the length of a typical bacterial operon - usually about 3 genes -, is about as long as the entire bacterial cell, if it is stretched out in its B-DNA double helical conformation!). In order this compaction to be effected, it is requested some kind of anisotropic flexibility or "bendability" of DNA, which is very much sequence-specific, and is different from the static "rigidity" of DNA. Whereas persistence length of DNA is relatively non-specific, and just has to do with its overall "rigidity" (on average, DNA has a persistence length of about 44nm, which is quite a bit longer than proteins - one way of thinking about this is that proteins tend to fold up into little spheres, or "blobs", and DNA is a bit more rigid), anisotropic flexibility is a measure of a particular sequence to be deformed by a protein (or some other external forces). Some sequences are both isotropically flexible and "bendable" - for example, the TATA motifs. Perhaps one of the best examples of this is the binding site for the Integration Host Factor (MF) - there are certain base pairs that are highly distorted upon binding of this protein. It is quite impressive that this protein induces a bend of 180

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degrees into a DNA helix. In other words, the curvature k at each sequence of the two strands of DNA helix must be very sharp in order the DNA double helix may assume its characteristic extremely compact form. Indeed, one consider that the DNA must be compacted more than a thousand fold in the cell, it is probably not that surprising that almost any protein that binds to DNA will bend it. Moreover, since the total curvature K of entire DNA double helix segment depends on the torsional stress which applies to DNA strands, and these strands forms hence a twisted curve, i.e. a curve of double curvature in three-dimensional space in a nucleosome inside a cell, the DNA double helix must coils by overwinding or underwinding of the duplex; in order supercoiling appears, the DNA must either be a closed circle, or the ends must be constrained. Notice that the supercoiled DNA runs much faster than the linear, whilst the relaxed DNA runs much slower than the linear. Moreover, the reduction in size taken up by the DNA when it is supercoiled is very important. There are two types of supercoiling: the plectonemic supercoils, which is unrestrained, and the toroidal supercoils, which is restrained by proteins; in fact, wrapping DNA around histone octamer introduces supercoils. Notice that solenoids can be further compacted and folded under various conditions into giant supercoiled loops in higher order chromatin associated with metaphase chromosomes; additional supercoiling and compaction generate the metaphase chromosome. The toroidal supercoiling plays a fundamental role in the compaction of DNA and chromatin. That explains mathematically why a supercoiled molecule is more compact than a relaxed one. The idea is that the more supercoiled a molecule is, the more it appears to cross over itself; hence the smaller the area it presents in the face of the fluid through which it is moving, and hence the faster it sediments. In other words, consider a rubber tube that has been coiled around a cylinder with its ends joined in such a way that all twist is relieved, then it jumps into an interwound helix coiled in the opposite direction when the cylinder is removed. If the are N right-handed turns in the first configuration, and if the pitch angle oti at which each helical turn inclines away from the horizontal is small, then in the second configuration there will be approximately N/2 turns going up and N/2 going down and the new pitch angle Ofe will be large. That means that DNA having a deficit in linking number supercoils into interwound right-handed double helix. That is closely related to an intrinsic mathematical property of torus knots. These knots are restricted to the surface of a torus, which is considered to be embedded in three-space in the most natural way. Pushing a knot inside a torus shows that the knot winds three times around the inner circumference marked by path a. Pushing the knot to the outside of the torus (lower right) shows that it winds five times around the outer circumference marked by a path b. More generally, any torus knot will wind p times around the inner circumference and q times around the outer circumference. For example, in the illustration at the top of this page, for the knot shown at the left p equals 2 and q equals 3, and for the knot shown at the right p equals 2 and q equals 5. Note that the paths a and b lie in the complement of the knot and have a common base point on the torus. The group of any torus knot can be presented by the homotopy classes [a] and [b], which are related by the single, simple equation [af = [b]q. This equation means that in the complement of the torus knot a path that is homotopic to a wound p times around the inside of the torus can be deformed into a path homotopic to b wound q times around the outside of the torus. It is straightforward to prove that the groups of two torus knots are the same if and only if the knots have the same pair of values p and q.

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Figure 36. Torus knols are a class of knots for which the group is particularly effective invariant. Only a torus knot and its mirror image cannot be distinguished by the torus knot group.

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Figure 37. Supercoiling of shape. The shape of DNA other; the quantity is called shows a relaxed DNA ring supcrcoiled DNA ring from

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a double-strand DNA ring deforms the ring into a more twisted and compact ring is strongly affected by the number of times one strand goes around the the linking number. It is a topological quantity. The upper electron micrograph from a bacterial virus called PM2. The lower micrograph shows a negatively the same virus. (See J. C. Wang. 1985).

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(a)

(b)

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(c)

Figure 38. (a) The linear form of the double helical model of DNA. (b) The relaxed closed circular form of DNA. (c) The plectonemically interwound form of supercoiled closed DNA.

4.4. Knots in the embryo A last remark about the development of embryo. In discussing the adhesion of blastomeres in early sea-urchin embryos, Balinsky has listed three mechanisms by which cells appear to stick together, (i) In "ordinary contact", long stretches of two plasma membranes remain in relatively close contact, but with a gap of some 140 A between them, this being presumably filled by some electron-light intercellular substance, (ii) In some cases contiguous cells seem to be held together by the formation of numerous interdigitating processes, (iii) Intercellular connecting bars appear, usually in groups forming the structures known as desmosomes. In the cleavage stages of the eggs of the mollusc Limnea peregra one can see what appears to be a new form of intercellular "contact body". During early cleavage the blastomeres are not everywhere in close contact with one another in the centre of the embryo, although a well-developed blastocoel has not yet been formed. In the angles of the spaces between cells, or occasionally bridging across the middle of a gap, one sometimes finds avoid or roughly

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spherical bodies, which are made up of highly folded regions of the cell membranes. The overall dimensions of the bodies measure one or two m\l, and a straight line through one of them would cross some 8-15 membrane profiles. In some cases it is difficult to be sure that the membranes which are folded together are derived from more that one cell, but in several examples this seems fairly certain, and it is thought that in general the bodies represent places in which the membranes of two (or more) cells are crumpled together to form what may be regarded as a "knot". Presumably such "knots" could play an important part in holding the cells together. 5. Geometrical and Topological Properties of the Double Helix Virtually every physical, chemical and biological property of DNA - its transcription, hydrodynamic behavior, energetics, enzymology and so on - are affected by closed circularity and the deformations associated with supercoiling. Understanding the mechanism of supercoiling and the consequences of this structural feature of DNA, however, presents problems of considerable mathematical complexity. Fortunately, there are two branches of mathematics that offer substantial help in this effort: topology, which studies the properties of structures that remain unchanged when the (geometrical) structures are deformed, and differential geometry, which applies the methods of the differential calculus to the study of curves and surfaces. In what follows we shall first describe a mathematical model of closed circular DNA and then discuss the implications of the model for the real DNA. For our purposes it is preferable to choose a model whose axis coincides with that of the double helix. In addition we specify that the ribbon must always lie perpendicular to the pseudodyads, or twofold axes of rotation, which are distributed along the double helix. This ribbon model follows the axis of DNA double helix and twists as the two chains of the molecule twist around that axis. In addition, because the sequences of atoms in the two-polynucleotide chains run in opposite directions the edges of the ribbon will be assigned opposite orientations. This model can be analyzed mathematically in a number of different ways. One way consists in studying the relation between the oppositely directed edges of the ribbon. When the ends of a ribbon are joined, each edge describes a closed curve in three-dimensional space. Furthermore, when the ribbon represents a closed circular molecule of DNA, a number of 360-degree twists are introduced before the ends of the ribbon are joined, and so the two curves described by its edges are linked. In other words, it is impossible to separate the curves without "cutting" one of them. If each loop in a linked pair represents a covalently bounded molecule, as is the case with the twopolynucleotide chains of the double helix, the two are said to be joined by a topological

bond. This is a peculiar type of bond in that although no part of one molecule is covalently joined to any part of the other, it is nonetheless necessary to break a covalent bond in order to separate the two. In mathematical terms the linking of two closed curves is a topological property: no matter how the curves are deformed (pulled, twisted and so on), as long as neither one is broken they will remain linked in exactly the same way. The linking number Lk is defined as a signed integer that describes a property of two closed curves in space. To separate a pair of curves without actually cutting them the value of Lk must be 0 (although the converse is not always true). If the curves in question are the edges of a closed ribbon with N turns in it, their linking number will remain unchanged when the

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ribbon is deformed. Notice that although the edge of the ribbon model of DNA were no chosen to coincide with the sugar-phosphate backbones of the double helix, the linking number of the ribbon will be exactly that of the backbones. Another way to analyze the ribbon model of DNA is by looking not at the relation between its edges but at the way the ribbon twists. For a ribbon whose axis follows a straight line the idea of a numerical value expressing twist is intuitively obvious. Here we shall adopt the convention that a night-handed twist of 360 degrees has a value of + 1 and a left-handed twist has a value of - 1. The definition of twist is less obvious, however, for a ribbon whose axis is not straight. Perhaps the best way to understand this concept is to imagine a small arrow placed perpendicular to the axis of the ribbon, pointing to one of its edges. As the arrow is moved along the twisting ribbon it rotates about the axis, and the twist of the ribbon can be defined as the integral of the arrow's angular rate of rotation with respect to the arc length of the axis curve. In the special case where the axis of the ribbon is confined to a plane this value can be measured simply as the number of rotations the arrow completes about the axis as it is moved along the ribbon. For example, when the ribbon models a closed circular piece of DNA 5,000 base pairs long that is relaxed (that is, its axis lies in a plane), the arrow makes one complete rotation for every turn of the double helix, and so the total twist Tw equals + 500, with the plus sign arising one again because the double helix is right-handed. For a relaxed circular piece of DNA 5,000 base pairs long, then, both the linking number and the twist are equal to + 500. From this example one might well assume that linking number is just another way of expressing twist, but that is not the case. Indeed, it is particularly important to understand the distinction between these two quantities. To begin with, linking is a topological property, whereas twist is geometrical: if a ribbon is deformed, its twist may be altered. Moreover, to compute the linking number (which is always an integer) the ribbon must be considered as a whole. On the other hand, twist (which may not be an integer) can be considered locally, and the twist values of individual sections can be summed to obtain the total for the ribbon. The realization that linking and twisting are distinct properties raises another question. Is there a geometrical significance to the difference between these properties, that is, to the difference between the linking number of a ribbon and its total twist? In 1968, it has been proved by J. H. White that the linking number of a ribbon and its total twist differ by a quantity that depends exclusively on the curve of the axis of the ribbon. This quantity is well known to mathematicians as the Gauss integral of the axis curve. In other words, assume that the axes of two closed ribbons follow the same curve in threedimensional space; then even if the ribbons themselves turn and twist in entirely dissimilar ways, their values of linking number and total twist will differ by exactly the same amount. At about the same time, it was suggested the name writhing number for the quantity by which the two differ. Thus for a closed ribbon in three-dimensional space the writhing number Wr equals the difference between the linking number Lk and the total twist Tw, or Wr = Lk - Tw. The writhing number of a ribbon is a powerful quantity whose value generally changes if the axis of the ribbon is deformed in space. Hence writhing, like twisting, is not a topological property of the ribbon but a geometrical one. The writhing number can be obtained by computing the Gauss integral, but is generally far easier to calculate it by evaluating the linking number and the total twist of the ribbon in question and then taking their difference. It is only in certain special cases

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that it is convenient to compute Wr directly. For example, if the axis of a ribbon lies entirely in a plane or entirely on the surface of a sphere, then it can be shown that Wr is zero. Substituting this value into the equation Wr = Lk - Tw gives Lk = Tw, which explains why in the example of the relaxed closed circular molecule of DNA both Tw and Lk were found to be + 500. Now consider what happens if the axis of this DNA molecule is made to writhe in such a way that its writhing number is no longer zero. When the writhing number of the molecule is made to change, the linking number remains the same (it can be altered only if one of the backbones of the double helix is broken) and so the twist must change. It is this relation that underlies the phenomenon of supercoiling. A ribbon's linking number, total twist and writhing number do not depend on the ribbon's location or orientation in space. They are also independent of scale, but if one axis of space is inverted (as it is the case when the ribbon is reflected in a mirror) or three axes are inverted (as is the case when the ribbon is inverted through a point), then the sign of all three quantities is changed. On the other hand, if any two axes are inverted, as happens if one looks into an ordinary microscope (that is, not a dissecting one), their sign are unchanged. In fact, any operation that turns a right-handed screw into a left-handed one without introducing other distortions will change the sign of the linking number, the total twist and the writhing number. It is also been shown that there is one other special mathematical operation that changes the sign of these quantities but leaves their magnitude unaltered: inverting the ribbon through a sphere. This result explains why the writhing number of a ribbon whose axis lies on the surface of a sphere is zero. Under this operation the closed curve described by the axis is transformed into itself. Although the equation Wr = Lk - Tw demonstrates that linking and twist are mathematically distinct, the physical difference between quantities may not yet be evident. It may be helpful, then, to consider what happens when a mathematical ribbon is wound around a cylinder in such a way that its surface is always flat against the cylinder. We shall call this pitch angle of the helix described by this ribbon a. In other words, a is the angle at which each turn of the helix inclines away from the horizontal, so that when a is small, the helix is shallow, and when a is large, the helix is steep. Now assume the ribbon is wrapped around the cylinder N times before its ends are joined in the most straightforward way. Then if the effects are ignored, it can be demonstrated that the linking number of the ribbon Lk equals Nsina. Therefore when the helix is stretched out so that the pitch angle a increases, the number of turns and thus the linking number remain the same, but the twist goes from a small value to a large one, clearly demonstrating the difference between linking and twist. Moreover, since a ribbon's writhing number is defined as the difference between its linking number and its total twist, the value of Wr for this ribbon is N - Ns'ma, or N(l - since). As this formula indicates, when a is small and the twist is small, the writhing is substantial, but when a is large and the twist is large, the writhing is minimal. The relation can be easily observed in a coiled telephone wire: when such a wire is unstressed, it assumes a highly writhed form with little twist; when its ends are pulled out, a highly twisted form that writhes only slightly is obtained. Consider how these findings apply to a real DNA, say to the polyoma-virus DNA. Remember that this DNA can be resolved through sedimentation into three components:

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I and n, which are circular, and HI, which is linear. It can be determined experimentally that the average linking number for a population of relaxed circular molecules of this DNA is about + 500. On the other hand, for a population of closed circular molecules (the supercoiled molecules that make up component I) the average linking number is about + 475. Thus, the closed circular molecules of the polyoma-virus DNA are underwound, having a deficit in their winding number of about 25. This finding suggests a way to define supercoiling. It is equal to ALk, the difference between the linking number of a molecule in the natural closed circular state and the linking number of the same molecule in the relaxed closed circular state (where the energy of deformation is at a minimum and the writhing number is zero). For example, for the DNA's of both polyoma-virus and the monkey virus SV40, ALk is approximately - 25. It can now be understood why a deficit in the linking number of a molecule of DNA causes the molecule to supercoil. A linear molecule of DNA in solution normally assumes a form known as the B configuration, in which the nucleotide bases are approximately perpendicular to the helical axis with 3.4 angstrom units between them and in which there are about 10 base pairs for each turn of the double helix. This is a configuration of minimum energy, and if the molecule is bent or twisted, its energy is increased. If a long molecule is simply circular, however, the diameter of the circle is large compared with the thickness of the double helix. Hence the curvature of the molecule is small and its energy is increased only slightly. As a result nicked circular molecules such as component II of polyoma-virus DNA hardly depart from the B configuration. The situation is quite different for a closed circular molecule with a deficit in linking number. To satisfy the condition that the value of Lk be less than that of a relaxed molecule (say 475 rather than 500) the double helix would have to be untwisted, a transformation that would substantially increase the deformation energy of the molecule. By supercoiling, however, the closed circular molecule minimizes the amount by which it departs from the B configuration. More precisely, as the analysis of the ribbon model revealed, one way that underwound DNA can reduce its deformation energy is by writhing. Since writhing and twist are interconvertible, it is apparent that by changing the extent of writhing it is possible to minimize the twist of a molecule, thereby minimizing the twisting component of its deformation energy. On the other hand, writhing always introduces some curvature, and so it increases the bending contribution to the energy of the molecule. Therefore the supercoiled configuration that the underwound DNA molecule assumes is one that minimizes twist while introducing the smallest possible amount of bending. To make a detailed analysis of the supercoiling of real DNA requires a fairly precise knowledge of all its elastic constants, not only end effects but also such matters as charge repulsion and thermal motion. The preceding arguments show quite clearly, however, why a closed circular DNA molecule, with a deficit in its linking number (or as is sometimes the case an excess) will writhe into a supercoil. In nature most closed circular DNA is negatively supercoiled, that is, the supercoiling results from a deficit in linking number. The analysis of the rubber-tube model explains why such molecules can be expected to assume the configuration of an interwound superhelix whose handedness, rather surprisingly, is the same as that of the double helix, namely right-handed. To create DNA molecules with different degrees of writhing it would be convenient to be able to make a nick in one of the backbones of a DNA double helix, relax the molecule

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by a few turns and then close the nick. Astonishingly, enzymes have been identified that do just that. Many enzymes have been discovered in a variety of sources, including different bacteria virus and the nucleus and mitochondria of animal cells. These nickingclosing enzymes, which are also called topoisomerases, generally require no energy source to function. They always act to reduce the supercoiling of a DNA molecule and thereby lower its energy. Nicking-closing enzymes have proved to be invaluable tools for studying the physical chemistry of DNA. To begin with, adding a topoisomerase to a solution of supercoiled DNA gradually reduces the supercoiling until all the DNA molecules are in or near the relaxed state, but much more ingenious applications have been devised for these enzymes. For example, suppose a high concentration of ethidium bromide is added to a closed circular DNA, causing it to supercoil in the direction opposite to its usual one. If a topoisomerase is then added as well, the molecules in the solution will become relaxed, but because of the large amount of intercalated ethidium bromide they will be considerably more underwound than they are in their natural state. Now, if first the topoisomerase and then the dye is removed, the molecules will compensate for the additional reduction of the linking number by supercoiling even more strongly in the usual direction. In this way supercoiled molecules can be created, molecules that have many more superhelical twists than are usually present. This type of energy storage and transfer may be central to the role of supercoiling in the cell. We can even conceive of a chemical engine that by going through a similar cycle or operations turns chemical energy into work. It has been found an enzyme in the bacterium E. coli that given an energy source such as adenosine triphosphate (ATP) will reduce the linking number of relaxed circular DNA, thereby increasing its writhing number and making it supercoil. The activity of an enzyme of this type, called a gyrase, is essentially opposite to that of the nicking-closing enzymes. So far gyrases have been found in a variety of microorganisms but not in any higher organisms. Nicholas Cozzarelli and his colleagues have shown that a gyrase seems to act by bringing two segments of the DNA molecule close together. The enzyme then cuts both of the backbones of one of the segments and passes the other intact segment through the resulting gap before it rejoins the originally cut backbone. It is easy to show that such a process would alter the linking number of the DNA in increments of two rather than one, and that is exactly what has been observed experimentally. How and why does supercoiling arise? In the SV40 virus and the polyoma virus the DNA is supercoiled because it is usually not naked in the nucleus of the cell. During replication the double helix is wound on nucleosomes (beads of protein consisting of eight histone molecules with one or two associated molecules). When the DNA molecule slips off this supporting structure, it supercoils (in much the same way that the coiled rubber tubing discussed above does). In fact that the linking number is always reduced in naturally occurring DNA implies that when DNA winds around the nucleosomes in a solenoidal coil, the coil is left-handed. It is difficult to determine the exact number of supercoils that are generated by each nucleosome, but it appears likely that the number will turn out to be between 1 and 2. Nucleosomes are found in association with DNA in all higher organisms; in lower organisms the origins of supercoiling are not completely clear.

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Figure 39. Closed circular DNA is modeled by a Misted ribbon whose ends have been joined, (a) Since the sequences of atoms in the two chains of the double helix run in opposite directions, it is also convenient to assign the edges of the ribbon opposite directions. When the edges are viewed as directed closed curves in three-dimensional space (/>), they are found to be mathematically linked; in other words, there is no way to separate them without breaking one or the other. This relation can be described mathematically by a linking number Lk whose magnitude expresses the number of times one curve is linked through the loop of the other and whose sign depends on the way the curves arc labeled. One way to compute the value of this quantity is to examine a projection, or two-dimensional representation, the two curves and to each point where one curve crosses over the other (but not where a curve crosses over itself) assign an index number according to the following rule: If a clockwise rotation is required to move the top piece so that it coincides with the bottom piece, then +1 is assigned to the crossing point, and if a counterclockwise rotation is required, then - 1 is assigned (c). (This convention is the reverse of the one normally employed in mathematics, but it ensures that the linking number of right-handed closed circular DNA will be positive.) IJt is then calculated by adding up the index numbers and dividing by 2 (the number of curves). The linking number obtained in this way is a signed integer equal to 0 if the two curves arc unlinked (
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fiv-

0 1* - 0

Kr- 4t,U • *l

Figure 40. Twisting of a ribbon can be assigned a numerical value by placing a small arrow on Ihe ribbon perpendicular lo ils axis and pointing to one of its edges. As the arrow moves along the ribbon it rotates about the axis, and the magnitude of the total twist Tw can be defined as the integral of the angular rate of this rotation with respect to the arc length of the curve described by the axis. Whether this quantity is positive or negative depends on whether the rotation of the arrow about the axis is respectively righthanded or left-handed. When the axis of the ribbon lies entirely in a plane as is shown here, then the total twist is easily computed, being equal to the number of rotations the arrow makes about the axis. The twist can be computed separately for different parts of the ribbon and can then be summed to obtain the total value.

Topological Knots

Models

Figure 41. Completion of replication of circular DNA molecules. Denaluration of the unreplicated terminus followed by supercoiling overcomes the steric and topological constraints of copying the terminus. At least with SV40 DNA. the final two steps (synthesis and decatenation) can occur in either order depending on experimental conditions. Parental strands arc in dark colors; daughter strands in light colors. (Inset) Electron micrograph of two fully replicated SV40 DNA molecules interlocked twice. This structure would result if synthesis was completed before decatenation. Topo H can catalyze decatenation of such interlocked circles in vitro. (Drawing adapted from S. Wasserman and N. Cozzarelli, 1986. Science, 232, p. 951. Micrograph from O. Sundin and A. Varshavsky, 1981. Cell, 25, p. 659.)

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Overwound region

Figure 42. Movement of the growing fork during DNA replication induces formation of positive supercoils in the duplex DNA ahead of the fork. In order for extensive DNA synthesis to proceed, the positive supercoils must be removed (relaxed). This can be accomplished by E. coli DNA gyrase and by cukaryotic type I and type II topoisomerases. t*i

m

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NEGATIVE SUPERCOILING helix opening facilitated

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Figure 43. Superhelical tension in DNA causes DNA supercoiling. (A) For a DNA molecule with one free end (or an nick in one strand that serves as a swivel), the DNA double helix rotates by one turn for every 10 nucleotide pairs opened. (B) If rotation is prevented, superhelical tension is introduced into the DNA by helix opening. One way of accommodating this tension would be to increase the helical twist from 10 to 11 nucleotide pairs per turn in the double helix that remains in this example; the DNA helix, however. resists such a deformation in a springlike fashion, preferring to relieve the superhelical tension by bending into supercoilcd loops. As a result, one DNA supercoil forms in the DNA double helix for every 10 nucleotide pairs opened. The supercoil formed in this case is a positive supercoil. (C) Supercoiling of DNA is induced by a protein tracking through the DNA double helix. The two ends of the DNA shown here arc unable to rotate freely relative to each other, and the protein molecule is assumed also to be prevented from rotating freely as it moves. Under these conditions, the movement of the protein causes an excess of helical turns to accumulate in the DNA helix ahead of Ihe protein and a deficit of helical turns to arise in the DNA behind the protein, as shown.

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6. Conclusions As we saw in previous section, knots have formed the basis of fruitful speculations in the realm of matter, and that inorganic and organic chemists and physicists claimed that the molecule they study and observe may form knots or links. In addition, section 2 has been devoted to show that, according to recent mathematical theories, physical space might be of string nature and that it might be knots-shaped. Almost a century after Einstein's accomplishment in fashioning general relativity, string theory give us a quantum-mechanical description of gravity that necessarily modifies general relativity when the distances involved become as short as the Planck length. Whereas general relativity asserts that the curved properties of the universe are described by Riemannian geometry, string theory asserts that this is true only if we examine the structure of the universe on large enough scales. On scales as small as the Planck length a new kind of geometry must emerge, one that aligns with the new physics of string theory. This new geometrical framework is called quantum geometry, and knotted structures are very likely one of its most fundamental ingredients We have given reasons why, in many contexts at the relativistic as well as at the quantum scales physics, 3+1 dimensional space of the special theory of relativity is the proper space for describing the physical world. However, further investigations led by supergravity and superstring theorists showed that the 11-dimensional topological space are the most physically relevant, notably because they allow for unifying gravity with the other fundamental forces in nature within the same mathematical framework. In the very recent years, it appeared clearly that quantum field theory, knot theory and topological and arithmetical invariants theory are intimately linked and involved in the same reality. One might say that knots are ideal topological configurations which optimises the energy needed by any physical, chemical or biological system in order to may evolve in space and time by avoiding its dissipation outside the system. In addition, knots seems to be involved in some kind of robustness property pertaining to many of those dynamical systems. For example, one of the inherent properties of vortex theory in modern fluid mechanics and molecular dynamics is that of transmutation, according to which knotted vortex atoms change their knot type if their energy is increased beyond a certain threshold, as do physical atoms change their atomic structure. Another example concerns the ideal fluids (in which there are no dissipative effects). Vortex or magnetic line topology is frozen in the ideal fluid while the structures of these objects, in continuous motion, can be highly distorted by the background flow. This means that if these tubes (the presence of tube-like structures at different length scales seems to be a generic feature of organized fluid patterns) are initially knotted or linked, they will evolve and deform in the ideal fluid by preserving the type of knot or link that ties them together, even through their geometry may become utterly complicated. This is a fundamental, intrinsic property of the governing equations (the Euler's equations). Topological properties of ideal fluids are therefore flow-invariant, and physical information expressed in pure topological terms is therefore bound to the conserved as well. Concerning finally the role of knots in living processes, it has to be once again stressed the participation of topoisomerases in nearly all cellular processes involving DNA. Because the enzymes affect the topology and organization of intracellular DNA,

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the primary effects of inactivating a topoisomerase are also likely to generate farreaching ripples. The regulation of the cellular levels of the enzymes themselves and the association of the enzymes with other cellular proteins are closely tied to the cellular functions of the enzymes. One major cellular function of the topoisomerases is to prevent excessive supercoiling of intracellular DNA. However, supercoiling is sometimes utilized in vivo to drive a particular region of intracellular DNA into a conformation suitable for a particular process. Initiation of DNA replication, for example, often requires that the DNA be in a negatively supercoiled state. Indeed, replication is the best known process that generates supercoils in intracellular DNA. The involvement of various topoisomerases in the removal of positive supercoils generated by replication is generally in accordance with their known in vitro specifities. Namely, eukaryotic DNA topoisomerases I and II, and bacterial DNA topoisomerases IV, can efficiently remove supercoils of either sense; bacterial DNA topoisomerases I and HI, and eukaryotic DNA topoisomerase HI, can remove negative supercoils, but not positive supercoils, unless a single-stranded region is present in the DNA. Bacterial gyrase is unique in its ability to convert positive to negative supercoils; depending on how fast the positive supercoils are generated and how fast they are converted to negative supercoils, gyrase can either prevent accumulation of positive supercoils in an intracellular DNA segment or keep the segment in a negatively supercoiled state. The DNA topoisomerases presumably co-evolved with the formation of very long and/or ring-shaped DNA molecules. To solve a variety of problems that are rooted in the double-helix structure of DNA, nature has created not one but three distinct enzymes. In eukaryotes, members of all three subfamilies of DNA topoisomerases have been found in the same cells; in bacteria, four members from two subfamilies participate in nearly all cellular transactions of DNA. The past decade saw much progress in the study of the DNA topoisomerases, but many questions remain. The key to answering to them may lie in the elucidation of interactions between the DNA topoisomerases and other cellular proteins. Complexes between these enzymes and transcription factors and chromosomal proteins illustrate new avenues yet to be fully explored. Furthermore, whereas the information available on topoisomerases-DNA interactions is substantial, that on interactions in the context of chromatin is scarce; whether eukaryotic DNA topoisomerase II has a structural role in the organization of interphase and/or metaphase chromosomes, for example, is yet to be settled. By the 1960s, it became clear fhat-alfhough the informational content of the genetic code was embodied in a linear array of bases-it was the three-dimensional structure of the DNA double helix that ultimately would govern its physiological functions. This is very likely the crucial point. As an illustration of this point, in perhaps the most striking biological example of "form dictates function", the two complementary parental strands of DNA must separate during semi-conservative replication in order to act as the templates for each of the two newly synthesized daughter strands. This discovery led to the realization that the structure of DNA, while elegant, burdened the cell with previously unimagined topological problems. Although these topological problems were originally recognized only for circular molecules, because of the long length of chromosomal DNA, we now know that they apply to linear genomes as well. The key for finding the solution to these problems seems to lies in the conformational, organizational and biological roles of the topoisomerases which, because of their extreme structural and functional complexity, still remains in part to be elucidated. All

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Rushing; T. B., Topological Embeddings, New York, Academic Press, 1973. Sauvage, J.-P., "Les Catenanes," Nouveau Journal de Chimie, 9 (5), 1985, 299-310. Schiicker, Th., "Knots and their Links with Biology and Physics," in Geometry and Theoretical Physics (J. Debrus, A. C. Hirshfelsd Eds.), Heidelberg, Springer, 1991. Schwarz, A. S., Quantum Field Theory and Topology, Heidelberg, Springer-Verlag, 1993. Seeman, N. C , "DNA Nanotechnology: From Topological control to Structural Control," in Pattern Formation in Biology, Vision and Dynamics, A. Carbone, M. Gromov, P. Prusinkiewicz editors, World Scientific, Singapore, 2000, 271-309. Seifert, H. and W. Threlfall, Lehrbuch der Topologie, Leipzig, Teubner, 1934. Simon, J., "Topological chirality of certain molecules," Topology, 25 (1986), 229-235. Simondon, G., L'individu et sa genese physico-biologique, Grenoble, Millon, 1995. Stasiak, A. et al., "Ideal knots and their relation to the physics of real knots," in Ideal Knots (A. Stasiak et al. eds.), Singapore, World Scientific, 1998. Sterwart, I., Life's Other Secret, Penguin, London, 1998. Strick, T. R., Allemand, J.-F., Bensimon, D., and V. Croquette, "Behavior of Supercoiled DNA," Biophysical Journal, 74 (1998), 2016-2028. Strogatz, S. H., Nonlinear Dynamics and Chaos, Perseus Publishing, Cambridge, MA, 1994. Sumners, D. W., "Knot theory and DNA," in New scientific applications of Geometry and Topology (D.W. Sumners, ed.), Proc. Sympos. Appl. Math., Vol. 45, Amer. Math. Soc, Providence, 1992. Taniyama, K., "Cobordism, Homotopy and Homology of Graphs in R3," Topology, Vol. 33, No. 3, 1994, 509-523. Thistlethwaite, M. B., "Knot tabulations and related topics," in Aspects of Topology. In memory of Hugh Dowker 1912-1982,1. M. James and E. H. Kronheimer eds., Cambridge University Press, 1985, 1-76. Thorn, R., "Topological models in biology," Topology, 8 (1969), 313-335. Thurston, W. P., Three-Dimensional Geometry and Topology, Princeton, Princeton University Press, 1997. Turaev, V. G., Quantum Invariants of Knots and 3-Manifolds, Walter de Gruyter, Berlin, 1994. Waba, D. M., "Topological stereochemistry," Tetrahedron, 41(1985), 3161-3212. Waddington, C. H., "Membrane knotting between blastomeres of Limnea," Exptl. Cell Research, 23 (1961), 631-633. Wang, J. C , "DNA topoisomerases," Ann. Rev. Biochem., 54 (1985), 665-697. Wang, J. C , "DNA Topoisomerases,'Mnrcw. Rev. Biochem., 65 (1996), 635-692. Wasserman, S. and N. R. Cozzarelli, "Biochemical Topology: Applications to DNA Recombination and Replication," Science, Vol. 232, May 1986, 951-960. White, J. H., An introduction to the geometry and topology of DNA structure, CRC Press, Boca Raton, CA, 1989. White, J. H., Cozzarelli, N. R. and W. R. Bauer, "Helical Repeat and Linking Number of Surface-Wrapped DNA," Science, Vol. 241, July 1988, 323-327. White, J., "Self-linking and the Guass integral in higher dimensions," Amer. Jour. Math., 91(1969), 693-728. Whitehead, J. H. C , "A certain region in euclidean 3-space," Proc. Natl. Acad. Sci. Usa, 21 (1935), 364-366. Whitney, H., "On regular closed curves in the plane," Comp. Math., 4 (1937), 276-284. Whittington, S. G., Sumners, De Witt and T. Lodge (editors), Topology and geometry in Polymer Science, Springer, New York, 1998. Witten, E., "Topological quantum field theory," Comm. Math. Phys., 117 (1988), 353-386. Zeeman, E. C , "Twisting spun knots," Trans. Amer. Math. Soc, 115 (1965), 471-495.

Part ffl

Mathematical and Logical Modeling in the Natural Science and Living Systems

"The exploration of topological space is the exploration of the last great frontier the human mind. These space have been created by humans for the purpose of understanding the world in which we live. But ultimately they lead to an understanding of our mind, for it can only be understood in terms of its creations. Topological spaces, then, are at once a form of art and a form of science, and as such they reflect our deepest intellect." J. Scott Carter, How Surfaces Intersect in Space, 1995.

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BEYOND MODELLING: A CHALLENGE FOR APPLIED MATHEMATICS PETER T. SAUNDERS Department of Mathematics, King's College, Strand, London WC2R 2LS, England peter. saunders@kcl. ac.uk

Abstract: Biologists and social scientists often carry out their research in ways that are quite different from those used by physical scientists. Their results are often different as well; the nature of their subjects makes it less likely that they can produce the firm and generally quantitative predictions that are standard in physics. As mathematics is being applied more and more outside the physical sciences, a new methodology is appearing that better reflects the nature of these other subjects. This is happening only slowly, chiefly because the relevant work is generally seen only in its context of trying to solve a particular problem rather than as a contribution to applied mathematics as such. The aim of this paper is to encourage progress by describing some results that have already been obtained, and by discussing explicitly some of the issues that arise.

Introduction Over the past twenty or thirty years, there has been a very great increase in the number of applications of mathematics to biology and the social sciences. There is now hardly a field that does not have a journal devoted to mathematical and theoretical papers. But while mathematics has been expanding into new and exciting areas, most of the work has been done using methods and methodology developed over three centuries in connection with the physical sciences. There have been useful contributions, but many important problems remain largely untouched, chiefly because we are not equipped to address them. Organisms and societies are both complex and highly structured, and we need techniques that are capable of dealing with such systems. Developing them is not primarily a matter of deriving new mathematical results, though that will be part of it. What is required is a new methodology, new ways of applying mathematics, and that makes this above all a challenge for applied mathematicians. There has already been progress, but it has been slow and largely uncoordinated. This is partly because while mathematicians are accustomed to learning new results, we are less used to thinking about new ways of applying those we already know. The chief reason, however, is that we do not generally set out to develop a new methodology in the way that we might decide to investigate a particular field or try to solve a known problem. What generally happens is that we are trying to solve a particular problem and the logic of the 281

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situation (combined, more often than not, with a lack of success by familiar methods) leads us to attempt something novel. Thus both those who are doing the work and those who are criticising it generally see it in its role as a possible contribution to a particular field rather than as part of the development of a new paradigm. They will not be expecting methodological problems, and will be taken by surprise when they appear1. Each example will tend to be seen in isolation, which makes it unlikely that we will learn from similar problems that have occurred in different contexts2. The aim of this paper is to explore the issues and to make them explicit. Whether we are engaged in developing and using the new methodology, or whether we are watching from the sidelines, it is important to understand what is happening. Like everything else in science, the new developments must ultimately be judged by whether they contribute to our understanding of the phenomena, but in arriving at this judgement we have to realise that both the questions we are posing and the kinds of answers we can accept may be different from what we are accustomed to. Most of the current applications of mathematics in biology are not novel in the sense that I mean here. Many, including some of the most useful, are not so much biology as physics or chemistry in a biological setting. In others, mathematics is applied directly to biology but in a way which at least looks like what happens in physics. For example, theoretical ecologists often write down and solve differential equations which are supposed to represent the interactions of different species with each other and with the environment. They do not claim that simple equations are precise models of reality. The expectation, however, is that they are close enough for the purpose, that while the details may be inaccurate they capture the general behaviour of the ecosystem in question. This approach has its own disadvantages, largely because we can think that we are closer to the paradigm of physics than we actually are, but if carefully used it can produce useful results. The applications described in this paper are different. They are not intended to be models at all, at least not in the usual sense. They are closer to what Pielou (1981) refers to as investigations. We do not set out to build a model of a whole system. Instead, we ask one or two specific questions about the system, and try to use mathematics to help us find the answers.

' A colleague in the humanities once suggested to me that the reason disputes within mathematics can be so fierce is that we can almost always reach a consensus by formal reasoning, and therefore find it very difficult to cope with situations in which we cannot. 2 See, e.g. Saunders (1999)

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A major difference with most conventional modelling is that we generally do not have a definite hypothesis about the mechanism that is producing the phenomena. That is not to say, however, that we ignore the mechanism altogether. On the contrary, we specifically assume that the phenomena arise out of a conventional mechanism and not, for example, from morphic resonance (Sheldrake, 1981). We often rely heavily on the assumption, explicit in the case of catastrophe theory and implicit or at least less rigorously translated into mathematics in others, that the mechanism is in some sense no more complicated than it has to be to produce the phenomena that we are observing and trying to understand. In conventional modelling, if our predictions turn out to be wrong, we generally conclude that the mechanism is not what we thought it was. In the new approach, our conclusion would more likely be that the mechanism is not as simple as we had thought. The phenomenon is not merely something we would expect a system of this kind to do; there is something else going on. This can be a very useful piece of information, because when faced with a complex system it is not always easy to know which features require special explanations and which do not. We can waste a lot of time trying to account for things that were bound to happen anyway. The approach can and does lead to testable predictions, though not always in quite the same way as in conventional modelling. To illustrate this, I shall discuss two early examples in detail, both depending on catastrophe theory. In the first, Zeeman (1974) predicted that there could be a wave associated with the formation of a boundary in a previously undifferentiated tissue. In the second, Bazin and Saunders (1978) predicted that an amoeba should modify a certain chemoattractant. Both predictions turned out to be correct. Naturally, that was gratifying to the authors, but the chief point is that results obtained using the new approach can indeed be held up to nature and tested, no less than those derived by conventional methods. In conventional modelling, and especially in physical science, we often construct a model and then treat it largely as an exercise in mathematics, only translating back into science at the end. In both the examples here, arriving at the prediction required mathematics and biology to be used together. When we are answering questions rather than making complete models, we generally find we have to keep the original problem in mind the whole time. This requires especially close cooperation between mathematicians and their colleagues in other disciplines. It also means that each will have to learn about the other's subject, because effective cooperation requires an overlap of knowledge between the collaborators.

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Catastrophe Theory Catastrophe theory was developed by Rene Thorn in the 1960s and first came to the attention of the general scientific community with the publication in 1972 of his book Stabilite structurelle et morphogenese. Briefly, what the theory says is the following. Imagine a system which can, in principle, be modelled by the sorts of differential equations that are commonly used in conventional modelling. Note that it only has to be possible in principle; the system may be far too large and complex for it to be practicable. Then under quite weak conditions, the number of qualitatively different configurations of discontinuities that can occur is independent of the size of the system and typically very small. If there are no more than four 'control variables' there are only seven possible 'elementary catastrophes'3. The applications generally have to do with the forms that can arise, as in the study of caustics, or in the question of when and where discontinuities will occur, as in the two examples discussed in this section. A distinctive feature of catastrophe theory is that the results we obtain are qualitative. Almost everything is specified only up to a diffeomorphism. While this may not be what applied mathematicians are accustomed to, it is generally an advantage, because in many applications it is qualitative results that we really want. Or perhaps I should say that given the nature of the data that are available, it is only qualitative results that we can reasonably expect, or can compare with the observations. In fact, this is not as different from conventional modelling as it sounds, because the equations we write down in biology are generally chosen because they have the right general form, rather than because we believe them to be a close representation of the mechanism of the system. No one sees the LotkaVolterra equations, or the many variations that are now being used, as the ecological equivalents of the equations of motion of the planets. And when we solve the equations we have written down, the actual numbers we obtain are seldom significant. What matters is what they tell us about the qualitative behaviour of the system: whether a species will persist, whether there will be oscillations, and so on. Thus we are actually using a quantitative model to find answers to questions that are essentially qualitative. This can be a useful strategy, but it has its drawbacks. When we write down equations we introduce many extra

3

For an accessible account of catastrophe theory and a more precise statement of the result, see Saunders (1980).

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assumptions into a model. The logistic equation dx/dt = rx (1-x/K) contains far more than the statement that the specific growth rate of a species tends to decrease as the population rises and that the population eventually levels out. For example, it also supposes that there is a fixed 'carrying capacity', K. Whether or not we really believe this to be the case, the fact that it is part of this standard model can lead us to include the idea in our thinking even when we are not using the equation in which the parameter occurs. When we use the logistic equation, or others of that kind, we are left with the problem of deciding whether our conclusions follow from our assumptions about the system, or whether they are merely artefacts of the particular equations we have chosen to model it. We may derive a large number of results about the equations, but how much they tell us about the real system is another matter. Qualitative results may be harder to obtain and do not give the same (generally misleading) impression of precision, but it is easier to identify the assumptions on which they depend and, therefore, to know how relevant they are to the system we are studying. When catastrophe theory first appeared it aroused a great deal of interest, but this soon turned to extreme scepticism, especially in the USA. The critics claimed that while the mathematics itself was correct, and while the theory could be applied to some problems in physics (such as phase transitions or the properties of caustics), it had nothing to contribute to biology and the social sciences (Zahler & Sussman, 1977). The attacks were directed against catastrophe theory in particular, but since the controversy concerned the applications rather than the mathematical foundations, the issue was broader than either side realised at the time. The real question was whether the only legitimate ways of applying mathematics are those that are standard in physics. As a result, when the critics won the day, they not only largely blocked the development of catastrophe theory, they also held back the growth of applied mathematics in general. As we shall see, catastrophe theory can indeed be applied in biology. It can produce results which are not obvious and which can be tested by observation or experiment. All the same, it is not difficult to imagine why so many mathematicians were suspicious and were so readily convinced that there was some sleight of hand going on. The theory seems to be offering something for nothing. You need little or no knowledge about the mechanism, and not much data, and yet you can derive results. This must be a violation of some sort of theoreticians' conservation law — more information is coming out of the analysis than was put in.

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In fact, this is not the case. If, for example, we write down and solve the equations that describe the fluid flow around an aircraft, then, within the accuracy of the model, we have completely determined the motion of the air at every location in the vicinity of the aeroplane. Catastrophe theory, in contrast, tells us only one thing, viz. that there can be a shock wave and, if there is, we expect it to be in the shape of a cusp. This is only a minute fraction of the information contained in the solution of the hydrodynamical equations. It may be the most interesting piece of information, but that's another matter. It is possible to answer specific questions with only a very small amount of input in the form of data and hypotheses. It's like driving through a tunnel rather than going over the top of the mountain. You don't get to see the whole panorama but you do reach your destination, and using much less energy. We may say that catastrophe theory is efficient, by analogy with the concept of an efficient estimator in statistics, i.e. one that gives you the most information about a particular parameter from a given amount of data. I have written of answering specific questions as though we could always find a way of addressing any question we might wish to ask. That, of course, is not the case. It could hardly be, for if it were we could build up a complete model by answering many questions one at a time. Nevertheless, it is sometimes possible, and when it is we should not be allow ourselves to be put off by those do not understand what we are doing. And on the whole, those we can answer to tend to be the important ones, and the same ones that we are generally trying to answer when we use other techniques. Even if what we discover is not what we most wanted to know, it is still information about the system and it may give us important clues that can help us in conventional modelling. For example, Seif (1979) fitted a cusp catastrophe to observations on patients with hyper- and hypothyroidism. He used this to explain some of the peculiarities of the treatment of patients with thyroid conditions. He then fitted a cusp surface numerically to the data, and from this he was able to infer that the number of secretory granules bound to a membrane should be two. This quantitative result did not follow from catastrophe theory alone, but would have been much more difficult to obtain without it. Primary and Secondary Waves The application that became the centre of the controversy about catastrophe theory was Zeeman's (1974) paper on travelling wave fronts. Zeeman claimed that when a frontier forms in a region that was previously undifferentiated, i.e. either homogeneous or with at most a gradient in properties along it, then this

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frontier does not first appear in its final position. Instead, it forms to one side, and then moves through the region before stabilising. The proof consists in showing that under quite plausible conditions catastrophe theory can be assumed to apply to the situation. We then suppose that the mechanism that creates the frontier is no more complicated than it has to be to produce the observed phenomena. This implies that the process involves a cusp catastrophe, which means there are two control variables, and the obvious pair are t, the time, and s, distance measured at right angles to the forming boundary. With these assumptions, and since we know that at an early time to the properties of the region are continuous whereas at some later time t] there is a discontinuity, we infer that there is a cusp in the s-t plane. The 'world-line' of any cell is a line s = const. Any cell whose world line crosses both branches of the cusped curve will undergo an abrupt change in its properties as it crosses the second branch. In general, different world lines will cross this cusp at different values of t, i.e. different cells will change from one state to the other at different times, and so the frontier between cells in one state and cells in the other will move as it forms. The only case in which this does not happen is when the axis of the cusp lies precisely in the t-direction. Only one world line crosses the cusp twice (in two coincident points) and this is where the frontier appears and remains. (See Fig. 1) Zeeman then argues that this is a non-generic case, i.e. that it occurs with zero probability. Hence, he concludes, the frontier always moves as it forms. How far it moves, we cannot say, and indeed there is nothing to say that it must move more than a very small amount, far too little to be of any interest. Within the model, the movement might be less than the length of a single cell, and in that case no observable motion whatsoever would be predicted. Perhaps it would have been better to say that a frontier may move. That is not the sort of result applied mathematicians are used to but it is sufficient for the purpose. For it is not at all obvious that a frontier forming in a previously undifferentiated region can move as it forms, not because of some additional factor but simply as a natural consequence of the process by which it appears. What is more, the result, even in this apparently tentative formulation, has two important consequences. The first is that if we see a frontier moving, there does not have to be some special mechanism, some chemical signal, some gene. We do not immediately have to ask what selection pressures during evolution have brought about the motion. It may be that it's just what systems of this kind typically do.

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to (a)

(b) Figure 1. Two possible orientations of the cusp in the s-t plane. In (a), the generic case, the boundary first appears at so at time to and stabilises at si at time ti. In (b), the non-generic case, the boundary appears at so at time to and does not move.

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The expression "of this kind" is important here. When Zahler & Sussman (1977) attacked Zeeman's work, they wrote: "Zeeman's 'proof consists of no more than the observation that, if the frontier did not move, that would be quite exceptional. To evaluate this kind of reasoning properly, notice that the same logic, if correct, would apply equally well not just to frontiers of tissues but to anything whatsoever. So, Zeeman's reasoning 'proves' that everything moves except for those exceptional objects that do not." In fact, this is a caricature of Zeeman's reasoning, which applies only to the mathematical model he had derived and hence only to frontiers that form in certain ways. It does not apply to those that are formed by other means nor to objects in general. No one was claiming that a stone wall must move as it is being erected. Even within developmental biology there are models that fall outside the class included in Zeeman's account. For example, if we suppose that a chemical gradient is established and then either one gene or an alternative is turned on depending on the local chemical concentration, we would expect the resulting boundary between the two types of tissue to form in its final position. To see the importance of Zeeman's result, we have to return to the biological problem that stimulated the work, which has to do with what he called primary and secondary waves. The idea is that sometimes a wave travelling through a region starts, at each point through which it passes, a process which will have an observable result some time later. This will appear as a second wave passing through the region, although in fact what is then happening at each point is independent of what is happening anywhere else. This is especially interesting if the primary wave is invisible, because then what we observe is a wave with apparently nothing to trigger it or to drive it, and with the unusual property that it cannot be halted by a barrier. For example, when an epidemic is spreading across a continent, we cannot see the virus as it passes from country to country. What we observe is the onset of the disease, which happens several days later. Yet by this time nothing is actually being transmitted, and if we now close international borders to travellers, the disease wave will proceed just as inexorably as if we had done nothing. To stop the spread of the disease we would have to act before the invisible primary wave passed through. What the result accomplishes, therefore, is the following. Suppose we see a wave of activity moving through a region. We want to know how this comes about, and so we might set out to discover the stimulus that sets it off, and how this influence is transmitted through the region: in the case of tissue, from cell to cell.

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We now know there is another possibility. We are led to ask whether a boundary has recently appeared in the region. If it has, then we know that it may have moved as it formed, even though no motion was actually observed, and what we are seeing now may be simply a secondary wave. In that case, there is no separate stimulus to find and nothing is being transmitted. We haven't proved that this is what is happening, but we can now set out to verify whether it is or not. For example, we can insert a barrier into the region and see if this stops the wave: if it does not, then it is in fact a secondary wave. Ultimately it will be observation and experiment that decide the issue, but it is the mathematics that suggests what observations and experiments we should undertake. In fact, Elsdale et al (1976) later found direct evidence for the existence of the wave that Zeeman had predicted. Some time after Zeeman's work appeared, Lewis, Slack and Wolpert (1977) suggested a model for the formation of a frontier using the particular equation

Here g is the concentration of some gene product, S is a 'signal substance' and the Kj are constants. In fact, this is one of the class that is included in Zeeman's analysis, except that it has only a single control variable, S. This is therefore a non-generic case, and without catastrophe theory to alert them to the need for a second control variable, Lewis et al failed to notice the possibility that the boundary might move as it forms. Saunders & Ho (1985) generalised (1) to include a second control variable (by the simple device of taking K3 as a parameter, not a constant) and derived a model to account for an otherwise puzzling pattern of transformations in Drosophila (the fruit fly) that had undergone treatment with ether in early development. This example illustrates a common misconception in mathematical modelling. We are all accustomed to writing down and solving equations in physics and obtaining precise answers. Unfortunately, this can mislead us into believing that an analysis using equations must always be better than one that does not. This is by no means always the case. Equation (1) looks plausible and it produces the desired behaviour, but it is not actually a model of any particular known system. It is also not structurally stable, which means that it misses some important features of the system it is meant to represent. In fact Saunders & Ho also wrote down an equation for illustrative purposes, but because they had used catastrophe theory first, they were in a position to ensure that their equation was structurally stable and therefore a better model of the system.

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Critical Variables This application (Bazin & Saunders, 19784; Saunders, 1980) arose out of an attempt to model a predator-prey interaction in a continuous flow system. This had turned out to be unexpectedly difficult, because the specific growth rate of the predator, the amoeba Dictyostelium discoideum, did not depend in a straightforward way on the density of the prey, the common gut bacterium E. coli. The relationship was neither smooth nor even single valued. Instead, the growth rate switched abruptly from one nearly constant value to another, even though the prey density was varying smoothly and over a wide range. The sudden jumps suggested that the response of the amoebae could be modelled using catastrophe theory. The simplest singularity that would serve was the cusp, which has two control variables, and Bazin and Saunders chose these to be the prey density, H, and the time t. They plotted H as a function of t, marked on the graph the points at which the abrupt changes in the state variable (the specific growth rate of the predator) occurred, and hence inferred the position of the cusp. See Fig 2, Fig 3. There is, however, a problem with this model. The sudden changes in the state variable occur as the trajectory enters the cusp, whereas the theory leads us to expect them to occur as it leaves it. It is possible to account for the observations on this model, but it requires a more complicated mechanism than we might expect. One that has been suggested (Bazin & Saunders, unpublished) requires the amoebae to be sensitive not only to the prey density but also to whether this is increasing or decreasing. That can account for the observations, but it's not an especially simple mechanism, especially from the point of view of the amoebae. Now while it is usual in microbial ecology to suppose that the specific growth rate of a predator depends on the prey density, for other organisms we might expect it to depend on the ratio of prey to predators, H/P5. Bazin and Saunders therefore repeated the analysis based on the cusp catastrophe, but using H/P instead of H as one of the control variables. The result is shown in Fig 3, and this time the abrupt changes do indeed occur as the trajectory leaves the cusp.

The reader may notice that this paper was published in the same journal as Zahler & Sussman's (1977) article claiming that catastrophe theory cannot be applied in biology, but a year later. For the record, it was submitted (and originally rejected) before Zahler & Sussman's article appeared. For humans, what matters is the amount of food per head of population. The situation with bacteria is different because the way they take up nutrient is more similar to a chemical reaction.

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Time (h) Figure 2. Prey density H as a function of time t. Dots indicate the places where abrupt changes in the specific growth rate of the predator occur, and the position of the cusp has been sketched in. From Saunders (1980).

Time (h) Figure 3. As Fig. 2 except that the ordinate is the prey/predator ratio H/P. From Saunders (1980).

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We now have a single question to be answered: To what do the amoebae respond, the prey density, H, or the prey/predator ratio H/P? The mathematical analysis suggests that if it is the former, they require a rather complicated mechanism, whereas if it is the latter, a simpler mechanism will do. We therefore infer that the critical variable is H/P. If we knew the mechanism, we might stop here, but in this case we look for confirmation. For this, we take the argument one step further. If the critical variable is H, then the amoebae must be responding to some substance which is secreted by the bacteria, probably folic acid, but if it is H/P then the substance must also be being modified by the amoebae. For if folic acid increases with H and decreases with P, it can serve as a measure of H/P. Bazin and Saunders therefore suggested that if folic acid is indeed the signal, then the amoebae must be modifying it. Thus the result led to a prediction which could be tested. As it happened, at almost the same time the paper appeared, Pan and Wurster (1978) reported that Dictyostelium discoideum do indeed inactivate folic acid, thus confirming the prediction. The mathematical result also explains what purpose this serves, which Pan & Wurster were not able to infer from their experiments. Dynamical Systems There are not many general statements we can make about systems in general, but one of them is that any system is, in some sense, stable. It has to be, because if it were not, we would not recognise it as a system. Unfortunately, while that explains why the systems we observe are stable, it tells us nothing about their other properties, nor even why there are systems at all . We can infer much more, however, if we concentrate on complex, non-linear systems. For if such systems are stable, they generally have a number of distinctive properties which we do not find in linear systems. For example, they are likely to have a number of equilibrium states rather than only one. These can be equilibrium points but they can also be trajectories. In either case, they are well defined and there is a finite number, not a continuum. About sixty years ago, the embryologist CM. Waddington (1940) identified a number of properties that are common to many organisms while they are developing. An organism does not have to have a perfect environment to develop into a recognisable individual of a given species, and it can survive many 6

Compare the principle of natural selection which allows us to understand why organisms are adapted to their environments (because if they were not, they would not survive) but does not tell us why they have the properties that they do, or indeed why there are organisms at all.

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perturbations while it is developing. Two embryos do not have to be genetically identical to turn into very similar adults. When an embryo is recovering from a perturbation, it does not return to the state it was in when it was disturbed; instead it continues to develop, eventually reaching more or less the state it would have been if there had been no perturbation. What is stable is not the state of the embryo but the developmental trajectory. Waddington introduced new terms to describe these phenomena: homeorhesis (similar flow) to describe a system that returns to a trajectory, chreod (necessary path) for the stable trajectory itself, and canalisation for the property that the end states typically form a discrete set rather than a broad spectrum7. He also devised the metaphor of the epigenetic landscape, a figure in which the developmental system is pictured as a mountainous terrain, with the valleys corresponding to the possible developmental pathways. Why do organisms have these properties? Was there selection for each of them through evolutionary history? It is hard to see how they could have originated through selection, although we might imagine that there would be selection for different kinds of stability once they had appeared. The simple answer is that because organisms, whatever else they are, are complex non-linear dynamical systems, they are almost bound to have the properties that Waddington identified. It would be surprising if they did not. That's not to say that there have to be organisms, though it does increase the likelihood. But if there are going to be organisms, than we would certainly expect them to have those properties; they are all part of a package, so to speak. So we do not need separate explanations for each of them. In fact, thinking of organisms as dynamical systems does not merely allow us to see why they have the properties that Waddington identified. We can derive others as well, as we would expect when we have the power of mathematics at our disposal and not just intuition. For example, the stability of the normal developmental trajectory and the existence of stable alternatives means that we would not expect large changes to occur as long sequences of small ones. Instead, we would expect that large phenotypic changes will occur abruptly as the developmental system is diverted into a new pathway, possibly by a small genetic change. Thus the mode of evolution that Eldredge and Gould (1972) called punctuated equilibria is precisely what we would expect to observe.

7

There can, of course, be small variations about the end states; the point is that there are identifiable types without a spectrum connecting them.

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The dynamical systems approach suggests that when species split there can be more than two successor species, which is contrary to what palaeontologists assume when they are trying to work out the course of evolution. We can also show that part of the cause of the irreversibility of evolution may lie in the properties of the developmental system. (Saunders, 1993). And all this without knowing much about the mechanisms involved. Self-Organisation Self-organisation has been recognised for a long time; the classical example, Benard convection, was described at the beginning of the twentieth century. Only recently, however, has it become a major research topic, as more work is being done on non-linear systems. It is being found that many non-linear systems are capable of self-organisation, generally as a bifurcation parameter passes through a critical value and a more or less uniform state becomes unstable. One of the best known examples is Turing's (1952) reaction-diffusion model of pattern formation. In fact, this paper is an example of the overlap that can exist between the different types of modelling. He gave it the title The Chemical Basis of Morphogenesis, and it does include an explanation of why he chose to concentrate on chemistry, rather than the elastic or other physical properties of tissue. He also intended to apply the model to the particular problem of phyllotaxis, though his tragic death meant that this was never done. All the same, Turing's real aim was to establish by an example the general point that a pattern can appear in a previously uniform region with no preexisting template, and through a comparatively simple mechanism. There are two reasons why he set out to do this. The first is that from the standpoint of developmental biology it is the initial symmetry breaking that is the major puzzle: once there is a rudimentary pattern it is comparatively easy to imagine how this might serve as the basis from which later ones can develop. Turing's chief motive, however, was deeper than that and really a part of the new approach being described in this paper. As he remarked to his student Robin Gandy, his ideas were "intended to defeat the argument from design." This term usually refers to the claims of eighteenth and nineteenth century natural theologians that since (so they believed) there is no natural process by which organisms could possibly be as well organised and their bodies as well adapted to needs as they are, we are forced to the conclusion that there must be a Creator. Most readers will be familiar with William Paley's metaphor about finding a watch on the ground and concluding from this that there must be a watchmaker somewhere.

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This argument was eventually refuted by Darwin's theory of natural selection, which provided a mechanism by which (apparent) design is possible without a designer. Ironically, however, modern neo-Darwinists also employ the argument from design. According to them, the only natural process that can produce organisation is natural selection, and so natural selection is the key to evolution, the 'blind watchmaker', as Dawkins (1986) has put it. What Turing showed, and recent work in non-linear dynamics amply confirms, is that this is not the case. Non-linear systems can self-organise, and while the patterns that appear in the development of an embryo may persist on account of natural selection, they may no more owe their origin to it than do Benard convection cells or the red spot on Jupiter. The dynamical systems approach can be used in other fields as well. For example, two fundamental concepts in psychology are the complex and the archetype. But while these concepts do seem to represent real phenomena, and many analysts have found them useful in their work, they have never been able to say exactly what they are, or even what sorts of entities they are. And in the case of the archetypes, which are supposed to be common to all humans, they cannot agree on when they appeared in evolution or indeed whether they have been there since the beginning and, if so, in what form. Saunders and Skar (2001), however, have pointed out that as the brain is a complex non-linear system, and as it is known to be involved in a constant process of organising information, we would expect that what psychologists call complexes should form by self-organisation. Moreover, because we all have similar brains and undergo similar key life experiences, such as being cared for by our mothers or some substitute mother figure, we would expect the complexes formed around certain typical developmental experiences to have the same general features. It also follows that there is no problem about when archetypes appeared; they appeared naturally as human brains and societies became sufficiently complex. Recognising that self organisation is by no means rare can also help us avoid a common error in science. We often insist that such and such a phenomenon does not occur not because it has been investigated in detail — or at all — but because it 'violates the laws of science', by which we really mean (or ought to mean) that it seems to run contrary to all our other experience of nature as collected and codified. 8 The term neo-Darwinism refers to the synthesis of Darwinian natural selection and Mendelian genetics. It is now the dominant theory of evolution, so much so that its supporters use the terms "evolution" and "natural selection" more or less interchangeably.

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Now this is often a useful strategy. We don't have to waste time analysing a proposed perpetual motion machine because the second law of thermodynamics tells is that it is impossible. The problem, however, is that it is all too easy to claim that something is impossible when it is not. That can happen because relevant scientific knowledge is not yet available, as in Lord Kelvin's objection that the Earth was not old enough for Darwinian evolution to have taken place. Darwin could not deal with this at the time; only after radioactivity had been discovered could Kelvin's point be refuted. More commonly, however, the problem is simply that we have not been clever enough at using the knowledge we already possess. For a long time, scientists and doctors in the west were sceptical about the efficacy of acupuncture because there seemed to be no possible mechanism by which it could act. Eventually, but only after the evidence that it works became too strong to be ignored, it was realised that there are ways in which it could happen that are not at all inconsistent with conventional western science. In general, it is difficult to know what sorts of forms can appear through selforganisation in a given system without going through the detailed calculations. On the other hand, simply to recognise that self-organisation is a common phenomenon can allow us to see that something is possible in principle, even if we cannot account for it in detail, and this should make us more cautious about dismissing it out of hand. We will not be so ready to jump to the conclusion that every pattern we observe in organisms must have been created by natural selection, or that regular piles of rocks on a plain must have been put there by extraterrestrials. Conclusion The application of mathematics to complex systems, especially in biology and social sciences, is leading to new ways of applying mathematics. Instead of setting out to construct a model which describes the whole system, we may ask one or two specific questions. Or we may look for the sorts of behaviour which systems of this kind typically exhibit. The new approach does not eliminate mechanism from explanation, even though it significantly alters its role. For while we do not base our work on a particular mechanism, we generally assume not only that there is a mechanism, but also that it is of the kind that a conventional modeller would be likely to imagine, and that it is no more complicated than it needs to be. In effect, we are studying the common properties of the class of the simplest mechanisms that are consistent with the observations.

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That mechanisms can fall into classes in this way can pose a serious problem for conventional modelling. We tend to assume that if we make a hypothesis about the mechanism of a system and if and if this leads to predictions which are borne out by observation, then that is strong evidence in favour of our hypothesis. If many mechanisms can produce the same phenomena, then the support for our hypothesis is much weaker. An example is provided by the work of Douady and Couder (1991). They found they could reproduce the characteristic pattern of spiral phyllotaxis — successive elements separated by the Fibonacci angle of about 137.5° — by allowing magnetised drops of fluid to fall onto a flat plate in a non-uniform magnetic field. They modelled their experiment using the appropriate (inverse fourth power) expression for the force between the droplets, and found that this gave the correct angle. But they also found they could get the same angle using many different expressions for the repulsion. One might have thought that for a model of phyllotaxis to produce the correct angle was almost conclusive evidence in its favour, but this turns out not to be the case. The process that determines phyllotaxis in plants, whatever it is, is certainly not the mutual repulsion between magnetic dipoles. Wherever the Fibonacci sequence, the golden section and the angle of 137.5° appear in nature, they are surely the result of the operation of some mechanism. Nevertheless, there seem to be a number of different mechanisms that can give rise to them, which gives them a status which in a real sense transcends individual mechanisms. There seems to be something more fundamental about this mathematical sequence than any of the mechanisms that can produce it. This is an example of the return of the formal cause as part of explanation in science (Saunders, 1989). We cannot really describe what is happening as a revolution9, partly because the classical methodology will continue to play an important role in modelling in the soft sciences, and partly because the distinction between the new and the old is not absolute; there has always been work with elements of both. Turing's reaction-diffusion model of pattern formation was presented primarily as a piece of conventional modelling of chemistry in a biological setting. Nevertheless, its primary significance was (and was recognised by Turing to be) the demonstration that the sorts of processes that occur in development can lead to pattern formation in a straightforward way. And indeed, it is the latter aspect that explains why it remains such an important paper, even though it is now

9

In the sense of Kuhn (1961).

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generally accepted that pattern formation by reaction-diffusion is not very important in development. A century ago there was a major change in applied mathematics when quantum mechanics and relativity appeared. Explanation was no longer necessarily in terms of mechanisms, and applied mathematics became less centred on differential equations. But what happened in applied mathematics was part of a larger change that was taking place in science. The present situation is different. Biologists and social scientists have always worked differently from physicists, as one would expect given the complexity of what they study, and mathematicians are now finding ways of using mathematics that reflect the differences. One of the most significant differences from the methodology appropriate to the physical sciences is that we do not set out to construct a model of the system we are studying. Even in quantum mechanics the aim is to have a model which is complete in the sense that it includes everything that we believe Nature allows us to know. If the model does not tell us which slit the photon passed through, that is because that is not a question that we are allowed to ask, at least not on our current understanding of the science. In chaotic systems too, the reason we do not have complete information about the future is not through a shortcoming of the model. It is a reflection of how things really are, or at least of how we believe things really are. Biologists and social scientists do not generally claim to have as complete an understanding of the systems they work with, and the way we apply mathematics to their subjects should acknowledge this. If we insist on doing only the sort of modelling that we are familiar with in physics, we are likely to convince ourselves that organisms and societies are much more like the solar system or a gas-filled container than they really are. If the only tool you have is a hammer, it is very tempting to see everything as a nail. For example, most contemporary evolutionists consider the organism as a collection of largely independent 'traits' each largely governed by a single gene10. This view cannot but be strengthened by the fact that only if we assume it to be the case are the evolutionists' mathematical models valid. The predominance of linear techniques in mathematics is not the cause of the reductionism that pervades biology and the social sciences, but it contributes to

No neo-Darwinist I know of will admit to believing the 'one gene/one character' assumption, but they do base their work on it. Perhaps 'one gene that makes any difference/one character' is a better description (Bateson, 1982; Saunders 1988). (I apologise for using such old references, but neoDarwinists are usually very reluctant to explain the assumptions they are making.)

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it. Even those who do not use mathematics in their own work are strongly influenced by the ideas that have proved so outstandingly successful in physics. Evolution theory and neo-classical economics are not part of Newtonian theory but they are part of the Newtonian paradigm. Similarly, much work in both ecology and economics depends on the assumption that the systems they are setting out to describe are at equilibrium, and we may ask whether this is because experience suggests they are or simply because it makes the mathematical analysis tractable. Of course tractability is important: there's little point in setting up a mathematical structure if it is too ill defined and too complicated to yield any results. But there is no point in having a tractable structure that doesn't represent the real system sufficiently closely. Indeed, it is worse than useless, because it will produce misleading results. The systems we deal with in biology and the social sciences are typically both complex and structured, and if we cannot model them in their entirety we can set ourselves the more modest task of using mathematics to learn something more about the system. This is, after all, closer to the way that people already working in those fields tend to proceed. They seldom have overarching theories from which they expect to deduce everything; instead there is a process closer to induction in which one tries to discover properties and gradually build from them a coherent picture. Mathematicians can look forward to the time when our subject has the same status in biology and social science that it has in physics. Our colleagues in the soft sciences are going to have to understand mathematics, just as physicists already do. But we are going to have to understand their subjects and their ways of working too, and develop with them a methodology that is appropriate for the task. References Bateson, P.P.G. (1982). Behavioural development and evolutionary processes. In Current Problems in Sociobiology (eds King's College Sociobiology Group). Cambridge University Press, Cambridge, pp.133-151. Bazin, M.J. & Saunders, P.T. (1978) Determination of Critical Variables in a Microbial Predator Prey System by Catastrophe Theory. Nature 275 (1978) 52-54. Dawkins, R. (1986). The Blind Watchmaker. Longmans, London. Douady, S. & Couder, Y. (1991). Phyllotaxis as a physical self-organized growth process. Phys. Rev. Lett. 68, 2098-2101.

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Elsdale, Y., Pearson, M. & Whitehead M. (1976). Abnormalities in somite segmentation induced by heat shocks to Xenopus embryo. J. Embryol. exp. Morph. 35, 625-635. Eldredge, N. & Gould, S.J. (1972). Punctuated equilibria: An alternative to phyletic gradualism. In Models in Paleobiology (ed. T.J.M. Schopf). Freeman Cooper, San Francisco, pp. 82-115. Kuhn, T. (1961). The Structure of Scientific Revolutions. Chicago University Press, Chicago. Lewis, J., Slack, J.M. & Wolpert, L. (1977). Thresholds in development. J. theor. Biol. 65, 579-590. Pan, P. & Wurster B. (1978). Inactivation of the chemoattractant folic acid by cellular slime molds and identification of the reaction product. J. Bacteriol. 136, 955-959. Pielou, E.C. (1981). The usefulness of ecological models: A stock-taking. Q. Rev. Biol. 56, 17-31. Saunders, P.T. (1980). An Introduction to Catastrophe Theory. Cambridge University Press, Cambridge. Saunders, P.T. (1988). Sociobiology — A house built on sand. In Evolutionary Processes and Metaphors (M.W. Ho & S.W. Fox, eds). Wiley, Chichester, pp 275-294. Saunders, P.T. (1989). Mathematics, structuralism and the formal Cause in biology. In Dynamic Structures in Biology (B.C. Goodwin, G.C. Webster & A. Sibatani, eds). Edinburgh University Press, Edinburgh, 1989, pp 107-120. Saunders, P.T. (1993). The organism as a dynamical system. In Thinking about Biology, SFI Studies in the Sciences of Complexity, Lecture Notes Vol. Ill (F. Varela & W. Stein, eds). Addison Wesley, Reading MA, pp 41-63. Saunders, P.T. (1999). Darwinism and economic theory. In Sociobiology and Bioeconomics (ed. P. Koslowski). Springer, Berlin, pp.259-278. Saunders, P.T. & Ho, M.W. (1985 ). Primary and secondary waves in prepattern formation. J. theor. Biol, 114 491-504. Saunders, P.T. & Skar, P. (2001). Archetypes, complexes and self-organisation. J. anal. Psych., 46, 305-323. Seif, F.J. (1979). Cusp catastrophe in pituitary thyrotropin secretion. In Structural Stability in Physics (eds W. Guttinger & H. Eikemeier). Springer, Berlin. Sheldrake, R. (1981). A New Science of Life: the Hypothesis of Formative Causation . Blond & Briggs, London.

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Thorn, R. (1972). Stabilite structurelle et morphogenese. Benjamin, Reading MA. (English translation by D.H. Fowler (1975): Structural Stability and Morphogenesis. Benjamin, Reading MA.) Turing, A.M. (1952). The Chemical Basis of Morphogenesis. Phil. Trans. Roy. Soc. 237B, 37-72. Waddington, C.H. (1940). Organisers and Genes. Cambridge University Press, Cambridge. Zahler, R.S. & Sussman, H.J. (1977). Claims and accomplishments of applied catastrophe theory. Nature 269, 759-763. Zeeman, E.C. (1974). Primary and secondary waves in developmental biology. In Some Mathematical Questions in Biology VIII, Lectures in Mathematics in the Life Sciences, vol. 7 (ed. S.A. Levin). American Mathematical Society, Providence, pp 69-161.

FONDEMENTS COGNITIFS DE LA GEOMETRIE ET EXPERIENCE DE L'ESPACE ALAIN BERTHOZ College de France Chaire de Physiologie de la Perception et de VAction 11, place Marcelin Berthelot, 75005 Paris, France

ABSTRACT: We try to describe and explain how are perceived motion and three-dimensional space, and how human beings perceive and control bodily movements. Reviewing a wealth of research in neurophysiology and experimental psychology, we argue for a rethinking of the traditional separation between action and perception, and for the division of perception into five senses. In our view, perception and cognition are inherently predictive, functioning to allow us to anticipate the consequences of current or potential actions. I assert in this paper that the brain is a biological simulator that predicts by drawing on memory and making assumptions. This interpretation of perception and action allows us to focus on psychological phenomena largely ignored in standard texts: proprioception and kinaesthesis, the mechanisms that maintain balance and coordinate actions, and basic perceptual and memory processes involved in navigation. We combine geometrical considerations of the components of perception and action with synthetic concepts borrowed from psychophysics and neurosciences to discuss the sensory receptors that enable us to analyze movement in space.

1. Introduction Ce texte est issu, en partie de deux cours sur le cerveau et l'espace au College de France que j ' a i donne en 1997 et 1998. Le deuxieme cours a ete consacre plus particulierement au probleme des fondements cognitifs de la geometrie et il a ete accompagne d'une serie de seminaires auxquels ont participe plusieurs des auteurs de ce livre (L. Boi, G. Longo). La geometrie est-elle le signe d'une realite exterieure a l'Homme ou est-elle incarnee, immanente ? Est-elle un geste ou une image ? Quels sont les liens entre l'espace physique et l'espace phenomenal, ou sensible, ou representationnel ? Y a-til une ou plusieurs geometries mentales ? Sont-elles categorielles ou metriques ? Comment, a partir de la fragmentation des informations sensorielles dans des espaces tres differents, une perception coherente des proprietes geometriques du monde se constitue-t-elle, et quelle est la contribution du mouvement a la perception tridimensionnelle ? Quels sont les referentiels utilises par le cerveau ? Le concept d'espace absolu a-t-il un sens ? Comment fonctionne la memoire de l'espace ? Comment se developpe la geometrie chez l'enfant et comment se degrade-t-elle par suite de lesions du systeme nerveux central ? Enfin, le controle du mouvement est-il fonde sur des invariants geometriques ou des regies de la 303

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dynamique. Voici quelques-unes des questions immenses que nous aimerions envisager mais qui sont encore hors de 1'analyse scientifique approfondie. Nous essaierons toutefois d'examiner quelques reponses ici. Je proposerai d'emblee une hypothese, un objectif: rehabiliter le corps sensible et Taction dans les theories sur les fondements de la geometrie. Je pense, en effet, que Taction est le fondement principal de notre connaissance du monde et que le cerveau projette sur le monde ses intentions, ses predictions, en relation avec les actions qu'il prevoit. J'ai soutenu cette these dans Le sens du Mouvement (Berthoz, 1997) et je 1'ai developpee dans La decision (Berthoz, 2003). Je pense aussi qu'il faut rehabiliter les theses d'Henri Poincare qui propose que le fondement de la geometrie est dans nos actions et « qu'imaginer un point dans l'espace c'est imaginer le mouvement qu'il faut faire pour l'atteindre ». Si l'approche formelle de la geometrie telle qu'elle a ete promue par Hilbert a fait ses preuves, elle ne peut nous cacher l'ancrage profond des concepts mathematiques dans le fonctionnement de notre cerveau. Ma theorie est done qu'il faut chercher la presence des concepts les plus elabores de la geometrie dans les operateurs qui sont apparus au cours de revolution et qui nous permettent aujourd'hui de manipuler mentalement, de simuler, le monde exterieur avec un codage allocentre de l'espace et non plus seulement egocentre comme c'est sans doute le cas chez tous les animaux jusqu'au singe. D 'Euclide a Poincare et Einstein Meme si Ton peut trouver en Egypte et dans les grandes civilisations du MoyenOrient des signes d'une pensee geometrique, c'est en Grece qu'est nee la premiere veritable grande theorie geometrique et, meme si Tarchitecte Thales de Millet fut le premier a formaliser le plan d'une ville selon des principes geometriques, tout le monde s'accorde a en donner la paternite a Euclide. Avant lui, de grands precurseurs avaient propose diverses theories sur la perception des objets, y compris la fameuse theorie de « Textramission » d'Empedocle qui suggerait que le cerveau eclaire les objets de son feu et que cette lumiere se reflechit en retour sur la retine. Mais c'est Euclide qui a donne le premier ensemble coherent et complet de principes et de regies d'ou est nee la geometrie qui porte son nom. II est hors de notre propos de retracer l'histoire de la geometrie mais il est important d'analyser en premier lieu, dans les temps modernes, la facon dont Poincare (Poincare, 1932 ; Poincare, 1970) en traite les fondements. La raison pour laquelle j'ai choisi Poincare, en particulier le texte de Science et Methode (Tedition de 1930 ; p. 97) est que Poincare fait une veritable rehabilitation du role du corps et de Taction dans Torigine de la geometrie. Comme on le sait, les theories qui en mathematiques rapprochaient la geometrie de Texperience sensible de l'espace ont ete balayees au debut de ce siecle par les disciples de Pasch et d'Hilbert qui ont reussi a creer une mathematique formelle dont Tobjectif etait de dissocier completement les mathematiques de Texperience sensible.

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L'espace absolu D'apres Poincare, ce mot est vide de sens : nous ne pouvons pas connattre la valeur absolue d'une distance. Poincare attribue aux comportements associes a l'espace, les actes de prehension, de capture, de parade, ce qu'il appelle « les evidences des verites geometriques ». II definit d'abord « ce petit espace qui ne s'etend pas plus loin que ce que mon bras peut atteindre » et que nous appelons aujourd'hui « l'espace de prehension ». II utilise les actions de parades pour definir le point geometrique. II distingue aussi deux espaces: l'un restreint a des coordonnees liees au corps ; l'autre, « etendu », est celui forme des divers points definis ainsi qu'il vient d'etre resume. Un point est done «la suite des mouvements qu'il convient de faire pour l'atteindre a partir d'une position initiale du corps ». La geometrie est ainsi fondee sur des gestes orientes vers des buts. Cette pensee est a rapprocher de celle exprimee recemment par le mathematicien G. Chatelet dans son ouvrage Les enjeux du mobile (Seuil, 1993 ; p. .31). « Ce concept de geste nous semble crucial pour approcher le mouvement d'abstraction amplifiante des mathematiques qui echappe aux paraphrases rationalisantes - toujours trop lentes -, aux metaphores et a leurs fascinations confuses et enfin, surtout, aux systemes formels qui voudraient boucler une grammaire des gestes : Godel a bien montre que des enonces rebelles - vrais, mais non prouvables - sont aussitot secretes par une syntaxe tant soit peu ambitieuse ». Et plus loin : « La veritable geometrie doit saisir l'instant ou l'espace frissonne enfin des virtualites qui l'habitent et nous invite a eprouver la dimension comme invention d'une articulation. Elle nous conduit pour ainsi dire par la main pour reapprendre le mouvement qui separe et lie a la fois, et pour savoir capter dans un simple fragment l'adresse et la continuite d'un geste » (p. 157). L'idee d'une relation profonde entre perception de la forme et action a ete aussi suggeree par Rene Thorn qui ecrit: « la forme biologique suggere une action » (Mathematical Models of Morphogenesis, 1983, p. 166). Poincare franchit alors une etape dans le raisonnement et dit que le mouvement est fondamental pour definir l'espace puisqu'un etre conscient qui serait fixe au sol ne connattrait pas l'espace. II expose ensuite longuement pourquoi il pense que l'espace est a trois dimensions et, ici encore, il justifie le caractere tridimensionnel de l'espace par le besoin de ranger les categories de comportement, d'atteinte ou de parade d'objets qui est un tableau a triple entree. Espace geometrique et espace sensible Poincare s'est aussi interesse a la difference entre l'espace geometrique et l'espace sensible qu'il appelle « representatif ». Dans La Science et I'Hypothese, il discute d'abord les differences. L'espace geometrique et l'espace representatif sont tres differents. II entre alors au cceur de notre question en disant : « mais, si l'idee de l'espace geometrique ne s'impose pas a notre esprit, si d'autre part aucune sensation ne peut nous la fournir, comment a-t-elle pu prendre naissance ?» «Aucune des sensations, isolee, n'aurait pu nous conduire a l'idee de l'espace, nous y sommes amenes seulement en etudiant les lois suivant lesquelles ces sensations se succedent».

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Ces lois sont issues des observations que nous faisons du changement des objets pendant nos mouvements. En effet, Poincare remarque que dans notre entourage les objets dont les deformations peuvent etre corrigees par nos mouvements sont les corps solides. II conclut alors : « S'il n'y avait pas de corps solide dans la nature, il n'y aurait pas de geometrie». Poincare se prononce aussi sur la difference entre la geometrie d'Euclide et celle de Lobatchevski: « L'experience ne peut decider entre les deux. En effet, les experiences ne peuvent porter que sur les corps et non sur l'espace ». II poursuit « Par selection naturelle, notre esprit s'est adapts aux conditions du monde exterieur, il a adopte la geometrie euclidienne car c'est la plus avantageuse a notre espece. La geometrie n'est pas vraie, elle est avantageuse». Ici, la geometrie est une expression d'un besoin naturel - un acte perceptif, dirait Janet - pour constituer le monde exterieur en y cherchant a distinguer des objets utiles a Taction. On ne peut eviter de rapprocher le texte de Poincare de celui ecrit par Husserl en 1907 sur le role des kinestheses (au risque de me faire designer comme "neo-husserlien") dans la constitution de l'espace percu. Husserl (Husserl, 1989), apres avoir rappele, comme le fait Poincare, l'importance des mouvements du sujet dans la constitution des proprietes percues des objets, ecrit: « Nous voulions considerer la chose visuelle et la constitution visuelle de la spatialite et de la localite, et voici que nous introduisons d'emblee les mouvements de notre corps et, a travers eux, les sensations de mouvement, qui n'appartiennent pourtant pas au genre des contenus visuels » (Choses et Espace - Legons de 1907, PUF, 1989, p. 194). Le point de vue d'Einstein On peut a ce sujet evoquer le temoignage d'une autre grand physicien de notre siecle, Einstein. Sa conception n'est pas tres differente. Dans son livre Conceptions Scientifiques (A. Einstein, 2001), il discute la facon dont est constitute notre conception de l'espace. « Une importante propriete de notre experience sensible et, plus generalement, de toute notre experience, est de l'ordre du temps. Cette propriete d'ordre conduit a la conception mentale d'un temps subjectif, un schema pour ordonner notre experience (...) Mais avant la notion de temps subjectif se trouve le concept d'espace et avant ce dernier se trouve le concept d'objet materiel; ce dernier est directement lie aux complexes des experiences sensibles (...). Poincare a justement insiste sur le fait que nous distinguons deux sortes de changements dans l'objet materiel: "des changements d'etat" et des "changements de position ". Ces derniers, disait-il, peuvent etre corriges par des mouvements arbitraires de notre corps ». II poursuit par une etude detaillee des conditions d'elaboration du concept d'espaces, et ecrit plus loin : « L'erreur funeste qu'une necessite mentale precedant toute experience est a la base de la geometrie euclidienne et du concept d'espace qui lui est lie, est due au fait que la base empirique sur laquelle repose la construction axiomatique de la geometrie euclidienne etait tombee dans l'oubli.

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Dans la mesure ou Ton peut parler de I'existence de corps rigides dans la nature, la geometrie euclidienne doit etre consideree comme une science physique, dont l'utilite doit etre montree par son application a l'experience sensible ». Les hypotheses de la perception phenomenale Un argument essentiel pour discuter les conceptions formalistes de la geometrie est la nature biologique de l'espace representatif, ou phenomenal comme le nomme Michotte (Michotte, 1962). Le fait que l'espace phenomenal est different de l'espace physique est suggere par de nombreuses illusions. Par exemple, on peut rappeler les illusions des chambres de Ames qui montrent que le cerveau deforme la realite en faisant des hypotheses de symetrie, de rigidite, et de regularite. Ces hypotheses sont aussi faites pour 1'interpretation des proprietes tridimensionnelles de forme et de courbure des objets en mouvement, 2. Le cortex visuel et la geometrie Nous allons ici rappeler brievement comment le systeme visuel, chez les Primates et chez l'Homme, traite les proprietes geometriques du corps, des objets et de l'environnement dans les premiers relais neuronaux des voies visuelles. Les operations neuronales dans les voix visuelles Notre systeme visuel est subdivise en analyseurs qui effectuent une segregation dans les proprietes de l'environnement et des objets. Deux grandes voies transmettent au cerveau les informations visuelles : la voie colliculaire et la voie corticale. Lorsqu'on analyse les mecanismes de la representation du monde visuel dans les voies retino-thalamo-corticales, le point le plus important est I'existence de deux voies principales issues des parties parvocellulaire et magnocellulaire du corps genouille lateral dans le thalamus. Ces deux voies different dans leur traitement des donnees visuelles par les aspects suivants : acuite, contraste, couleur, sensibilite au mouvement et a la vitesse de celui-ci. Lorsqu'on analyse les traitements qui sont effectues au niveau de l'aire VI, ou aire de Brodman 17, premier relais cortical visuel, les deux voies parvocellulaire et magnocellulaire se projettent dans des zones differentes de VI, appelees respectivement « blobs » et «interblobs ». Les neurones des blobs, sur lesquels se projette la voie parvocellulaire, sont sensibles a la couleur et insensibles a I'orientation. Le systeme magnocellulaire se projette sur les neurones des interblobs qui sont selectifs a I'orientation et a la direction du mouvement et insensibles a la couleur. Dans l'aire corticale suivante de cette chaine de traitement, l'aire V2 ou aire de Brodman 18, on trouve une organisation en bandes de plusieurs millimetres de large qui sont de deux sortes : "minces" ou "etroites", et "larges". Les blobs s'y projettent sur des bandes etroites ou les neurones n'ont pas de selectivity a I'orientation et 50% sont sensibles a la couleur. Les champs recepteurs y sont plus

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grands que dans VI. Les interblobs se projettent sur les bandes pales de V2. Les neurones de ces bandes sont selectifs pour une orientation mais pas pour la direction, et 50% sont de type "end stopped" : ils sont actives par 1'extremite d'une ligne, des coins, ou des zones des images qui ont une courbure tres forte, et detectent aussi la direction de la fin d'une ligne. On trouve done, des cette aire, un traitement des proprietes geometriques des lignes. Les interblobs se projettent sur des bandes larges de V2 dont les neurones ont une selectivity a l'orientation mais pas "end stopped". Ils codent la disparite binoculaire et participent a la perception de la profondeur stereoscopique. II existe dans l'aire V2 des neurones selectifs a l'orientation des frontieres de contraste aussi bien qu'a des contours illusoires. Ces neurones preferent des barres longues avec des orientations obliques comme on les trouve dans les triangles reels mais aussi dans des figures de Kanizsa ou, malgre 1'interruption des cotes du triangle, nous percevons un cote illusoire alors que les neurones de VI ne repondent qu'aux vrais contours mais pas aux contours illusoires ou a des formes cachees dans des nuages de points. Dans VI, des qu'une ligne est interrompue, le codage de la ligne par les neurones cesse. On trouve dans V2 des neurones qui repondent a une ligne formee de points. Les neurones de V2 ont des proprietes de detection de contour qui sont influencees par l'orientation de la tete par rapport a la gravite et recoivent done des entrees vestibulaires. Cette propriete pourrait contribuer a l'invariance perceptive. Dans les premieres stations du traitement, une segregation est done realisee dans l'analyse de la couleur, du mouvement, de l'acuite, du contraste. Au niveau suivant, une segregation est realisee entre forme, couleur, mouvement et profondeur. De facon generale, il semble que le systeme magnocellulaire soit implique dans la perception 3D et qu'il assure la segregation entre la figure et le fond, entre les objets et l'environnement. Un aspect important de cette construction geometrique de l'unite des objets dans l'environnement est de trouver des "linking features", des caracteres qui relient des elements de figure et assurent la perception de l'unite de l'objet lorsqu'il est fragmente. Le systeme parvocellulaire voit la couleur et peut l'utiliser pour detecter des bords mais, alors que le systeme magno est incapable d'un examen stable et durable (il s'inactive apres quelques secondes de vision), le parvo peut maintenir l'image pour l'examiner en detail. Le magno serait plus primitif et le parvo plus recent. Le parvo est tres developpe chez les primates et permet d'examiner la forme, la couleur et la surface des objets et done, de leur assigner des attributs. Les proprietes du parvo et du magno se distribuent respectivement selon les voies visuelles dites dorsale et ventrale. II se produit done une segregation tres precoce des proprietes des objets visuels dans les voies primaires. Toutefois, une autre theorie a ete proposee recemment pour interpreter les faits empiriques concernant l'activite des neurones dans VI, V2, ... La couleur et la forme pourraient etre codees dans un meme neurone par le decours temporel de la frequence de decharge des neurones (Eskandar, Optican et al., 1992). Ce codage serait "separable", ce qui veut dire que, si besoin en est, les

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caracteristiques de couleur et de pattern peuvent etre retrouvees dans la forme du decours temporel de la decharge du neurone. Pour ces auteurs, le "liage perceptif' qui assure la recombinaison des proprietes des objets, qui sont separees dans les premiers relais visuels, n'est done pas assure par une « synchronisation » comme le pretendent d'autres theories. La separation observee serait done due a un codage different des combinaisons de proprietes. Le controle centrifuge de la perception visuelle : le cerveau projectif La vision n'est pas un processus a un seul sens, centripete, de la retine vers le thalamus, puis VI, V2, V3, V4, etc... Chacun de ces centres recoit aussi des influences centrifuges : un anatomiste a dit un jour que VI recoit plus d'informations du reste du cerveau que de la retine. Nous savons qu'une situation semblable est vraie pour les noyaux dits vestibulaires dans le tronc cerebral. On peut ainsi, chez 1'Homme, montrer que les aires visuelles primaires sont activees lorsqu'un sujet imagine un objet dans le noir: cette activite, observee grace a la camera a emission de positons, est due a des projections des centres de la memoire vers les aires visuelles primaires. Notre cerveau projectif peut done modifier ce que nous percevons en fonction de ses memoires ou ses projets d'actions qui evoquent des preperceptions. J'ai developpe les consequences de ce fait dans La decision (Berthoz, 2003). Un autre exemple de cette action centrifuge est celui de 1'intention d'action et de signaux moteurs sur 1'activite des neurones visuels, par exemple la direction du regard. La decharge de neurones dans le thalamus visuel ou dans les premiers relais visuels corticaux est en effet modifiee par la position de l'ceil pendant les saccades oculaires, en particulier dans les aires du cortex parietal 7a, VIP, LIP. Plusieurs effets ont ete decouverts ou suggeres : a) Des changements de coordonnees transformeraient le codage retinien de la position d'une cible en un codage dans un referentiel lie a la tete ou meme a l'espace. Recemment, dans notre laboratoire, l'enregistrement des neurones de l'aire VIP, a permis de dissocier, chez le singe, les activites des neurones codes dans un referentiel lie a la retine et a l'espace grace a une procedure de test des champs recepteurs d'un neurone tres rapide qui permettait de tester le champ recepteur associe a chaque saccade, et done ses modifications en fonction de la direction du regard. La nouveaute de cette decouverte reside dans le fait que si dans d'autres aires corticales "visuelles" l'invariance spatiale est codee par des populations de neurones, on a affaire ici a une propriete de chaque neurone. Toutefois cette interpretation est fondee sur l'idee que le cerveau reconstruit la position des cibles en coordonnees spatiales pour controler les mouvements des yeux. Cette idee est contestee par certains qui proposent que le cerveau ne code que des positions relatives de l'ceil par rapport aux cibles et ne travaille done pas dans un espace absolu mais uniquement par une detection d'erreurs.

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b) Le champ recepteur des neurones ne serait pas fixe comme on le pensait, mais se deplacerait avant meme la saccade pour anticiper la direction vers ou le regard se dirige (Colby and Duhamel, 1996). c) La direction du regard est prise en compte au niveau de l'aire MST pour assurer l'invariance perceptive de la direction de notre trajectoire pendant la locomotion. En effet, des enregistrements de neurones suggerent que la distorsion du flux optique, introduite par le fait de regarder de cote pendant la marche, est compensee automatiquement a ce niveau par des signaux moteurs corollaires de la position de l'oeil dans l'orbite (Andersen, Bradley etal., 1996). Cette influence des signaux du regard ne se produit pas seulement au niveau du cortex visuel. Les signaux dits « vestibulaires » sont eux-memes influences par la direction du regard des les premiers relais sensoriels vestibulaires, comme nous avons ete les premiers a le montrer. L'hypothese que j'ai proposee dans plusieurs articles est que la direction du regard est en elle-meme une reference autour de laquelle est organisee la geometrie de notre espace percu. 3. Ontogenese de la geometrie chez l'enfant L'enfant ne possede pas la capacite de traiter des proprietes geometriques des la naissance. Ces proprietes s'etablissent au cours de l'enfance et de l'adolescence ; elles ne se developpent pas de la meme facon chez les garcons et chez les filles. Les deux voies centrifuges et centripetes suivent des processus de developpement chez le jeune animal, et sans doute chez le jeune bebe, qui sont tres differents. Des mecanismes moleculaires guident done ce developpement, independamment de toute action exterieure. Les voies centripetes sont mises en place plus tardivement, au cours de la premiere annee, en meme temps que se developpe le cortex cerebral. Les donnees de la neuroanatomie et de la neurophysiologie du systeme visuel eclairent les etapes du developpement des fonctions d'analyse du cortex cerebral, mais il est interessant d'examiner aussi cette question a un niveau plus global, celui de la Psychologie experimentale. Les theses de Piaget: la succession topologie, geometrie projective, geometrie euclidienne Piaget s'est interesse au developpement de la perception de la geometrie. Dans le livre La Representation de I'Espace chez I'Enfant, ecrit avec Inhelder, Piaget soutient une these principale : l'enfant construit d'abord une representation des proprietes topologiques de l'espace avant d'en comprendre les relations metriques. II critique a la fois Kant et Poincare. « Kant, dit-il, concevait deja l'espace comme une structure a priori de la "sensibilite", le role de l'entendement consistant simplement a soumettre des donnees spatiales perceptives a une suite de raisonnements susceptibles de les debiter indefiniment sans en epuiser le contenu. (...) Poincare, de meme, lie la formation de l'espace a une intuition sensible et rattache ses vues sur la signification du groupe des deplacements au jeu des sensations proprement dites, comme si l'espace sensori-moteur fournissait

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l'essentiel de la representation geometrique et comme si l'intellect travaillait sur du sensible deja tout elabore au prealable »(p. 11). Piaget dit qu'en realite l'enfant construit effectivement, des le debut de son existence, un espace sensori-moteur lie a la fois aux progres de la perception et de la motricite « qui prend une grande extension jusqu'au moment de l'apparition du langage et de la representation imagee (c'est-a-dire de la fonction symbolique en general). Cet espace sensori-moteur est lui-meme greffe sur divers espaces organiques anterieurs (postural, etc..) mais dont il est loin de constituer un simple reflet (...)». «Puis ensuite seulement vient l'espace representatif dont les debuts commencent avec ceux de l'image et de la pensee intuitive, contemporains de l'apparition du langage ». Alors, poursuit Piaget, se produit un phenomene tres curieux... «tout en profitant des conquetes de la perception et de la motricite, (lesquels fournissent, sur leur plan, l'experience de ce que sont par exemple une droite, des angles, un cercle et un carre, des systemes perceptifs,...) la representation precede ab initio comme si elle ignorait tout des rapports metriques et projectifs, des proportions, etc. La representation est obligee de reconstruire l'espace a partir des intuitions les plus elementaires, tels que les rapports topologiques de voisinage, de separation, d'enveloppement, d'ordre, etc. (...) mais en les appliquant en partie deja a des figures projectives et metriques superieures au niveau de ces rapports primitifs et fournies par la perception ». « Faute de preter attention a ce divorce entre la forme de connexions representatives initiales et le contenu perceptif (...), on s'imagine que l'intuition geometrique s'appuie directement sur les donnees sensori-motrices ». Piaget distingue alors l'activite « representative », apparue plus tardivement, et l'activite « perceptive » fondee sur les activites sensori-motrices et il montre le rejaillissement de la premiere sur la seconde. II denonce alors l'equivoque entre les rapports du « representatif » et du « perceptif » du au fait que l'adulte ayant perdu tout souvenir des etapes anterieures s'imagine alors que chaque perception utilise des l'origine des systemes de coordonnees, ou les rapports de verticalite et d'horizontalite, en realite tres complexes qui ne sont acheves que vers 8 a 9 ans. II distingue plusieurs periodes dans le developpement. - La premiere (jusqu'a 4 mois) est caracterisee par une « non-coordination » des divers espaces sensoriels entre eux. Seules les proprietes topologiques de voisinage, de separation, d'ordre, d'entourage ou de d'enveloppement (par exemple, le nez entoure du visage) sont identifiees. Piaget rappelle que Poincare avait identifie un continu "empirique" base sur le voisinage. A ce niveau, dit Piaget, le bebe ne percoit que des rapports spatiaux elementaires qui caracterisent cette partie de la geometrie appelee "topologie", etrangers aux notions de forme rigide, de distance de droite, d'angles, etc. ainsi qu'aux rapports projectifs et a toute mesure. Cette periode ne comporte « que des rapports pre-perspectifs et pre-euclidiens » qui s'apparentent aux relations topologiques elementaires. Mais il s'agit d'une topologie perceptive et motrice et surtout radicalement egocentrique.

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- La seconde periode (4-5 a 12 mois) est caracterisee par la coordination entre vision et prehension qui permet la perception des formes. Mais, dit Piaget « contrairement a I'hypothese centrale de la theorie de la Gestalt, nous croyons que des bonnes formes elles-memes (ou formes euclidiennes simples) se developpent, avec 1'image, en fonction de l'activite sensori-motrice, mouvements du regard, exploration tactile, analyse imitative, transpositions actives (...). La Constance des grandeurs est liee, par exemple, a la coordination des mouvements controles perceptivement. La decentration perceptive de cette periode associee a l'activite motrice aboutit a la constitution de rapports metriques et projectifs ». - Pendant la troisieme periode (seconde annee) les changements de points de vue, les deplacements contribuent a l'elaboration d'une perception des mouvements des objets les uns par rapport aux autres (allocentriques). La seconde moitie de cette periode, « en marquant le debut des coordinations interiorisees et rapides qui caracterisent Facte complet d'intelligence, voit apparaitre l'image mentale en prolongement de rimitation differee (...). De purement perceptif, I'espace devient done en partie, representatif. Ce n'est alors que vers 7-8 ans qu'un espace intellectuel sera construit, qui sera capable de l'emporter definitivement sur I'espace perceptif. C'est la motricite qui est le facteur commun entre ces deux constructions, representative et perceptive. L'image est, pour lui, une imitation interiorisee qui procede par consequent comme telle de la motricite ». Ces analyses de Piaget portent toutes sur le primat de la topologie sur la metrique. II remarque la demarche inverse qu'auraient faite le developpement de la perception d'une part, et la science de la geometrie d'autre part, et ecrit: « on comprend que relevant des conditions elementaires de Taction, ces rapports fondamentaux aient echappe si longtemps a la science geometrique qui a debute avec la mesure et ne s'est engagee qu'extremement tard dans la recherche des notions primitives ». Nous avons, pour illustrer ces theories, pris quelques exemples comme 1'apparition de la notion de droite projective ou de droites affines. Piaget remarque que la droite n'est pas, en effet, une notion topologique car pour transformer une simple ligne (seule envisagee par la topologie) en une droite, il est necessaire d'introduire un systeme ou bien de points de vue, ou bien de visee (notion que Ton retrouve dans l'analyse de Husserl). Le dernier niveau envisage par Piaget apres la construction de I'espace topologique, celui de I'espace projectif, est celui de I'espace euclidien. Celui-ci se construirait par la coordination des points de vue des objets de I'espace entre eux, ce qui conduit a la construction de systemes de coordonnees. Pour Piaget, l'image n'est done jamais que l'imitation interieure et symbolique d'actions d'abord anterieurement executees, puis simplement executables, ce qui evoque evidemment le concept d'« affordance» de Gibson. (Gibson, 1966 ; Gibson, 1977).

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Critiques de la chronologie de Piaget Recemment, plusieurs critiques ont ete faites de la chronologie de Piaget. Le developpement de la rationalite ne peut se reduire a la substitution majorante des structures nouvelles, qu'elles soient symboliques ou sub-symboliques, mais se developper c'est aussi, et souvent, inhiber une structure concurrente. Les auteurs ont tente de relier l'apparition des fonctions superieures a la capacite qu'a le cortex prefrontal d'inhiber des comportements automatiques et primitifs en quelque sorte. J'ai moi-meme insiste dans mon livre Le sens du mouvement (Paris, 1997) sur le role fondamental, et mal connu, de l'inhibition comme facteur de modulation et d'organisation de l'activite. Les theories recentes attribuent a l'enfant des capacites perceptives tres precoces (avant 4 mois) et contestent souvent le constructivisme de Piaget. Cette precocite a ete montree pour plusieurs operations perceptives : a) la Constance de la forme et de la taille, la perception de l'inclinaison. II est interessant de constater que des donnees recentes de neurophysiologie chez le singe confortent ces observations. En effet, des neurones du cortex parietal repondent, chez cet animal, a l'inclinaison des surfaces ; b) la reconstruction d'un objet cache (paradigme dit « d'occlusion »); c) la perception des formes, des contours illusoires, des visages ; il a ete propose, a ce sujet, que le developpement de la geometrie etait du a l'interet de la detection des formes geometriques en mouvement pour F evaluation du risque que representent les objets de l'environnement; d) la perception du mouvement biologique. Le developpement de la geometrie de l'espace de navigation L'espace de navigation et la memoire des deplacements sont, en plus de l'espace de prehension, de manipulation, l'un des espaces les plus importants dans le comportement. La navigation et la representation de l'environnement peuvent etre realisees par des strategies cognitives differentes comme, par exemple, les strategies de simulation des mouvements et des reperes sur une route, qui different de revocation d'une carte. Une tache interessante est celle, par exemple, de reperer des objets dans un environnement et memoriser leur place. C'est la tache que j'appelle « les ceufs de Paques ». Lorsque les enfants ne trouvent pas tous les oeufs, il faut que les parents se rappellent ou ils ont cache les oeufs non trouves. La conclusion est que les enfants de cet age, comme cela a ete aussi montre pour les rats, utilisent les indices geometriques. Les adultes utilisent des combinaisons d'informations geometriques et non geometriques. II est done possible que chez F Homme, il y ait deux modules independants (Cheng, 1987 ; Cheng and Gallistel, 1984 ; Helmer and Spelke, 1994) pour le codage des informations egocentrees et allocentrees, les dernieres utilisant la geometrie des relations de l'environnement, meme a un age tres precoce. L'imagerie cerebrale suggere qu'un module specialise existerait dans le parahippocampe (Epstein and Kanwisher, 1998) et dans le cortex frontal pour le traitement des caracteristiques geometriques de l'environnement. La perception de la geometrie des formes, dans la mesure ou elle est liee a Faction, implique une identification des objets en relation avec leur usage ou leur

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signification. Or, le developpement des mecanismes neuronaux peut 8tre sous l'influence de l'environnement et, en particulier, de l'environnement social. C'est ce qu'a etudie Luria. Inspire par les travaux de l'ecole russe, en particulier ceux de Vitgosky, il a compare la perception des formes geometriques chez des populations agricoles et urbaines. Sa these principale est que I'attribution cognitive de la signification a des formes geometriques est intimement liee au degre de culture de ces populations et qu'il ne faut pas oublier la dimension historique et culturelle dans l'etude des mecanismes de la perception de la geometric 4. La geometrie et le mouvement La perception des formes geometriques, loin d'etre comme on le pense souvent, une propriete statique de traitement d'images, est profondement liee a la perception et au controle du mouvement. Elle est multimodale. Elle est une decision perceptive et pas seulement un traitement passif. Elle evalue des grandeurs pertinentes pour Taction en cours. Les aires medio-temporales (MT ou V5) et medio-temporale superieure (MST), qui font suite aux aires VI, V2, V3, V4 dans le traitement seriel de 1'information visuelle, sont specialisees dans le traitement du mouvement. Recemment, une autre aire specialised dans le traitement du mouvement (KO) a ete decouverte dans le cortex occipital chez l'Homme par imagerie cerebrale. Cette region est situee a peu pres au meme niveau que la region MT et V5 chez l'Homme mais plus posterieure et mediane le long de la surface occipitale. Cette region est particulierement sensible aux frontieres de mouvement, appelees en anglais "kinetic boundaries", que Ton peut provoquer en fabriquant une image dans laquelle des bandes paralleles se deplacent a des vitesses differentes ou meme opposees, comme les voies d'une autoroute. Nous avons considere ici principalement les aires MT et MST parce qu'elles ont fait l'objet d'etudes qui permettent de relier la perception chez l'Homme avec les activites des neurones enregistrees chez le singe de facon remarquable. Proprietes des neurones de MT Les champs recepteurs de MT sont legerement plus grands que ceux des aires precedentes mais nettement plus petits que ceux de MST. Les neurones de VI sont en general accordes sur une direction du mouvement visuel. lis repondent a un bord qui se deplace avec une certaine orientation. lis ne peuvent toutefois pas coder, a travers la petite fenetre sur le monde que constitue leur champ recepteur, de fa§on non-ambigue la direction car ils ne mesurent que la composante du mouvement perpendiculaire au bord. C'est le probleme dit « de 1'ouverture ». Les neurones de MT ont une direction preferentielle ; le probleme de 1'ouverture est done leve a leur niveau. De plus, alors que les neurones de VI repondent de la meme fa§on au mouvement dans leur direction preferee, meme s'il y a dans le champ recepteur un autre mouvement dans une autre direction, les neurones de MT, eux, seront fortement inhibes s'il y a un autre mouvement dans une autre

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direction. Cette propriete dite « de suppression contraire » est peut-etre importante pour enlever du bruit dans la reconstruction des surfaces en mouvement dans laquelle MT est particulierement impliquee. L'aire MT joue aussi un role dans la perception de la profondeur selon laquelle sont disposees des surfaces en mouvement, la segmentation de ces surfaces, par exemple ces neurones repondent a des ensembles coherents de points qui se deplacent. II s'agit done d'une aire ou les traitements visuels les plus elabores pour la reconstruction de la geometrie des objets et de l'espace en mouvement sont executes. La reconstruction de la forme tridimensionnelle avec le mouvement Les objets de l'environnement ont une forme dans l'espace tridimensionnel. Or, leur image se projette sur la surface spherique de la retine. Comment le cerveau reconstruit-il la forme 3D ? Le mouvement est utilise par le cerveau pour cette reconstruction. Nous avons presente une demonstration de notre laboratoire sur cette propriete et avons decrit des experiences faites recemment grace a l'imagerie cerebrale par emission de positons qui revelent les aires corticales impliquees dans cette reconstruction de la forme a partir du mouvement. On a utilise des perceptions «bistables» e'est-a-dire des stimulations visuelles qui peuvent evoquer alternativement une perception 2D ou 3D. Des enregistrements de neurones chez le singe, en utilisant aussi des presentations de stimuli visuels bistables (cylindres), ont montre que les neurones de MT modifient leur activite lorsque le percept (evalue chez le singe par des methodes psychophysiques) change. Proprietes des neurones de MST Les neurones de MT se projettent sur MST ou sont accomplies d'autres transformations et analyses des messages sur le flux optique. Les neurones de MST ont des champs recepteurs plus grands que MT. II y a done dans ces voies un agrandissement progressif des champs recepteurs. lis ont egalement une selectivity directionnelle, mais ils repondent aussi a de nombreuses autres caracteristiques du flux optique : la rotation, l'expansion et la contraction, le « curl», la spirale, la divergence, le mouvement laminaire, etc. L'idee que le cerveau contient des neurones selectifs pour ces aspects geometriques du mouvement operateurs avait ete suggeree par la psychophysique. L'existence d'un repertoire de formes dynamiques ainsi detectees par les neurones de MST correspond peut-etre a la distribution des vecteurs de mouvement dans le flux optique pendant les mouvements naturels. Si, par exemple, nous avancons en ligne droite en fixant un point au sol, le flux optique aura, autour du point de fixation, la forme d'une spirale. Les neurones de MST ont une capacite remarquable a continuer a signaler ces composantes du mouvement visuel quelle que soit la localisation du stimulus dans le champ recepteur qui, rappelons-le, est tres large. On appelle cette propriete « l'invariance de position et d'echelle ».

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L'influence des signaux moteurs du regard Les signaux moteurs du regard modifient les decharges des neurones de MST, qui est done un centre de convergence pour les messages visuels et des messages dits « extraretiniens », endogenes, venus d'autres parties du cerveau et associes a Taction. Par exemple, la decharge des neurones de MST ne cesse pas, contrairement a celle des neurones de MT, lorsque l'ceil poursuit une cible visuelle et que la cible disparait de facon transitoire. C'est un signal d'origine motrice qui entretient la decharge pendant un bref moment; ce signal n'est pas necessairement une decharge corollaire de la commande motrice : il peut aussi etre une prediction de la position ou du mouvement de la cible. Le cerveau est un predicteur. Ces signaux lies au mouvement du regard sont aussi utiles pour le maintien de l'invariance de la direction pergue du mouvement propre pendant la navigation. Gibson avait propose l'idee que pendant une translation vers 1'avant une portion particuliere du champ visuel, appelee «le foyer d'expansion de l'image retinienne », peut etre utilisee pour determiner la direction du mouvement. Ce foyer d'expansion est dans la direction de la marche si les yeux et la tete ne bougent pas et sont parfaitement dans l'axe sagittal du corps. Si les yeux bougent, alors un mouvement laminaire se produit dans le flux optique, en sens contraire du mouvement des yeux, et s'ajoute a l'expansion, creant un flux optique extremement complexe. Or, nous continuons a percevoir la direction de notre marche en avant, par consequent celle du tronc, meme si la tete et les yeux bougent. On a recemment demontre chez le singe que les neurones de MST peuvent etre impliques dans la correction du flux optique par les mouvements du regard de facon a retablir un flux optique qui corresponde a la direction de la marche. L'invariance perceptive est done peut-etre realisee a ce niveau du traitement. De meme, ces neurones re§oivent des entrees vestibulaires et la compensation peut done etre aussi realisee pour des mouvements de la tete. Autres aires corticales impliquees dans la perception des formes geometriques La sensibilite a la rotation a ete trouvee au-dela de MT, dans l'aire parietale PG. On a suggere que ces neurones combinent une mosai'que de detecteurs de direction circulairement disposes ; ils pourraient etre de veritables detecteurs de "curl". Mais les neurones du cortex parietal detectent aussi des proprietes complexes qui ne sont pas necessairement liees au mouvement dans un plan fronto-parallele mais sont peut-etre impliquees dans la detection des objets ou des surfaces.

5. La perception des objets Les questions que pose la remarquable capacite du cerveau a reconnaitre des objets sont nombreuses : existe-t-il un systeme unique pour la reconnaissance de tous les objets dans le monde ou y a-t-il plusieurs systemes suivant la categorie des objets (visage, train, chien, arbre, etc.) ? les objets naturels sont-ils plus facilement percus que les objets artificiels ? comment s'effectue la classification des

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objets ? comment est reconstruite I'unite de I'objet, on dit aussi comment se fait le "liage", apres la decomposition qu'effectue la vision en ses composantes de mouvement, couleur, forme, contrastes, etc. ? les objets sont-ils codes dans des referentiels lies a I'objet ou a l'observateur ? quelles sont les relations entre le codage de I'objet et les actions que l'observateur envisage de faire ? comment est assuree I'invariance de I'objet percu lorsqu'il se deplace ou lorsque l'observateur change de point de vue ? le codage de I'objet est-il dependant du point de vue ou abstrait ? comment interagissent les descriptions verbales de I'objet et les descriptions spatiales ? La litterature concernant ce domaine, surtout pour la perception visuelle, est considerable. Nous citerons trois aspects seulement. Le premier concerne la decomposition des objets en formes prototypiques ou composants. Une des theories recentes sur cette question, est celle de Biederman (Biederman, 1987). Le deuxieme aspect concerne la perception des visages. Le visage est, en effet, un objet d'un interet primordial qui est a la fois geometriquement tres bien defini, porteur de messages qui suscitent la crainte aussi bien que la tendresse, tout a la fois predateur et proie, principal interprete des emotions et des intentions, porteur de l'histoire de chacun d'entre nous mais aussi reflet des conventions et des grands courants de la societe ; son dessin peut etre le temoin de troubles profonds de la cognition. On peut a ce sujet evoquer les theories modernes qui proposent l'idee que I'unite de I'objet est realisee par la synchronisation temporelle de l'activation des structures dans lesquelles sont memorisees ou activees les composantes de I'objet au meme moment. Dans ces theories, I'unite de I'objet est done assuree par le temps. La decomposition de I'objet en formes prototypiques et en composantes Les resultats recents concernant la reconnaissance visuelle des objets suggerent qu'il n'existe pas, dans le cerveau, de systeme general de reconnaissance des objets a usage multiple. Au contraire, on trouve des representations multiples des objets, formees dans diverses parties du cerveau, chacune specifique aux transformations requises soit pour la perception soit pour Taction. La reconnaissance des membres prototypiques d'une categorie d'objets, le codage des transformations des objets ou des parties des objets, le codage de la taille, de l'orientation, de la forme, de l'identification des membres individuels d'une classe d'objets homogenes, et la planification des mouvements qui sont, en general, associes aux differents objets familiers, dependent de representations differentes formees dans des centres nerveux nombreux et grace a des ensembles de liaisons neuronales differentes. II existe, de plus, des contraintes severes sur la capacite qu'a le cerveau de reconnaitre des formes. Par exemple, nous ne pouvons pas identifier un visage a l'envers. Cet effet n'existe pas chez le singe qui a l'habitude de voir des visages suivant de multiples orientations. De meme, le jeune enfant semble reconnaitre les visages, meme a l'envers. La polarisation de la perception est done acquise en meme temps que se mettent en place, comme nous l'avons vu, les mecanismes qui

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permettent l'identification et la memorisation de configurations complexes avec la mise en place de fonctions corticales superieures. II est lie a l'identification de "configurations" des elements dans la reconnaissance des visages. L'identification des objets est done parfois independante du point de vue ; elle semble "centree sur l'objet lui-meme" dont les proprietes ont ete en quelque sorte "generalisees" comme disent les psychologues; mais elle est parfois dependante du point de vue. Toutefois, les travaux de Psychologie cognitive n'ont pas encore elucide completement cette question. D. Marr (Marr, 1982) a ete un des premiers a proposer l'idee selon laquelle le systeme visuel reconstruit progressivement les objets a partir de leur analyse decomposed par ses premiers relais. Nous avons choisi d'analyser la theorie recente proposee dans ce sens par Biederman. II propose que la perception des objets est fondee sur la decomposition des images en composantes qu'il appelle des « geons ». La premiere decomposition serait effectuee dans les premiers relais VI, V2, V3, etc. Elle assure 1'extraction des contours de l'objet grace aux informations de luminance, texture, couleur, etc. Elle donne une information de contour par des lignes. II faut noter I'existence d'une theorie recente parallele qui suggere la possibility que soient extraits non seulement le contour mais le squelette d'objets qui serait particulierement utilise pour l'analyse des objets non rigides en mouvement comme, par exemple, les animaux. Puis se produirait une detection des proprietes dites «non accidentelles » telles que la colinearite, la symetrie, la courbure continue, le parallelisme, etc., et un decoupage en zones principalement dans la partie concave des formes. Cette double analyse permet de decomposer la forme de l'objet en composantes. L'etape suivante serait la comparaison des composantes avec des representations presentes dans la memoire. On voit que la memoire est essentielle dans ce processus d'analyse. Ces memoires sont composees d'elements simples appeles « geons » (pour ions geometriques). Un geon est un volume defini par le deplacement d'une courbe fermee dans un plan le long d'un axe suppose etre a angle droit par rapport a la surface plane comme des cylindres, des cones, des cubes, des parallelepipedes, etc., qui correspondent aux criteres gestaltistes de « bonne forme ». Une bibliotheque d'environ 36 geons permettrait d'identifier un nombre considerable d'objets largement superieur a tous les objets que nous rencontrons ou que nous souhaitons memoriser et reconnaitre. Une des consequences importantes de la theorie de geons est que le principe de la Gestalt s'applique a chaque geon et non pas a la figure dans son ensemble. La consequence en est que ce sont les composants qui sont stables en cas de bruit perceptif. Le probleme est de savoir si la neurophysiologie moderne confirme cette decomposition en formes elementaires. Nous avons vu dans les cours precedents que Ton distingue actuellement deux grandes voies neuronales : la voie dite « dorsale » et la voie dite « ventrale » dans l'elaboration des informations visuelles

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et les relations avec Taction. Un accord est maintenant fait sur Pimportance de la voie ventrale dans 1'identification des objets. II est aussi suggere que les deux hemispheres droit et gauche analysent des proprietes differentes des objets. Certains pretendent en effet que l'hemisphere droit s'interesserait aux proprietes globales de forme des objets et travaillerait dans un repere spatial, alors que l'hemisphere gauche ne serait concerne que par les attributs locaux de chaque objet, entre analyse des proprietes spatiales de coordonnees et proprietes categorielles, comme le propose Kosslyn. Role du cortex infero-temporal dans la reconnaissance des objets L'aire infero-temporale (IT) est une region du cerveau qui, chez le singe, est situee dans la zone juste anterieure au sillon occipital inferieur et s'etend jusqu'au pole temporal et du fond du sillon temporal superieur (STS) au fond du sillon occipito-temporal. II correspond a peu pres aux aires 20 et 21 de Brodman. II est divise en aires TEO et TE pour certains et en PIT, CIT et AIT pour d'autres, mais de nombreuses subdivisions ont depuis ete introduites dans cette zone du cerveau. TEO recoit des projections organisees de facon topographique des aires V2, V3 et V4 et, en retour, se projette aussi sur ces aires. Les traitements qui se font dans TEO peuvent done influencer les etages precedents. TEO se projette sur de nombreuses aires du cortex temporal mais aussi vers les FEF, l'aire 46, l'amygdale, l'hippocampe, etc., et des structures sous-corticales telles que la formation reticulee, le colliculus superieur. Cette partie du cerveau est connue pour son role dans la reconnaissance des objets, la formation des habitudes, la memoire associative, et des deficits de ces fonctions ont ete observes chez des patients porteurs de lesions. Ces aires continuent la chaine des traitements visuels et sont disposees fonctionnellement de facon sequentielle. La taille des champs recepteurs le long de cette chaine augmente de 2 a 3 degres jusqu'a 30 a 50 degres, suggerant que des proprietes locales des objets sont progressivement integrees dans des proprietes plus globales de l'environnement. Les neurones de IT sont sensibles aux attributs visuels comme la couleur, l'orientation, la texture la forme. Mais, bien que deja dans V4 on trouve des neurones sensibles a la forme, e'est dans IT que Ton trouve une grande proportion de neurones qui repondent a des variations de formes complexes, a des objets synthetiques, des descripteurs de forme parametrique, ou des fonctions de Walsh qui sont des patterns visuels produits mathematiquement. De plus, cette region contient des neurones specifiquement sensibles aux visages ou a la main. Certains neurones repondent a un objet meme si on en change les attributs comme la couleur, le mouvement ; par contre, ils sont sensibles aux contrastes, ce qui est coherent avec le fait que la reconnaissance d'une forme est liee aux ombres. IT contient done tous les elements necessaires pour la reconnaissance des objets mais la question est de savoir si les objets sont represented par quelques neurones « gnostiques » ou par l'activation conjointe de petits groupes de neurones qui

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represented chacun les differents attributs ou les geons de l'objet, ou encore s'ils sont representes de facon implicite par des combinaisons de decharges dans des populations larges de neurones. II semble actuellement qu'au moins deux sortes de code soient presentes. D'une part, des neurones qui codent pour des composants de la forme ou des parties et qui peuvent contribuer a l'identification d'un objet par leurs combinaisons en fonction des relations spatiales des composants ainsi definis, ce qui fait penser aux geons de Biederman, mais aussi des neurones qui codent les aspects globaux des objets avec une selectivity particuliere a la configuration des elements. De plus, par des methodes d'imagerie optique de portions du cortex IT chez le singe, on a recemment suggere que dans une zone d'environ 1 mm2 de surface sont presents des neurones qui codent les aspects successifs d'une forme ou d'un visage, par exemple plusieurs points de vue. Ceci suggere que les attributs codes pour une meme forme sont places dans des regions bien delimitees et de facon tres organisee. Le cortex temporal inferieur coopere avec le cortex perirhinal et enthorinal, pour assurer la memorisation et le rappel des formes visuelles lorsqu'il faut comparer une forme nouvelle avec une forme deja vue. Des neurones pour ajuster la main a la forme des objets Des travaux recents apportent des donnees sur la detection de la forme dans des taches de prehension, autrement dit, le couplage entre perception et action. Rappelons que, lorsque nous attrapons un objet, le cerveau traite en parallele les commandes de la trajectoire du bras et celles qui controlent les doigts et permettent d'adapter la forme de la main a la taille des objets. Or, on trouve dans l'aire intraparietale inferieure (AIP) des neurones qui dechargent en relation avec la forme de l'objet qui est saisi par 1'animal. La plupart des neurones de type visuomoteurs preferaient un objet particulier a la fois pendant la manipulation et la simple fixation visuelle sans manipulation. Ceci est vrai lorsque l'objet a une forme canonique proche des geons de Biederman (cylindre, cone, sphere, etc.). lis jouent sans doute un role important pour ajuster le pattern de mouvement de la main aux caracteristiques spatiales de l'objet pendant la manipulation. Ceci correspond bien aux deficits observes chez les patients porteurs de lesions de cette zone qui ne peuvent pas ajuster la forme de la main a l'objet. D'autres neurones repondent a l'orientation de l'axe d'une barre lumineuse dans l'espace. Certains d'entre eux associent le codage de la forme de l'objet (cylindre, sphere) a celui de l'orientation de son axe. Ceci pourrait correspondre aux observations chez des patients porteurs de lesions parietales qui ont une perception biaisee de l'orientation des objets. Dans d'autres aires parietales, on trouve des neurones sensibles a la fois a des surfaces et a une orientation particuliere de la surface. Par consequent, le cortex parietal est implique dans la perception des proprietes 3D des objets liees a la stereopsie (beaucoup des reponses observees exigeaient la vision binoculaire).

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Le systeme cortical dorsal est done divise en deux voies distinctes. L'une a un relais dans l'aire V5 (MT) et se termine dans V5a (MST) et l'aire VIP. Elle est impliquee dans la vision du mouvement. L'autre voie contient un relais dans l'aire V3-V3a et se termine dans des aires autour des sillons parieto-occipital et intraparietal. Les aires V6, LIP, et 7a sont concernees par la representation de la position en coordonnes egocentriques alors que IPS est important pour le traitement des informations de haut niveau sur la stereopsie et la forme 3D. L'aire AIP est essentielle pour le contr61e du mouvement de la main pendant la prehension. Elle influence l'aire F5 du cortex premoteur qui envoie les commandes motrices au cortex moteur afin d'accomplir la manipulation des objets. L'avenir dira si cette correspondance remarquable entre les predictions de la theorie de Biederman et les resultats de la neurophysiologie tient la route mais on voit ici comment les deux approches theorique et experimentale nous conduisent a mieux comprendre les fondements cognitifs de la perception de la geometrie a defaut encore de nous reveler les fondements cognitifs de LA geometrie. 6. La neuropathologie de la cognition de la geometrie Le deficit de la reconnaissance des formes visuelles (Griisser and Landis, 1991) chez l'Homme est appele « agnosie visuelle ». Bien que deja Thucydide decrivit en 430-429 avant J. C des agnosies chez des patients apres une epidemie de peste a Athenes, et bien qu'Hippocrate fit allusion a cette maladie dans son livre, les premieres descriptions modernes furent faites par Quaglino en 1867. Munk en 1877 remarqua que des chiens pouvaient voir des objets mais ne les reconnaissaient pas et nomma ce deficit "cecite corticale". C'est Freud qui inventa le terme "d'agnosie visuelle" et utilisa ce concept comme Wernicke. La periode de 1880 a 1900 vit fleurir les travaux de Charcot, Wilbrand, Dejerine et des ecoles de neurologie franchise et allemande. Particulierement pertinent pour les questions que nous avons etudiees dans ce cours, on peut noter le travail de Pick qui decrivit, en 1908, un patient qui, a la suite d'une atrophie du lobe occipital, ne pouvait plus reconnaitre des objets de grande taille bien que son acuite visuelle et sa capacite a reconnaitre la couleur soient normales. Ceci est sans doute la premiere identification de ce qui est appele aujourd'hui "l'agnosie simultanee". De nombreuses revues recentes resument les decouvertes sur la pathologie de ces fonctions. Notons toutefois l'interessante distinction faite par Liepman en 1908 de ce qu'il appela l'agnosie "dissolutionnelle" qui comporterait trois niveaux : un deficit de la fusion des signaux sensoriels a l'interieur d'une seule modalite ; un deficit de la memoire liee a une seule modalite; enfin un deficit de l'association des "images", ou plutot des representations de l'objet dans differentes modalites sensorielles. II etait, dans le cadre de ce cours, hors de notre propos de resumer un siecle de recherche et d'observations cliniques. Nous avons toutefois rappele la classification du livre de Landis et de Griisser. Ces auteurs proposent de diviser les agnosies en 5 niveaux qui correspondraient a des niveaux d'integration des informations

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sensorielles : la sensation ; la perception ; l'aperception ; l'association ; la cognition. Wundt fut le premier a definir le niveau d'analyse des objets qu'il appela «aperceptif». Lissauer, en 1890, proposa de diviser le processus de reconnaissance en deux etapes : l'etape de conscience d'une impression sensorielle ou « aperception », et l'etape d'associations d'autres notions avec le contenu de la perception ou « association ». Ce concept a ete l'objet d'un nombre considerable de discussions et a ete reformule de nombreuses fois. Landis (p. 204) note que, chez tous les patients presentant un deficit aperceptif, la perception des qualites visuelles est intacte ou montre des deficits mineurs. Ceci est en contraste saisissant avec leur deficit de reconnaissance, de discrimination, de comparaison ou de copie des formes visuelles simples. Par exemple, ces patients ne reconnaissent pas des figures faites avec des points, ils ont du mal a effectuer la difference entre la figure et le fond, et semblent done avoir un deficit de la perception des formes au sens de la Gestalt. Milner precise a propos du malade DF : « la perturbation de la reconnaissance des formes chez DF ne vient pas d'un deficit de canaux sensoriels particuliers (les detecteurs de contours par exemple) mais d'un deficit qui intervient a un niveau plus eleve ou au-dessus d'un mecanisme general d'extraction des primitives de la forme ». Ces malades ont une capacite a detecter le mouvement. Landis remarque que certains des patients de Golstein, Efron, Adler, et le patient qu'il a etudie utilisent meme des strategies curieuses de tracage des formes par une action de poursuite visuo-motrice pour reconstruire la forme. Ceci serait l'equivalent des strategies de route utilisees par certains sujets pour memoriser un trajet. Bien que tres peu de donnees anatomiques detaillees soient disponibles sur les patients qui souffrent d'agnosie aperceptive, il semblerait possible que cette affection implique les voies temporo-occipitales et done plus particulierement le systeme magnocellulaire. 7. La geometrie et le controle du geste et de la posture Comment intervient la geometrie dans le controle du mouvement ? Jusque dans les annees 1980, les neurophysiologistes ont ete interesses par les bases neurales des reflexes et ont surtout cherche a comprendre comment etaient resolus des problemes de dynamique : par exemple, comment, a partir de la detection de 1'acceleration de la tete, le cerveau pouvait produire des signaux qui controlaient la position du regard, ce qui suppose des integrateurs au sens mathematique, e'est-adire des operateurs qui transforment l'acceleration en vitesse et puis en position. Puis vint l'idee que le cerveau devait aussi resoudre des problemes de geometrie. Par exemple, les informations de mouvement de la tete, codees par les capteurs vestibulaires selon les trois plans des canaux semi-circulaires, dans un referentiel euclidien trirectangle, doivent subir des transformations pour produire la contraction correcte des muscles oculaires et cephaliques. On proposa alors d'assimiler le cerveau a un tenseur qui effectue des transformations de coordonnees dans des espaces differents et resout les problemes dits de « transformation inverse » (Pellionisz and Llinas, 1980).

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On proposa aussi que le cervelet est l'organe qui contient les operateurs neuronaux pour realiser ces transformations inverses. Une formulation plus recente de ces memes hypotheses a ete donnee suggerant que le cervelet est un veritable modele interne des proprietes mecaniques des membres. Grace a ce modele interne neuronal, une veritable simulation interne de la geometrie et de la cinematique de la trajectoire du mouvement ainsi que les corrections appropriees peuvent etre realisees avant meme que le mouvement soit execute. L'idee que le cerveau effectue des transformations visuo-motrices geometriques complexes pour le controle du mouvement a ete a l'origine de nombreuses etudes recentes. Mon objectif n'est pas de decrire en detail ces mecanismes, mais, en suivant le cadre general des theories, d'inverser completement l'approche traditionnelle. En effet, si Ton se contente de croire que le cerveau va mesurer toutes les informations des sens (vision, informations vestibulaires, mesures des longueurs et des vitesses d'etirement par les muscles, etc.), les informations a traiter seraient considerables, heterogenes, codees dans des referentiels tres differents avec des proprietes geometriques et dynamiques differentes. Compte tenu des centaines de degres de liberte a controler, le probleme serait probablement insoluble ou au moins demanderait beaucoup de temps et ne permettrait pas la necessaire rapidite des gestes, la prediction, et 1'anticipation. De plus, les systemes biomecaniques des membres sont surdetermines, il y a plusieurs trajectoires possibles pour arriver a un point donne de l'espace. Comment done le cerveau choisit-il la solution particuliere ? Quelles sont les contraintes biomecaniques, neurales, qui limitent le nombre de solutions, etc. ? Est-ce la dynamique qui est controlee ou est-ce la geometrie ? Quelles solutions la nature a-t-elle trouvees pour simplifier la complexite de la geometrie du corps ou encore, comme le disait Bernstein (Bernstein, 1967), reduire le nombre de degres de liberte ? La correspondance entre les espaces des capteurs sensoriels, par exemple le fait que les plans des canaux semi-circulaires sont les memes que ceux des directions preferentielles des neurones du systeme optique accessoire ; la stabilisation de la tete qui devient, grace a cette stabilisation, une plate-forme de guidage pour la coordination des mouvements (Pozzo, Berthoz et al., 1990 ; Berthoz and Pozzo, 1988) ; la geometrie du squelette, par exemple les blocages articulaires (Graf, de Waele et al., 1994), (Berthoz, Graf et al., 1991) et le fait remarquable que le cerveau connait les regies des mouvements naturels possibles ; la geometrie des muscles, par exemple les muscles bi-articulaires ; les synergies musculaires qui constituent un repertoire de mouvements tres simples et des contraintes cinematiques dans 1'execution de ces synergies et de leurs relations, par exemple la loi de 1'antiphase entre bras et avant-bras (Lacquaniti, Soechting et al., 1986) ; les bases neurales des synergies, l'anatomie des collaterales d'axones code la geometrie ; les lois simplificatrices, par exemple, la loi de Listing (Hepp, 1994 ; Haslwanter, 1995 ; Misslisch, Tweed et al, 1994 ; Hepp, 1995) et la loi de la puissance 1/3 qui lie la vitesse tangentielle et la courbure de la trajectoire d'un geste (Viviani and Flash, 1995 ; Wann, Nimmo-Smith et al., 1988) ; le choix, par

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le systeme nerveux, de variables globales et non pas de variables locales, par exemple les mouvements du bras qui sont controles par la definition de l'azimut et de l'elevation, la dissociation entre le controle de la distance et de la direction, etc. Le controle de la posture et de I'equilibre Les mecanismes fondamentaux du controle de la posture ne sont pas encore connus. L'ecole reflexologique a suppose qu'une chaine de reflexes assurait la regulation posturale et la resistance aux perturbations. A 1'oppose de ce point de vue, des theories « globalistes » utilisant le concept de schema corporel propose par les neurologues du debut du siecle, ont soutenu que la posture et I'equilibre sont controles de « haut en bas » par un mecanisme cortical (Gurfinkel, 1994). Le debat est encore ouvert car meme les variables controlees ne sont pas encore connues. Certains suggerent que c'est la position du centre de gravite ou du centre de masse dans le poly gone de sustentation qui est controlee ; d'autres suggerent que c'est la geometrie du squelette ; d'autres enfin ont propose que le cerveau controle des relations entre des variables cinematiques qui correspondent a des synergies posturales simples, par exemple la strategic dite « de la cheville » qui fait tourner le corps autour de la cheville comme un pendule inverse, et une autre dite « de la hanche » qui le fait se plier autour du bassin. Le controle postural resulterait d'une combinaison de ces strategies elementaires (Nashner, 1985). Une derniere suggestion, enfin, est que le cerveau controle la geometrie (Lacquaniti, 1997), c'est-a-dire les rapports entre les angles que font les segments corporels entre eux. Par exemple, la longueur totale et l'orientation par rapport a la verticale de chaque membre seraient controlees avec precision. L'idee est que ces deux variables, orientation et longueur, sont controlees independamment par le cerveau et que les neurones des voies dorso-spino-cerebelleuses les codent separement. La caracteristique remarquable de ce codage est que ces deux parametres, orientation et longueur du membre, definissent la position de la patte dans des coordonnees polaires globales sans tenir compte du detail de la configuration du membre qui permet d'atteindre cette posture particuliere. Elles definissent done une posture sans se preoccuper de la facon dont elle est executee. La litterature contient des propositions theoriques sur la facon dont le cerveau pourrait realiser ce type de controle. La theorie du point d'equilibre Une theorie dite du « point d'equilibre » a ete decrite dans un cours precedent. Rappelons que l'idee de Bernstein fut que la position d'un membre peut ne pas etre specifiee en tant que telle mais peut etre codee implicitement en definissant la relation des tensions de deux muscles antagonistes autour d'une articulation. Le « point d'equilibre » qui sera atteint lorsqu'on impose aux muscles une certaine force ou tension determine la position. Cette theorie est interessante car elle implique que le cerveau peut se contenter de controler une seule variable qui est la relation ou le rapport des deux tensions. Si Ton approfondit encore la theorie, on

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decouvre que le cerveau pourrait se contenter de controler un seuil de sensibilite des motoneurones car, en controlant ce seuil, on controle la contraction du muscle. (Feldman and Levin, 1995). Comme on le comprend peut-etre par ces explications trap breves, la theorie du point d'equilibre suppose que la geometrie soit codee de facon implicite et non pas explicite. II n'est pas necessaire d'avoir dans le cerveau des tenseurs et des representations cartographiques de la geometrie du squelette, le cerveau utilise des variables intermediaires pour transformer des instructions globales (azimut, elevation, orientation, longueur, etc.) en activite musculaire. Ceci expliquerait aussi pourquoi un meme point peut etre atteint avec des configurations differentes. Nous travaillons nous-memes sur l'idee que le cerveau utilise des variables composites (Hanneton, Berthoz et ai, 1997) dont la theorie a ete proposee par Slotine. Posture et geometrie En mesurant les angles que font, chez le chat et chez 1'Homme, les angles des pieds par rapport a la jambe, de la jambe par rapport a la cuisse, de la cuisse par rapport au tronc, on observe qu'ils sont lies deux par deux par des relations lineaires pendant des reajustements posturaux. Ceci se traduit par le fait que, si Ton porte sur un diagramme tridimensionnel ces relations d'angles, les points experimentaux restent dans des plans. Cette covariation des angles des articulations de la jambe suggere qu'il se produirait une transformation intermediaire de type inverse qui transforme les cordonnees de l'extremite (ce qui est controle) en un ensemble de commandes qui reglent les angles des segments corporels entre eux selon des rapports tres fixes. Lacquaniti (Lacquaniti, 1997), qui a fait ces observations, suggere que ce comportement est caracteristique des systemes dynamiques gouvernes par des attracteurs chaotiques. La contrainte planaire pourrait emerger d'un isomorphisme entre le modele interne du schema corporel, le mouvement reel du membre et sa perception par la proprioception. Alors, l'orientation specifique du plan dans l'espace des articulations devient signifiante non seulement d'un point de vue du controle moteur, mais aussi d'apres une perspective sensorielle. En resume, l'equivalence motrice implique que se produise une recalibration continue de l'espace sensori-moteur par 1'intermediaire d'associations sensori-motrices variables. La theorie du minimum de secousse. Son caractere morphogenetique Une autre theorie, concue pour expliquer la generation de trajectoire pendant un geste, suppose que la variable controlee par le systeme nerveux ait simplement un minimum d'energie (minimum de secousse, la secousse etant la derivee de 1'acceleration). II suffirait, d'apres cette theorie due a Flash et reprise par Flash et Viviani (Viviani and Rash, 1995), que le cerveau connaisse les points de depart et d'arrivee du geste, ainsi qu'un ou plusieurs points intermediaires (appeles « via points» en anglais), pour que la trajectoire soit definie. Cette theorie est interessante car elle suggere que le cerveau ne connatt pas la trajectoire

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geometrique du geste qui est planifie. II n'y aurait pas dans le cerveau de planification de la courbe du geste. Celle-ci resulterait de la mise en oeuvre du principe de minimisation de la secousse avec ces quelques contraintes de position. C'est en cela que Ton peut dire que le minimum d'energie est morphogenetique. Notons que la theorie du point d'equilibre a de semblables proprietes. Pour le moment cette theorie n'a ete testee que sur des mouvements de la main et aucun travail experimental n'a ete realise sur la posture. 8. La simulation mentale de la geometrie des trajets au cours de la navigation On a decrit, depuis les auteurs grecs presocratiques, les strategies mentales que le cerveau peut utiliser, pour memoriser les mots, les evenements, ou eventuellement les objets. Par exemple il utilise une methode, "mnemotechnique", qui consiste a les ranger mentalement dans des villes, des palais, des maisons, ou parfois a les disposer sur des echelles ou dans des pots comme un apothicaire (Yates, 1984). Les representations internes de l'espace sont done utilisees par la memoire et la recuperation des donnees s'effectue grace a une veritable navigation mentale dont les regies de fonctionnement cognitif sont sans doute les memes que celles de la navigation mentale que nous faisons lorsque nous essayons de retrouver un trajet. Pour effectuer cette interaction entre la memoire et l'espace, le cerveau peut utiliser differentes strategies cognitives (Tolman, 1949) (Tolman, 1948) (Garling, Book et al., 1982 ; Allen, Siegal et al., 1978). On trouvera des resumes recents de la litterature dans les ouvrages de (Denis, 1989), (Golledge, 1999), (Burgess, Jeffery et al., 1999), (Kosslyn, 1980 ; Kosslyn, Alpert et al, 1993). Par exemple nous pouvons nous rappeler une route que nous avons parcourue de la porte d'un hotel a une salle de conference ou de notre maison a notre bureau de plusieurs facons. La premiere de ces strategies cognitives peut etre appelee la "strategie de route" qui consiste a se souvenir des mouvements, des tournants, des translations, que nous avons effectuee et a les associer a des reperes visuels que nous avons remarques. Cette memoire des routes suppose des mecanismes assez complexes qu'il faut decomposer pour esperer les comprendre un jour. Une deuxieme strategie cognitive pour se rappeler un trajet a ete appelee par les psychologies qui Ton etudiee "strategie de survol". Elle consiste a evoquer une carte de l'environnement vue de dessus et a suivre notre trajet sur cette "carte mentale". La strategie de survol, qui consiste a visualiser une carte, ou a en faire une description propositionnelle, est utilisee en particulier lorsque nous cherchons a nous souvenir de grandes distances. Elle a ete etudiee par l'imagerie. On doit a Kosslyn de nombreux travaux sur cette facon de manipuler mentalement la representation d'un environnement (voir aussi Mellet, Tzourio et al., 1995). Une troisieme strategie a ete decrite (Thorndyke and Hayes-Roth, 1982). Lorsque, par exemple, nous essayons de nous rappeler ou est le bureau d'un de nos

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collegues dans un batiment. Dans ce cas, le cerveau construit une representation interne en quelque sorte transparente du batiment qui n'est ni une carte, ni vraiment une route constituee de sequences de lieux et de mouvements. De nombreux auteurs ont elabore des classifications de ces strategies mentales. Par exemple Touresky et Redish (Touretzky and Redish, 1996) ont propose de diviser les strategies de navigation en quatre sortes : les deux precedentes (route et carte), plus deux autres qu'ils appellent la navigation par "taxons" (il s'agit de l'approche directe d'un but ou de l'evitement d'un obstacle, c'est la plus primitive des navigations); et la navigation "praxique" qui serait une navigation purement endogene due a une sequence de mouvements programmes en interne (peut etre la trajectoire d'une ballerine sur la scene serait un exemple de cette navigation). lis ont propose dans le livre de Burgess et al. (1999) cite plus haut, une synthese de cette theorie. (Trullier, Wiener et al., 1997) ont aussi propose une classification des differentes sortes de navigation en quatre categories. La strategie de route Considerons maintenant la strategie que nous avons appelee de "route". Cette strategie est tres importante a de nombreux egards. Par exemple le mathematicien Henri Poincare a propose que cette simulation mentale de nos deplacements dans l'espace intervient dans les fondations de la geometric II ecrit: « localiser un point dans l'espace, c'est simplement se representer le mouvement qu'il faut faire pour l'atteindre ». II precise ensuite que lorsqu'il dit que nous nous representons le mouvement il veut dire que nous nous representons les "sensations musculaires" qui sont associees a ce mouvement et qui n'ont aucun caractere geometrique. Nous avons propose (Droulez, Berthoz et al., 1985 ; Droulez and Berthoz, 1991 ; Droulez and Berthoz, 1988 ; Droulez and Berthoz, 1990) l'idee que le cerveau ne represente pas l'espace sur des cartes spatiotopiques seulement. Nous avons suggere que, pour produire et controler des mouvements, le cerveau utilise un processus de reactualisation dynamique, utilisant un mecanisme de "memoire dynamique". Cette memoire consiste en fait en une reactualisation permanente de la localisation de Taction sur une carte par des signaux reentrants corollaires de signaux de commande motrice, ou combinant des signaux proprioceptifs avec la decharge corollaire. Selon cette theorie, le cerveau n'aurait pas besoin de construire une representation de l'espace absolu, en coordonnees spatiales, pour controler un mouvement de l'ceil, du bras ou un deplacement locomoteur. II suffit que le mouvement de l'effecteur, ou meme sa vitesse soit reinjecte sur la carte sensorielle ou le point de l'espace que Ton veut atteindre est code en coordonnees "sensori-topiques" c'est a dire dans l'espace que les roboticiens appellent « l'espace des capteurs », et par consequent ne presupposent aucune representation de l'espace (Poincare, 1970). Par consequent Poincare nous propose l'idee que les fondations de la geometrie sont dans les mecanismes biologiques de Taction (voir Berthoz, cours au College de France, Annuaire 1997-1998). Einstein a approuve cette theorie qui a ete critiquee par des logiciens comme Hilbert et les theories de

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la logique formelle qui ont domine les mathematiques dans les quelques dernieres decades (voir a ce sujet l'analyse de G. Longo dans Intellecticd). Nous devons rehabiliter les idees de Poincare et d'Einstein, nous devons reintroduire le corps en action dans les theories concernant la memoire de l'espace. II faut substituer une theorie dynamique basee sur la memoire de Taction a des theories ou domine l'idee que le cerveau ne travaille que dans un univers d'abstraction cartesien deconnecte du corps sensible. Cette idee a ete developpee dans (Berthoz, 1997). De fa£on tres generale, cette theorie suppose que Ton peut trouver des preuves de l'utilisation par le cerveau des derivees successives de la position (vitesse, acceleration) ou meme combinaison de celle-ci comme nous 1'avons recemment suggere (Hanneton et al. 1998). Autrement dit cette theorie suppose que Ton puisse demonter que le cerveau utilise, et peut etre aussi memorise, des variables dynamiques du mouvement comme la vitesse par exemple, de preference a la position. II y a plusieurs avantages a travailler dans l'espace des vitesses. D'abord pour s'affranchir des problemes de condition initiale, ensuite on s'affranchit des problemes de derive. II est aussi connu que la stabilite est meilleure etc. Enfin un certain nombre de capteurs sensoriels (comme les capteurs vestibulaires, mais aussi la voie du systeme visuel qui code le mouvement, ne detectent que des derivees de la positon (vitesse, acceleration etc.). Notre equipe a essaye de mettre au point de nouveaux paradigmes pour etudier chez l'homme, les mecanismes de la memoire des trajets et du guidage de la locomotion par des simulations mentales de la navigation. Nous donnerons cidessous quelques exemples de ces travaux. La simulation mentale des trajets Une premiere question concerne la reactualisation des orientations et de la position d'un objet pendant la locomotion. En effet nous ne pouvons nous rappeler un trajet que si nous avons fait une reactualisation de la position des objets pendant notre route, un liage perceptif en quelque sorte entre les differents points de vue d'un meme objet. Cette question des changements de point de vue avait ete etudiee par Rieser (Rieser, 1989), mais il n'avait pas considere les processus mentaux qui accompagnent ces changements de points de vue. Amorim (Amorim, Glasauer et al., 1997) a propose l'idee qu'il y a deux mecanismes possibles qui nous permettent de reactualiser mentalement la position et l'apparence d'un objet dans l'espace pendant la locomotion. Considerons par exemple une personne qui marche dans une piece les yeux fermes et qui essaye de se souvenir d'un objet dans la piece pour en memoriser l'orientation, la position et la forme. Amorim et al. (1997) ont montre que le cerveau peut adopter deux tactiques. L'une, "sequentielle", consiste a se concentrer, pendant la marche, sur les mouvements du corps et, une fois arrive a destination, a la fin de la trajectoire, a effectuer mentalement la reactualisation de la position, de l'orientation et de la forme de l'objet. Mais Ton peut aussi effectuer

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la reactualisation des proprietes de 1'objet de fagon continue pendant la marche (strategie continue). Ces deux tactiques cognitives sont toutes deux de type "egocentrees" en ce sens qu'elles referent 1'objet exterieur par rapport au corps propre du sujet, mais elles different par l'ordre dans lequel sont effectuees les operations mentales. II est done clair que pendant la locomotion aveugle le cerveau utilise une simulation mentale des relations avec l'environnement pour guider la locomotion. Ceci a plusieurs consequences que nous allons brievement decrire. La simulation mentale d'une trajectoire guide la locomotion : le pointage locomoteur Les paradigmes de pointage locomoteur consistent en general a demander a un sujet de marcher en l'absence de vision vers une cible prealablement memorisee. Ceci correspond done aux grands paradigmes classiques de la physiologie experimentale qui sont ceux soit du pointage manuel, soit du pointage oculomoteur. Thomson (1983) a demontre que le cerveau peut reactualiser de fagon tres precise les cartes mentales d'un environnement visuel lors de la locomotion. Ses ont ete dupliquees par un grand nombre d'auteurs. Toutes ces etudes ont confirme qu'il etait possible a un sujet d'atteindre une cible vue prealablement jusqu'a des distances d'une vingtaine de metres, meme si le sujet effectue des detours assez complexes. Plusieurs theories ont ete proposees pour expliquer cette capacite. On peut imaginer que le sujet possede une carte mentale comme une carte de geographie et que lors de la locomotion le cerveau procede a une reactualisation de la position du corps sur cette carte. On peut egalement imaginer que le cerveau ne fonctionne pas du tout avec une representation metrique. La litterature de psychologie cognitive montre que les sujets utilisent des strategies completement differentes. Certains sujets dits « visuels » semblent effectivement reconstituer une image de la piece dans laquelle ils evoluent et d'une fagon encore mysterieuse parviennent a reactualiser la position de leur corps par rapport a cette image. Ces sujets ont tendance a incliner la tete vers la cible, comme s'ils la regardaient. D'autres sujets n'ont pas du tout cette strategie et semblent utiliser des informations proprioceptives ou des informations liees a la commande motrice. Nous avons repris ces paradigmes en posant quelques questions simples. En effet si le cerveau peut effectuer cette reactualisation des cartes mentales il ne peut utiliser, les yeux fermes, que les informations d'origine vestibulaires, ou les donnees de la propioception des jambes, ou encore des decharges corollaires des commandes motrices. Laquelle de ces trois sortes d'informations sont-elles utilisees ? Les informations vestibulaires sont-elles utiles dans cette navigation aveugle pour la manipulation mentale des representations de l'environnement ? Nous avons, pour repondre a ces questions, employe des methodes que je voudrais rappeler tres rapidement. II est possible, grace a des cameras reliees a un ordinateur, d'obtenir en trois dimensions les deplacements des differents segments

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du corps et ensuite d'en deriver des parametres de vitesse, d'acceleration et de distance. Nous avons dans le passe utilise ces methodes dans I'etude de la locomotion pour montrer qu'au cours d'une simple marche, un sujet qui se deplace a la tete stabilised dans l'espace de facon remarquable en rotation (Berthoz and Pozzo, 1988 ; Pozzo, Berthoz, and Lefort, 1990). Et nous avons ensuite specule sur le fait que cette stabilisation de la tete permet precisement de simplifier le traitement des informations vestibulaires de deplacement et d'utiliser les capteurs vestibulaires pour traiter les problemes complexes que pose la differentiation gravito-inertielle, cette ambiguite venant du fait que la gravite est detectee par les otholithes. Nous avons ici demande a des sujets porteurs de marqueurs d'effectuer la tache de l'experience de Thomson. Des sujets avec des lesions des capteurs vestibulaires peuvent realiser cette tache avec une tres grande precision (Glasauer, Amorim et al., 1993). Ceux-ci ne sont done pas necessaires pour ce pointage locomoteur balistique. Nous avons emis l'hypothese que cette contribution des capteurs vestibulaires deviendrait peut-etre apparente si nous demandions au sujet de marcher suivant une triangulation (dessiner sur le sol un triangle d'environ 3 m de cote, a parcourir d'abord les yeux ouverts puis les yeux fermes). II y avait ainsi en plus d'un probleme de distance la realisation de deux rotations. C'est une tache de "retour au gite" dans le noir. Dans une tache comme celle-ci, la question theorique que Ton peut se poser est de savoir si la distance parcourue est traitee par le cerveau par les memes mecanismes que la direction de la tete. Existe-t-il une dissociation entre la distance et la reorientation de l'espace (d'ou l'interet des deux marqueurs permettant de reconstruire Tangle de la tete par rapport au corps) ? Nos experiences ont d'abord montre que juste avant de tourner le coin, la tete commence a indiquer la nouvelle direction avant meme que le sujet soit arrive a ce coin. Lorsque nous tournons autour d'un coin, la tete anticipe et va en quelque sorte diriger le corps dans la nouvelle direction. Cette propriete reste conservee meme dans le noir. Cette avance de phase de la direction de la tete par rapport a la trajectoire du corps a ete montree en mesurant la vitesse tangentielle de la tete le long de la trajectoire. Ceci est pour nous l'indice que dans le guidage de cette locomotion, pendant cette tache memorisee, le cerveau ne se contente pas en quelque sorte de suivre les pieds, mais la locomotion est guidee par une simulation interne de la trajectoire. C'est cette poursuite de la trajectoire qui entraine, pensons-nous, une orientation de la direction du regard vers la nouvelle direction attendue (il y a prediction). Nous avons en effet enregistre les mouvements des yeux. II y a une veritable anticipation par le regard (Grasso, Prevost et al., 1998). La formule que je voudrais proposer est « nous allons la ou nous regardons », c'est le regard qui guide la locomotion et c'est une poursuite interne d'une trajectoire mentale qui guide le regard. Cette affirmation apparemment triviale est importante, car en robotique le probleme du guidage des robot a de grandes vitesses se heurte encore a des difficultes non negligeables. Aussi etonnant que cela puisse paraitre, cette idee tres

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simple que la navigation d'un robot, a partir d'une simulation interne d'une prediction des deplacements futurs dans l'espace, puisse etre guidee non pas par ses roues mais par son regard (en l'occurrence des cameras orientables) n'avait pas ete «implementee » sur les robots mobiles. Nous nous sommes aussi pose la question des developpements de cette capacite predictive. Nous avons essaye de regarder ce qui se passe chez les enfants (Grasso, Assaiante et al., 1998). II n'y a pas d'anticipation chez l'enfant de trois ans, mais elle est presente chez l'enfant de huit ans. Chez les patients presentant des lesions labyrinthiques bilaterales, la realisation de la tache les yeux fermes ne presente pas de deficit pour le premier segment, la premiere rotation s'effectue bien (Berthoz, Amorim et al., 1999). Par contre, un deficit important apparait lors de la deuxieme orientation (lorsqu'il y a cumul des directions). Ceci nous suggere que ces informations sont utilisees par le cerveau dans ce type de tache. II existe une grande variabilite dans les resultats en fonction du type de lesion. Les donnees montrent que chez les patients vestibulaires uni ou bilateraux, il se produit tres peu d'erreurs en distance, mais beaucoup en direction (reorientation). D'ou l'idee d'une dissociation possible entre le controle de la distance et celui de la direction, suggere il y a longtemps notamment par Paillard. Dans une autre serie d'experiences sur la locomotion aveugle, nous avons propose une marche non plus en triangle, mais en cercle. On montre au sujet un cercle dessine sur le sol puis on lui demande de reproduire la trajectoire en effectuant deux tours et de s'arreter dans la bonne direction. Le sujet doit done faire une tache de production de trajectoire memorisee, avec un aspect generation d'une forme de trajectoire, une tache de reproduction d'une distance et une tache de reorientation finale. Le sujet devait en plus accomplir une tache d'arithmetique mentale (compter a l'envers a partir de 200). II est ensuite possible de mesurer la performance en prenant des indices, comme la distance totale parcourue, Tangle final de la tete par rapport a Tangle de depart, et un indice important pour retrouver la capacite du cerveau a produire une trajectoire qui est le rayon moyen. Pendant ces taches locomotrices, qu'elles soient effectuees dans la lumiere ou en obscurite, la tete anticipe d'environ 300 ms sur le mouvement du corps. Chez les patients ayant des lesions vestibulaires on constate une segmentation de la trajectoire en petits elements. Mais surtout les resultats suggerent qu'il existe bien une dissociation entre le controle de la distance et le controle de la direction (la distance peut etre correcte, mais non la direction par exemple) car les patients ayant des lesions vestibulaires bilaterales peuvent reproduire la distance (car leur systeme somatique du controle du pas qui met en oeuvre les membres inferieurs est intact), mais perdent Torientation. La memoire vestibulaire Si le cerveau construit une memoire des trajets a partir des donnees des sens, il faut montrer que ces informations peuvent reellement etre utilisees pour

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revaluation de nos deplacements dans un espace de grande dimension et que ces informations peuvent etre stockees en memoire spatiale et ensuite reutilisees pour memoriser un trajet ou conduire une action d'orientation. Ceci est bien connu pour la vision. On sait aussi que les informations proprioceptives peuvent etre memorisees et que, dans le cas de trajets rectilignes elles peuvent conduire a une navigation assez precise (H. Mittelstaedt and M.L. Mittelstaedt, 1992). Toutefois il nous fallait demonter que les informations vestibulaires pouvaient etre memorisees et utilisees pour guider une reproduction de trajet ou un acte de pointage oculomoteur ou manuel. Ceci a ete demontre par plusieurs experiences que je resumerai rapidement ici. a) Reproduction de mouvements de rotation et de translation passifs Nous avons demontre, avec I. Israel, que le cerveau peut utiliser les informations vestibulaires de rotation donnees par les canaux semi-circulaires pour la memoire des rotations. Nous avons aussi montre que cette memoire des rotations peut guider des mouvements des yeux vers des cibles dont la position est memorisee, et que des patients ayant des lesions corticales du cortex vestibulaire, du cortex frontal et prefrontal montrent des deficits dans cette tache de saccades vestibulaires memorisee (Andre-Deshays, Israel et al., 1990 ; Israel, Fetter et al., 1993 ; Israel, Rivaud et al., 1995 ; Israel, Rivaud et al., 1991). Les informations vestibulaires de translation detectees par les otolithes peuvent aussi etre utilisees par le cerveau pour la memoire des deplacements (Berthoz, Israel et al., 1995 ; Israel and Berthoz, 1989). Par exemple nous avons utilise un robot mobile sur lequel le sujet pouvait etre soumis a des translations passives. Les sujets pouvaient ensuite prendre le controle du robot et induire son deplacement en variant la vitesse. lis etaient assis sur le robot les yeux fermes, par consequent sans aucun indice visuel concernant leur mouvement. Le robot se deplacait de quelques metres vers l'avant (5 a 10 metres), distance appelee "distance imposee". Apres quelques secondes d'arret du robot, le sujet devait a son tour lui imposer un deplacement vers l'avant d'une distance egale a celle qu'il avait parcours pendant le deplacement impose. La seule information disponible pour le sujet concernant cette distance devait avoir ete derivee de la mesure de 1'acceleration du robot par les capteurs vestibulaires (otolithes) puisqu'il etait dans le noir complet et n'avait aucune information visuelle. En effet nous avons verifie que 1'evaluation du temps de parcours n'etait pas indispensable (par exemple en imposant plusieurs distances lors de parcours d'une meme duree). La reproduction de la distance fut tres precise. Le cerveau peut done evaluer, a partir de 1'acceleration lineaire, une grandeur qui permet cette reproduction de la distance. L'explication la plus triviale de ce processus est qu'il y a, dans le cerveau, des mecanismes qui accomplissent cette "integration", au sens mathematique, entre acceleration et deplacement. Une grandeur representant le deplacement serait memorisee et comparee, pendant la tache de reproduction, avec une autre mesure integree du deplacement detectee par les otolithes.

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Toutefois la question est de savoir dans quelle mesure le cerveau memorise reellement des distances ou un mouvement. En d'autres termes, comme l'a propose Poincare, ce qui est memorise est peut-etre le mouvement que nous devons faire pour aller d'un point de l'espace a un autre et non un nombre dans un espace cartesien. Effectivement si nous varions le profil de vitesse du mouvement passif impose pendant la translation du robot (en changeant les profils de vitesse et d'acceleration) les sujets tendent a reproduire les profils de vitesse comme si ce qu'ils avaient memorises etait un pattern dynamique de mouvement. Si cela est vrai alors le fait qu'ils reproduisent une distance est un effet secondaire d'une reproduction precise du profil de vitesse. Dans ces conditions la memoire du mouvement peut etre une memoire cruciale pour la reproduction des distances. Un nouveau paradigme : la reproduction de trajectoires memorisees pendant la navigation dans un corridor virtuel. Nous avons recemment elabore un nouveau paradigme (Sreung, DEA Paris VI, 1997, Viaud Delmon, These Sc. Paris VI, 1999, Berthoz et al. 1999) destine a accomplir plusieurs objectifs. D'abord permettre de tester la memoire des trajets pendant des taches de navigation et poser la question de la nature des informations qui sont memorisees ; ensuite permettre la dissociation des informations visuelles, vestibulaires, de decharge corollaire, et proprioceptives pour pouvoir varier leur poids respectif, creer eventuellement des conflits et nous permettre d'observer 1'effet de l'adaptation au conflit sur la memorisation de l'espace parcouru; enfin donner au sujet une tache active de navigation qui mette en jeu les circuits qui construisent la cognition spatiale. Les sujets etaient assis sur une chaise tournante (une simple chaise qui pouvait tourner sur elle meme sous l'impulsion des pieds du sujet). lis etaient soumis a deux situations differentes. La premiere situation est une condition de "navigation virtuelle" : des images visuelles d'un couloir comprenant plusieurs segments rectilignes joints par des angles etaient presented dans un casque de realite virtuelle dont le champ monoculaire etait d'environ 50 degres par 40 degres. L'ordinateur imposait aux images dans le casque un mouvement de translation a vitesse constante dans le couloir virtuel correspondant a une vitesse de locomotion normale. Le sujet etait done soumis a une translation visuelle passive dans le couloir virtuel, mais la rotation de sa trajectoire dans le couloir virtuel etaient actives, il les realisaient avec son corps. En effet, la rotation horizontale de la tete du sujet etait mesuree par un systeme a ultrasons qui ensuite pouvait modifier 1'image du couloir dans le casque de realite virtuelle. Lorsque le sujet tournait sur sa chaise la trajectoire dans le couloir virtuel effectuait une rotation. Par exemple, lorsque le sujet arrivait dans le couloir virtuel a l'extremite d'une branche rectiligne du couloir s'il ne faisait rien avec son corps il heurtait le mur. Pour tourner dans le monde virtuel il fallait

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qu'il tourne dans le monde reel sur sa chaise. Nous avons teste les sujets dans deux conditions : sujet assis et debout. Dans ce dernier cas, les sujets devaient tourner leur corps pour effectuer la rotation. II s'agit done d'une "locomotion virtuelle" assise ou d'une locomotion virtuelle debout dans laquelle seules les rotations sont reelles. L'avantage de ce dispositif est que Ton peut modifier les relations entre le monde virtuel (visuel) et le monde reel (la rotation sur la chaise ou le corps debout). Par exemple pour tourner de 90 degres dans le monde virtuel le sujet peut etre oblige de tourner de 120 degres dans le monde reel. Ceci permet de briser la coherence des traitements des informations visuelles, somatiques et vestibulaires. Le rapport est done normalement 1. Nous avons employe deux rapports de 0.66 et de 1.5. Le rapport de 0.6 correspond au fait que si le sujet tourne la chaise de 90 degres la rotation dans le couloir virtuel est de 59.4 degres. II est done possible d'etudier l'adaptation a ce conflit et pour decouvrir la nature de l'interaction qui se produira entre les deux "mondes". L'avantage de ce paradigme est qu'il est possible de construire une "trajectoire", dans le couloir virtuel, qui est la somme du deplacement passif a vitesse constante imposee par l'ordinateur et les rotations du sujet. La deuxieme condition permet d'etudier la memoire des trajets. Nous l'avons appelee "navigation mentale". Dans cette condition, le sujet est assis, ou debout, et ferme les yeux. II doit mentalement simuler la navigation visuelle a laquelle l'ceil a ete precedemment soumis. Lorsqu'il pense qu'il est dans le couloir rectiligne, il ne bougera done pas ; lorsqu'il pense qu'il doit tourner au bout du couloir, il doit tourner le corps et la tete sur sa chaise ou debout. On obtient ainsi une "trajectoire" de cette navigation mentale qui permet d'avoir une representation des processus mentaux du sujet. Les sujets accomplissent la tache de navigation visuelle sans aucun probleme aussi bien lorsque le gain est 1 que lorsqu'il y a conflit. Cette experience a donne plusieurs resultats remarquables. D'abord chez 53 sujets ayant subi la navigation avec un conflit (gain 0.66 ou 1.5) aucun sujet n'a ete conscient de ce conflit. Ceci est probablement du au fait que les plages de rapports utilises ne sont pas tres au-dessus de la gamme des rapports qui sont compenses par des mecanismes sous corticaux (comme une variation du gain du controle du regard lorsque nous mettons nos lunettes est inconsciemment compensee par des mecanismes cerebelleux). Lorsqu'on demande au sujet de reproduire, les yeux fermes, les angles de rotation qu'ils ont effectues dans la situation de navigation visuelle on constate que les sujets reproduisent les angles de rotation dans le monde qui a l'amplitude la plus importante dans le rapport de gain entre vision et rotation du corps. Ceci nous suggere que, dans le cas d'un conflit comme celui-ci, le cerveau ne memorise pas une combinaison ponderee des informations mais semble choisir, operer une selection, entre les deux principales sources d'information et attribuer une priorite a l'une d'entre-elles, celle dont l'amplitude est la plus elevee. Un des meilleurs

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temoins de ce fait est le dessin que realisent les sujets apres T experience. Celui-ci montre bien que c'est l'une ou l'autre des informations qui est memorisee et non un melange des deux. Les consequences de ces observations sont importantes pour comprendre les strategies cognitives associees au controle de Taction. Le cerveau ne se contente pas de traiter passivement les informations des sens. Suivant le contexte et la tache, il selectionne et decide d'attribuer un role preponderant a telle ou telle information. Cette hierarchie de selection dependant du contexte et du but de Taction a ete suggeree (Nashner and Berthoz, 1978) dans le cas de Tadaptation aux prismes (Melvill Jones and Berthoz, 1985) et la compensation des deficits vestibulaires (Berthoz, 1985 ; Berthoz, 1989). Cette capacite de jouer sur un repertoire de strategies est sans doute dependant d'un jeu subtil entre Tinhibition et Texcitation dans le systeme nerveux central (Berthoz, 1996). Un systeme appartenant au repertoire sensorimoteur peut etre choisi qui se substitue au systeme deficient. L'adaptation requiert une decision. Un troisieme resultat important de ces experiences concerne la nature des caracteristiques de la trajectoire qui est memorisee. Lorsqu'on demande au sujet de dessiner la trajectoire, on decouvre que la precision de la reproduction sur un plan en coordonnees cartesiennes, done sous forme d'une carte, n'est pas tres bonne (bien que dans le cas de conflit elle reflete le choix entre memorisation de la vision ou des perceptions somato-vestibulaires). Si le rythme general de la trajectoire est bien memorise, au contraire il y a une grande variation d'un essai a l'autre pour un meme sujet. Par contre, si on prend la derivee de cette trajectoire, e'est-a-dire la vitesse angulaire, par exemple, on obtient un trace qui est nul lorsque la trajectoire est rectiligne, mais la vitesse angulaire de la tete varie en forme de cloche lorsque le sujet tourne sur sa chaise. Le resultat remarquable est la similarite des courbes ainsi obtenues lorsqu'on compare ces courbes pour la condition de navigation virtuelle visuelle et la navigation mentale dans le noir. Malgre de grandes variations d'un essai a l'autre, et d'un sujet a l'autre, la forme de ces courbes est tres reproductible entre ces deux conditions de navigation. Pour ecarter la possibility que cette ressemblance soit due a une memoire motrice de la rotation du corps (le sujet se rappellerait les commandes motrices qu'il a donne pour tourner), nous avons introduit entre la session de navigation visuelle et la reproduction mentale un changement de la raideur de la chaise tournante (dans le cas ou le sujet est assis). Cela n'a pas modifie le resultat et les deux courbes de vitesse pendant la navigation visuelle et la navigation mentale, restent tres proches. Ces donnees nous suggerent, une fois de plus, comme Tavaient aussi suggerees les donnees de Berthoz, Israel, Georges-Francois, Grasso, and Tsuzuku, (1995), que le cerveau memorise un aspect dynamique de la trajectoire et pas seulement le deplacement d'un point sur une "carte cognitive". Ces etudes suggerent Texistence d'une classe particuliere de memoire spatiale qui pourrait etre appelee "memoire topokinetique" ou "topokinesthesique"

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(Berthoz, 1999) qui implique plusieurs structures du cerveau (le cortex parietal, le cortex parieto, insulaire, 1'insula, le cortex cingulaire, la formation hippocampique, le cortex frontal dorso lateral etc.), et done un systeme implique dans le processus de navigation active. Par memoire topokinesthesique, nous voulons dire une memoire du mouvement construite pendant la navigation et associee avec une serie d'evenements ou de reperes visuels. II s'agit d'une memoire motrice particuliere qui met en relation des actions dans l'environnement et des indices sensibles, des configurations d'indices. Cette memoire a un rapport avec la memoire episodique mais concerne une sequence d'episodes. C'est une memoire episodique sequentielle. Un script retroactif des relations entre le corps et le monde environnant. Un des caracteres les plus importants de cette memoire est qu'elle concerne des evenements auto-induits par une recherche active d'un environnement. II est bien connu que les cellules de lieu de l'hippocampe et les cellules de direction de la tete ne construisent leurs proprietes que si l'animal explore activement l'environnement. Ce mecanisme contient par consequent des secrets importants pour comprendre le role de l'activite. 9. Bases neuronales de la memoire des trajets a) Donnees de la neuropsychologic La memoire des deplacements est done un processus mental complexe qui associe l'ensemble des souvenirs perceptifs et les souvenirs de nos actions et dans lequel intervient le langage (voir Denis, 1989). De nombreux travaux ont ete consacres recemment aux bases neurales de la memoire des trajets. II est hors de notre propos de les resumer ici et nous renvoyons aux ouvrages de synthese mentionnes au debut de ce texte. Nous donnerons quelques indications utiles pour notre demonstration. On trouve dans les donnees de la neuropsychologie des indications interessantes a ce sujet. Des dissociations claires entre les differentes strategies ont ete trouvees chez divers patients porteurs de lesions cerebrales. De nombreuses etudes ont montre des deficits de memoire topographique et une dissociation a ete trouvee entre la capacite de reconnaitre des reperes familiers et celle de decrire des routes bien connues avec des reperes (Whiteley and Warrington, 1978 ; Incisa della Rocchetta; Cipolotti et al. ; 1996 ; Pallis, 1955 ; Paterson, 1994). Parfois des patients peuvent decrire leurs routes et reconnaitre des reperes, mais se perdent quand meme parce que les reperes ne contiennent plus, pour eux, des informations de direction (Haecan, Tzortzis et al., 1999). L'hippocampe n'est pas necessairement la seule zone du cerveau impliquee dans ces deficits de la memoire spatiale. Habib and Sirigu (1987), par exemple, ont montre que des lesions chez les patients ayant des deficits de la memoire topographique etaient limites au parahippocampe et au subiculum mais ne s'etendaient pas a l'hippocampe, ce qui rejoint les travaux recents de l'imagerie cerebrale. De plus le patient R. B., qui avait une amnesie retrograde severe apres la lesion de l'hippocampe, n'etait pas perdu dans son voisinage (Zola-Morgan, Squire et al., 1999).

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Le patient etudie par Incisa della Rochetta, Cipolotti et Warrington (1996) avait des deficits severes pour decrire des routes familieres dans son environnement mais aucune difficulte a identifier des pays d'apres les cartes de leurs frontieres et a nommer une ville dans un pays lorsqu'on l'identifiait par un point. La memoire episodique et semantique etaient atteintes chez ce patient. La suggestion est que ce patient avait un deficit specifique pour la connaissance categorielle d'objets inanimes (collines, batiments et autres objets). II semble done exister un codage specifique des objets topographiques distinct d'autres classes d'objets et une separation entre les objets qui demandent qu'on les atteigne par la locomotion et d'autres objets situes dans l'espace de prehension par exemple. II est interessant de remarquer que cette dissociation rappelle la distinction enter les diverses categories d'espace (personnel, extrapersonnel, environnemental sur lesquels neurologues et psychologies ont souvent insiste (voir a ce sujet Griisser and Landis (1991)). b) Donnees de I'imagerie cerebrale Pour etudier les bases neurales des strategies de navigation mentale deux experiences ont ete realisees (Ghaem, Mellet et ai, 1997). La question abordee etait la suivante : s'il y a reellement des mecanismes differents dans notre capacite a memoriser les deplacements en utilisant une strategie de type route et de type carte, peut-etre pourrions-nous observer chez l'homme l'activation de structures differentes lorsqu'on donne aux sujets ces deux types de taches. Nous avons propose d'abord une tache de type route. Les sujets se promenaient a pied dans la ville d'Orsay sur un trajet inconnu, et on leur indiquait des reperes visuels (station essence, porche, etc.). On leur demandait ensuite d'effectuer une locomotion mentale le long de ce trajet. Pour essayer de verifier que ces sujets effectuaient bien une route mentale, nous avons utilise des travaux montrant qu'il faut le meme temps pour imaginer un deplacement que pour le realiser effectivement. Dans de nombreuses taches, il existe en effet une isochronie entre le deplacement effectue et le deplacement mental (Decety and Jeannerod, 1995 ; Decety and Michel, 1989 ; Decety, Jeannerod et al., 1989). La comparaison des donnees temporelles a ete effectuee (il y a bien sur une compression dans le temps, cette isochronie n'est pas valable pour toutes les distances). Cette locomotion mentale a de nouveau ete effectuee par le sujet enregistre par la camera a emission de positons. II lui etait alors demande de se rendre mentalement d'un point de repere a un autre le long du trajet qu'il avait memorise. Trois conditions ont ete etudiees : repos, imagination simple des reperes (tache de memoire visuelle), trajet mental. Les deux hippocampes, le gyrus cingulaire posterieur et des regions du gyrus temporal median ainsi que du gyrus precentral ont ete activees de facon significative. Le deuxieme depouillement a consiste a comparer les regions activees pendant la locomotion mentale a celles activees pendant le repos. II y a plus de regions activees, le gyrus pre-frontal dorso-lateral, des regions hippocampiques gauche et droite, le noyau pre-cuneus situe dans le

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lobe parietal, le gyrus cingulaire, l'AMS activee en association avec la preparation du mouvement, des regions du gyrus occipital, du gyrus fusiforme et des regions pre-motrices. On effectue ensuite une comparaison entre la condition de simulation mentale du trajet et la condition d'imagerie visuelle des reperes. On constate cette fois que les regions actives sont I'hippocampe gauche, le precuneus et l'insula. Cette activation de I'hippocampe gauche est un peu surprenante car une grande partie de la litterature attribuait plutot a I'hippocampe droit son role dans les representations de l'espace. Ceci fait actuellement l'objet de comparaisons. Le noyau pre-cuneus est une structure interessante. II a ete montre que cette zone, situee au voisinage de l'aire 7 de Brodman, est activee dans plusieurs types de taches. II serait « I'oeil de I'esprit » (« minds'eye » ) ; cette expression est due a Fletcher, Frith et al. (1995). Ce noyau serait requis pour l'imagerie consciente visuelle et serait fondamental pour revocation de souvenirs episodiques impliquant une imagerie visuelle et non pas une imagerie semantique. II serait implique dans de nombreuses taches visuo-spatiales: le traitement mental des attributs (caracteristiques visuelles et spatiales des objets) visuo-spatiaux, le traitement de l'attention spatiale, lors de l'apprentissage, du rappel (a partir de la memoire) ou de la reconnaissance de figures geometriques complexes, et enfin lors de l'imagerie visuelle et visuo-spatiale, lorsque nous imaginons des objets ou des lieux. Le precuneus a ete active dans une autre experience recente (Mellet, Tzourio, Denis, and Mazoyer, 1995) en utilisant le paradigme de l'tle de Kosslyn (paradigme de psychologie cognitive dans lequel les sujets doivent explorer soit visuellement soit mentalement la carte d'une tie). L'exploration visuelle de l'ile a ete comparee a I'exploration mentale. Pendant I'exploration visuelle, les auteurs ont trouve une activation des aires visuelles primaires, des gyri occipitaux superieur et inferieur, des gyri fusiforme et lingual, du cuneus et du pre-cuneus, des gyri parietaux superieurs bilateraux et des aires pre-motrices. Pendant I'exploration mentale, seules quelques zones sont activees, comme le cortex occipital superieur, l'AMS et le vermis cerebelleux. Une deuxieme experience a ete menee sur I'exploration mentale d'une carte (Mellet, Bricogne et al., 2000). Cette fois les sujets devaient imaginer mentalement une carte et devaient accomplir trois sortes de taches. Ces donnees preliminaries suggerent done qu'un reseau different de centres corticaux sous-tend la navigation mentale a l'aide de cartes et a l'aide de memoire de routes. Des travaux semblables ont ete menes recemment avec d'autres taches de navigation qui eclairent ces mecanismes (Maguire, Frith et al., 1998 ; Maguire, Frackowiack et al., 1997 ; O'Keefe, Burgess et al., 1998), mais il faudra sans doute encore des experiences et des idees theoriques pour en comprendre bien les composantes. c) Donnees de la neurophysiologie chez Vanimal: des rotations, de la direction et de la position de la tete dans l'espace Les donnees recentes de la neurophysiologie suggerent une segregation des systemes neuronaux qui traitent des informations sur les mouvements de la tete

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pendant la navigation. Trois systemes a la fois distincts et en interaction ont ete decouverts a ce jour. lis concernent respectivement le codage des rotations, de la direction et des lieux ou de la position. Codage des rotations Les rotations de la tete mesurees par le systeme vestibulaire sont transmises aux noyaux vestibulaires ou elles sont combinees avec les informations sur le mouvement visuel donnees par le systeme optique accessoire. Nous savons maintenant qu'une voie conduit ces informations vers le cortex cerebral a travers le thalamus sensoriel et qu'une structure du lobe parieto-temporal (le cortex parietoinsulaire) contient des neurones (Griisser, Pause et al., 1982) qui codent les rotations de la tete chez le singe. Cette decouverte recente a ete confirmee par des travaux d'imagerie cerebrale chez l'homme qui ont montre I'activation de cette region du cortex par la stimulation calorique (Bottini, Sterzi et al., 1994), et galvanique (Lobel, Kleine et al., 1998). II est interessant de constater que cette derniere etude a montre I'activation d'une region du cortex frontal qui correspond a l'aire 6 qui est aussi activee dans des taches de memoire spatiale et qui donne lieu a des syndrome de negligence d'origine frontale. Le codage de la direction Dans plusieurs parties du cerveau chez le rat (Ranck, 1984 ; Taube, Muller et al., 1990), il existe des neurones appeles neurones de direction de la tete. Ces neurones presentent une decharge maximale lorsque l'animal a la tete dans la direction particuliere d'un lieu connu. Cette direction est controlee, entre autres, par des indices visuels. (Sharp, Blair et al., 1995) ont enregistre ces neurones dans le postsubiculum et dans le noyau antero-dorsal du thalamus. lis ont eu l'idee de dissocier la frequence de decharge de ces neurones dans deux conditions : ils ont pris d'une part les donnees qui sont obtenues lorsque l'animal se tourne dans une direction dans le sens des aiguilles d'une montre (ou le contraire). Alors que dans le postsubiculum les deux courbes sont parfaitement superposees, il n'en est pas de meme dans le noyau antero-dorsal du thalamus. Ces auteurs ont bati a partir de cela une theorie selon laquelle le systeme neuronal possederait la capacite de predire en quelque sorte la position future de la tete. Cette anticipation neuronale pourrait etre a l'origine de 1'anticipation que nous avons trouve dans des taches de locomotion (voir ci dessus les travaux de Grasso et al). Une deuxieme question est de savoir s'il est possible de trouver dans ces structures des activites neuronales refletant la propriete du cerveau de mettre en correspondance un lieu avec un deplacement. Si Ton regarde les moments ou le rat a la tete a 45°, on s'apercoit que la plupart des decharges ont eu lieu dans une region particuliere de l'arene. II y a done association dans le codage par ce neurone d'une direction et d'une zone de la piece. Par ailleurs, si Ton examine la relation entre la frequence de decharge du neurone et la vitesse angulaire avec laquelle la tete a tourne, on constate que ce neurone decharge plus particulierement lorsque le

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rat tourne la tete vers la gauche. II y a done codage d'une direction, d'une localisation et d'un certain domaine de vision. Chez le rat libre de ses mouvements en enregistrant simultanement des neurones dans I'hippocampe et des neurones de direction de la tete dans le thalamus on constate que si Ton manipule l'indice, le neurone de direction de la tete du thalamus effectue une rotation du meme angle et dans les memes conditions (le codage reste invariant par rapport a l'indice visuel). II y a done un couplage entre le codage dans les neurones de I'hippocampe de la place par rapport a l'indice visuel et le codage de direction de la piece. Cartes cognitives et codage de lieux Une theorie essentielle en ce domaine est celle proposee par O'Keefe et Nadel. Ces auteurs attribuent a I'hippocampe un role crucial pour la constitution et la manipulation de "cartes cognitives". Leur theorie est basee sur les proprietes des neurones de lieu qui codent la position du rat dans une piece familiere quelque soit la direction de sa tete. Cette theorie a ete toutefois defiee par d'autres auteurs (voir le livre recent de Burgess et al., et celui de Ono et al. (1999), Rolls and O'Mara, (1992), Berthoz (1973), Eichenbaum and Wiener (1989)). Les neurones de I'hippocampe qui codent les lieux peuvent aussi etre actives par les signaux vestibulaires. Nous avons cherche s'il etait possible de trouver dans I'hippocampe des neurones sensibles aux mouvements chez le singe. Nous avons trouve des neurones actives par des mouvements particuliers (exemple : translation vers la porte, mais non pas en s'eloignant de la porte) (O'Mara, Rolls et al., 1992). Nous avons reproduit ces experiences chez le rat avec un robot mobile (Gavrilov et al.) et montre d'abord que le rythme theta de I'hippocampe est modifie avec la vitesse de la rotation de la tete, et ensuite que les neurones de I'hippocampe peuvent etre sensibles au mouvement, mais associent ce mouvement avec une partie de la piece dans laquelle se trouve l'animal. Autrement dit, le codage du mouvement dans I'hippocampe est associe au codage des lieux. Nous avons aussi pu montrer chez l'homme, par enregistrement de l'activite par IRM fonctionnelle, que la stimulation vestibulaire calorique activait la region de I'hippocampe sur laquelle se projette le cortex vestibulaire par 1'intermediate du cortex cingulaire. II existe un degre supplemental de complexite dans le codage des lieux (Booth and Rolls, 1998). Un grand nombre de neurones de I'hippocampe ne codent pas seulement la position de l'animal dans un lieu donne, mais egalement des conjonctions, des associations entre differents indices de l'espace (visuels, texture, olfactif). On a meme propose (Tamura, Ono et al., 1992) que certains de ces neurones sont sensibles a des objet naturels importants pour le singe et a des signification de ces indices pour l'animal (plaisir, danger). Chez l'homme, les structures impliquees dans le codage de la direction n'ont pas ete trouvees. Toutefois un travail recent (Vallar, Lobel et al., 1999) a ete effectue concernant un aspect particulier de cette question. II est en effet connu que les patients souffrant de lesions du cortex parietal droit (principalement) presentent

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le syndrome dit de negligence spatiale qui s'accompagne d'un decalage de la perception du droit devant. Nous avons montre que la perception egocentrique de l'axe sagittal du corps est associee a l'activation d'un systeme impliquant les aires du cortex parietal postero-inferieur et premoteur frontal bilaterales, mais principalement dans I'hemisphere droit. Plus precisement les aires activees etaient le gyrus occipital superieur et le sillon precentral gauche, d'une part, le sillon intraparietal, le gyrus angulaire et le gyrus frontal inferieur droit de l'autre. Ces aires sont aussi connues pour etre impliquees dans les lesions qui entrainent la negligence. Dans une autre experience d'imagerie cerebrale, nous avons montre les differences entre les aires activees lorsque le sujet resout un probleme en coordonnees allocentriques ou egocentrique (Galati, Lobel et al., 2000). Ainsi la manipulation mentale des systemes de reference est maintenant a notre portee. Nous sommes done loin de comprendre completement les mecanismes neuronaux qui permettent notre navigation et les bases neurales de la memoire des trajets et des differentes strategies cognitives qui sont disponibles. II nous faudra aussi comprendre les strategies individuelles dans le traitement mental de l'espace et , en particulier, les differences entre les sexes (V. Delmon, Ivanenko et al., 1997), ou le role de facteurs lies a l'emotion (V. Delmon, Ivanenko et al., 1999). Mais la combinaison des approches comportementales, neuropsychologiques, d'imagerie cerebrale et de neurophysiologie animale combinees a la modelisation permettra sans doute d'avancer dans ce champ si interessant des sciences de la cognition.

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THE REASONABLE EFFECTIVENESS OF MATHEMATICS AND ITS COGNITIVE ROOTS GIUSEPPE LONGO Laboratoire d'lnformatique CNRS and Ecole Normale Superieure 45, Rue D'Ulm, 75005 Paris (France) www. di. ens.fr/users/longo

"At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it." [Galileo Galilei, 1632 (Opere, p. 298)]. "The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a measure of its value. Mathematics, as a tree which freely develops his top, draws its strength by the thousands roots in a ground of intuitions of real representations; it would be disastrous to cut them off, in view of a short-sided utilitarism, or to uproot them from the ground from which they rose." [H. Weyl, 1910].

Summary: Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity; we invented it and used it to single out key regularities of space and language. This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years. Its unity is also stressed: in this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by this both its autonomous generative nature and its effectiveness. But effectiveness is also due to the fact that we re-construct the world by Mathematics; we organise knowledge of space and language by Mathematics, and give meaning by it to their structuring. But, what is "meaning", for us living and historical beings? What does "mathematical intuition" refer to? We will try to propose an understanding of these crucial aspects of the mathematical praxis, often disregarded as "magic" or as beyond any scientific analysis. Finally, some limits of the remarkable, but reasonable effectiveness of Mathematics will be sketched, in particular in reference to its applications in Biology and in human cognition.

Introduction In the twentieth century, two major foundational paradigms have been splitting man and the world around him from one of its major conceptual constructions, Mathematics. On one side, formalism proposed the perfect rigor of mechanical rules as the core for certainty, effectiveness and objectivity of Mathematics: stepwise deductions along finite strings of symbols, perfectly independent of meaning, were meant to reconstruct completely mathematical reasoning or even propose a method for mathematical creativity (to be transferred, possibly, into digital computers). The philosophical background (and the practical aim, in the latter case - Artificial Intelligence) was based on the idea of a possible mechanical implementation of "the Universal Laws of Thought": once these were all formally described in a symbolic notation, without the 351

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ambiguities of meaning, we could fed a machine with them and completely simulate reasoning, action or general human behaviour. On the other hand, the naive platonistic reactions to this bold program, explained the failure of the formalist program as well as the certainty and objectivity of Mathematics by assuming independent ontologies; usually, this perspective as well refers to absolute "laws of thought" or to the perfect geometry of absolute "mathematical spaces", but as meaningful structures of truth, as "essence" which underlies the world, "per se". In either cases, Mathematics was separated from our "being in the world", from our forms of knowledge as embedded in our living and historical beings, where "understanding" is based on non-arbitrary proposal and descriptions constructed while interacting with that very world. Then, time to time, some leading colleagues came up with exclamations of surprise: "how is it possible that this game of meaningless symbols (or, alternatively, this perfectly independent ontologies) happens to say something, indeed a lot, about the "concrete reality" surrounding us? What an amazing miracle!". So, after inventing or accepting a schizophrenic split, some tried to recompose it by referring to magic or metaphysical, inexplicable connections (even ... "not deserved"?!, see [Wigner, I960]) and forgot that human knowledge should be analysed in a scientific perspective and not in terms of miracles nor in search for "absolute knowledge". Mathematics is the result of a "knowledge process", as one of the ways we relate to the world, while constructing our own "cognitive ego": our intelligence is an ongoing and active organisation of phenomena while trying to make them intelligible to us, it is co-constructed while structuring phenomena. We cannot separate Mathematics from the understanding of reality itself; even its autonomous, "autogenerative" parts, are grounded on key regularities of the world, the regularities we see and develop by language and gestures. In this paper we will try to hint to the unity of human conceptual constructions, as well as to the rich way in which various forms of knowledge are articulated. They constitute a network of meaningful attempts to understand the world, all rooted in "acts of experience", as active forms of interpretation and reconstruction of reality, deeply embedded in our cognitive and historical lives. Mathematics is one of these forms whose peculiar nature, by its "conceptual invariance and stability" as we will stress, is not independent from the others and, by this, it can "say something", indeed a lot - but not "everything" - about them and the world: it is reasonably effective. 1. The Formalist Foundation of Arithmetic and the Role of

Geometry in Mathematics 1.1 From the Geometry of space to Logical Truth Let's first briefly recall a story which begins with a major crisis: the loss of certainty in the absolute of physical space, whose geometry was identified to Euclidean geometry. For more than two thousands years, the "pure intuition of space per se" guaranteed certainty and foundation to the many constructions of Mathematics: the "synthetic a priori" of geometry was behind the objectiveness and effectiveness of the drawing of the geometric sign, of the proof carried on "more geometrico", on the perfect planes and spaces of Euclidean geometry. But... what happens if the world were curved? With this motivation, Gauss analysed another "possible world": the idea was to look at a surface "per se", not as embedded in an Euclidean space. That is, to analyse the

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surface "without leaving it", by moving, say, along its geodetics, as lines of (relative) minimal length. The connection to Euclid's fifth axiom was explicitly made by Lobachevskij and Bolyai. Riemann pursued Gauss's aims and developed the (differential) geometry of curved surfaces: he thus invented the fully general notion of "manifold" (topological, metric, differentiable manifolds... in today's terminology). Riemann actually considered the curvature of the physical space as related to the presence of physical bodies ("their cohesive forces", as said in his Habilitation): the notion of Cartesian dimension, a "global" property of space, is topological, while the curvature and the metric is a local property, which may depend on the "cohesive forces of matter", an amazing anticipation of Einstein's relativity1. As a matter of fact, by the analysis of the geometric structure of space (as ether), Riemann meant to unify action at distance (heath, light... gravitation). Clearly, Riemann greatly contributed, by his n-dimensional differential geometry, to demolish the absolute and certainty of Euclidean space, just a special case of his general approach. Yet, he tried to re-establish knowledge as related to our understanding of the (physical) world. For him, geometry is not "a priori" by its axioms, but it is its grounded on certain regularities of physical space, to be singled out and which have an objective, physical, meaning (continuity, connectivity and isotropy, for example). In Riemann's approach, we actively structure space, as manifold, by focusing on some key properties, which we evidentiate by "adding hypotheses". Thus the foundational analysis consists, for Riemann, Klein, Clifford, Helmoltz ... in making these regularities explicit and in spelling out the transformations which preserve them. However, this "relativized" neo-kantian attitude turned out to be unsatisfactory for many, since it dangerously involved an analysis of the "genesis of concepts", as apparently originating from our more or less subjective presence in the physical world. In fact, these concepts, as for instance those of differential form, group and curvature are objective because they are invariants that we actively single out of the hysical world; that is, they become concepts as a result of the interaction, at the phenomenal level, between us and the world. How to re-establish then absolute certainty and objectiveness, after the shocking revolution of non-Euclidean geometries, while avoiding this "implication of the subject" into knowledge? First, avoid any reference to space and time, the very reference that had given certainty, for so long, not only to geometry, but also to algebra and analysis. For Descartes and Gauss, say, algebraic equations or the imaginary numbers are "understood" in space: this is so in analytic geometry and in Argand-Gauss interpretation, over the Cartesian plane, of V-l (of the complex numbers, thus). But, if our relation to space is left aside in order to avoid the uncertainties of the "many geometries" and the shaky sands of human cognition, then language remains, in particular the logical laws of thought that the English school of algebra had already been putting forward, in a minimal language of signs ([Boole,1854]). Language, considered as the locus of the manipulation of (logical) symbols, with no reference to phenomenal space, nor, in general, to forms of experience.

1 Riemann's Habilitation is of 1854, under Gauss' supervision [Riemann, 1854; it. transl. 1999]; more remarks were made in the '60s. A broad analysis may be found in [Boi, 1995], [Tazzioli, 2000].

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Frege best represents this turning point, in the foundation of Mathematics. His search for "absolute knowledge" was meant to exclude, first, any hint to "intuition" or "empirical evidence", any analysis of the "knowledge process" (see [Frege, 1884]), in particular in reference to phenomenal space and time. In his work, the fight against Stuart-Mill (empiricism) is explicit, the one against Herbart (psychologism), who had largely influenced Riemann, and to Riemann himself is evident: empiricism and psychologism are Frege's worst enemies. We must acknowledge the depth of Frege's analysis. His scientifically plane style as well: he established a novel standard of rigor by his "language of formulae", where universal and existential quantification (the "for all x ..." and "there exists x ..." so relevant in Mathematics) are finally handled in a uniform and sound way, in contrast to the mathematical practice up to that time, or most of it (the formal gaps in Cauchy's work, say, are well-known). The too elementary "laws of thought" of the British algebraists (Boole, Babbage) are thus enriched by the so-called (first order) quantification: Mathematical Logic is born, as the search for "unshakeable certainties" in absolute assumptions and laws of deduction. Absolute, but meaningful: the assumptions and the laws must have a "logical meaning"; even arithmetical induction, a technical proof principle for number theory, has a logical meaning for Frege. Arithmetical computations are logically valid deductions. The reference is again to a platonic realm of logical and absolute truth, independent of man: "pure concepts", without conceptor. The search for foundation (and certainties) in the interaction between us and the world, starting with physical space, is abandoned: Mathematics is brought back to sit in platonic realms, detached from us and the world. What a surprise when some will rediscover that it is very ("unreasonably") effective, in Physics in particular, as if it where constructed to say something about the world, in a suitable language for us. 1.2 Formalism and linguistic stratifications An alternative path towards grounding Mathematics away from human reasoning about reality, was proposed by Hilbert's foundational work, once again for good reasons and by a strong proposal. Suppose that you have a proof of existence and, maybe, of uniqueness of the solution of a system of (differential) equations or of a finite basis for certain algebraic systems; yet, assume that you cannot give the solution explicitly, as an elementary function, say, nor construct it as the limit of suitable (Fourier) series nor provide effectively the finite basis. Where does this solution, this finite set, live? In which platonic/logical realm? Hilbert has an original and robust proposal for an answer to this: provide a finite axiomatic frame for your proof, with finitary (effective) deduction rules, then the "existence property" in your theorem may be guaranteed by a proof of consistency (noncontradiction) of the axiomatic theory. In '900, he poses, as a key open problem, the proof of consistency of Arithmetic (and thus, by Cantor-Dedekind construction of the reals, of Analysis). A proof to be carried on in a "potentially mechanizable" fashion, in order to reduce certainty to the finite manipulation of symbols, with no reference to (actual) infinite nor to (possibly geometric) meaning. This is Hilbert's conjecture of the finitistically provable consistency of Arithmetic (and other key formalised theories). The project is strong and revolutionary: two dangers are avoided at once. Infinitesimal analysis had introduced the infinite to analyse the finite: since the XVIII

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century, powerful tools had been developed to describe physical, finite, movement around us (speed, acceleration) by actual limits, some sort of dangerous metaphysical infinities (based on Leibniz's monads, say). In spite of the work of Cantor, the foundation of infinity remained uncertain and flawed by paradoxes (contradictions). By the formalist program, and its developments, the situation could be reversed: in particular, once proved, by finitistic tools, the consistency of theories that formalise the infinite, such as (Zermelo's) Set Theory, then, by the consistency of the theory, the "existence" issue is solved, including the existence of infinite objects (sets, functions, actual limits ...). As a matter of fact, in Hilbert's program, mathematical existence is provable consistency of the intended theory, and nothing else. No need to dream of platonic realms, at least in the foundational work: once consistency is proved by finitary tools, the working mathematician could happily live in Cantor's paradise of infinite and ideal objects (in his practice of mathematics, Hilbert was far from being a formalist!). Moreover, the shaky reference to space (Euclidean, non-Euclidean? physical?) may be avoided as well: as for foundational purposes, geometries are just finite sets of (provably consistent) formal axioms, which may be interpreted in many ways, and the interpretations are irrelevant to deductions and proofs. And this is so, since deduction rules are applied mechanically, i.e. according only to the syntactic structure of the formulae (well-formed strings of symbols) in the assumption, with no reference to their meaning (logical, geometric ...). Then the bold enterprise of the formalist finitism began, grounded on one further and crucial idea of Hilbert's. He proposes to carry on the foundational analysis in a mathematised "metalanguage", whose object of study is the object language of the formal theories. Moreover, these theories, at their purely syntactical level, with no reference to metalanguage nor meaning, should be able to describe completely Mathematics. That is, any formalised assertion of it should be decided by finitistic deductions from the axioms. And here is Hilbert's conjecture of "completeness" of the key axiomatic theories, such as formal arithmetic. In summary, by finitistic metamathematical tools one should be able to prove consistency and completeness of the core of Mathematics. Hilbert's "linguistic stratification" (language, meta-language) is a remarkable way to organise the "discourse of Mathematics", perhaps comparable to Euclid's proposal to organise physical space. Yet, as much as Euclid's, this is not an absolute. That is, Hilbert's proposed distinction between theory and metatheory is not the only frame within which one may approach the foundational problems: other conceptual construction may violate this organisation and the blend of levels (and of meanings) may require an analysis that goes well beyond it. Indeed, the failure of this paradigm is the very reason for the incompleteness phenomena, as we will briefly hint below. The fact is that the formalist paradigm for mathematical knowledge (both foundation and praxis) marked the century and many still beleive that a "sufficient collection of (settheoretic?) axioms", once fully formalised, may allow a complete deduction of Mathematics ... yet, this very same people show a great surprise when, in spite of this assumed automatism (or "meaning independence") of the discipline, it helps us in understanding and giving meaning to the world. A few soon reacted to Hilbert's program, such as the "lone wolf" among Hilbert's students, Hermann Weyl, who conjectured in Das Kontinuum, 1918 (!), the incompleteness of formal arithmetic (§.3 in [Weyl,1918]). He also stressed in several

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places that the idea of mechanisation of Mathematics trivialises it and misses the reference to meaning and structures. Unfortunately, Weyl calls this crucial reference to meaning, "the mathematician's insight" or "intuition", with no further explanation; a reference to the "unspeakable" that we have to overcome: below, we will try to understand what these words may mean. Besides Weyl (and Poincare and a few others), Wittgenstein is another thinker who criticised Hilbert's program. For him "Hilbert's metamathematics will turn out to be a disguised Mathematics" [Waismann, 1931], since "[A mefamathematical proof] should be based on entirely different principles w.r. t. those of the proof of a proposition ... in no essential way there may exist a metamathematics" (see Wittgenstein, Philo. Rem., § 153; quoted in [Shanker,1988]), and ... "I may play chess according to certain rules. But I may also invent a game where I play with the rules themselves. The pieces of the game are then the rules of chess and the rules of the game are, say, the rules of logic. In this case, I have yet another game, not a metagame" [Wittgenstein, 1968; p. 319]. As for arithmetic, the key theory for finitistic foundationalism, these remarks may be understood now in the light of Godel's Representation Lemma [Godel, 1931]: by this very technical result, one may encode the metatheory of arithmetic into arithmetic itself, thus the "rules of the metagame" are just viewed as ... rules of the "arithmetical game". Moreover, many proofs which entail the consistency of arithmetic, such as (Tait-)Girard proof of "normalisation" of Impredicative Type Theory ([Girard et al., 1989]), need a blend of metalanguage and language; or even purely combinatorial statements, such as Friedman's Finite Form of Kruskal's theorem, provably require the same entangled use of metatheory, theory and semantics, by the impredicative notions involved: an indirect confirmation of Wittgenstein's philosophical insight (see [Harrington et al., 1985] for the mathematics; a discussion and more references are in [Longo, 1999a]). These theorems are some of the many recent examples of more or less "concrete" incompleteness results of formal arithmetic; that is, they are interesting arithmetic statements whose proof requires essentially "non formalizable" (not effective-axiomatic) tools. They are not self-referential "tricks" such as Godel's independent statement of arithmetic, a simple proof-theoretic translation of the Liar's Paradox2. The point is that, in the arithmetic proofs of these interesting (concrete) statements, meaning is essential; more precisely, at a certain point of the proof, in order to go "from one line to the next", one has to refer to some variables as interpreted by sets, to others as interpreted by their elements (use of some form of an impredicative second order comprehension axiom), or to the well-ordering of integer numbers (second order induction), or to similar "concepts" which provably cannot be formalised arithmetic, in a finitistic way. Computers get stuck, but human beings, by referring to the wellordered structure of integer numbers in space or time, by "seeing" sets and elements, as meaningful and conceptually distinguished notions, have no problem in understanding and carrying on the proof, even though these are provably not formalizable in a finitistic, mechanisable fashion (or by symbols which run independently of meaning). Thus, meaning steps in and the formalist analysis has been such a strong paradigm as to prove this for us, by these and the many other unprovability results in formal theories 2

Godel's incompleteness theorem is an immense achievement as for technical inventions and insights, essentially by the tools used in the proof of the Representation Lemma: godelization, recursive functions ... . The "unprovable" statement "per se" has no mathematical interest, in contrast to the many recent examples, as those quoted above.

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(the independence of the "continuum hypothesis" and of the "axiom of choice", whose analysis motivated formal Set Theory are further examples, see any text in Set Theory or [Longo, 1999c] for more discussions and references to this even more severe "debacle" of formalism). It is time now to go further and stop believing in the absolute of the Hilbert-Tarski foundational frame (theory, metatheory and semantics); that the finitistic formal level and the metalinguistic, yet mathematical, analysis can say everything about Mathematics and its foundation or that the close structuring of mathematical concepts, in this specific way, coincides with the foundational analysis of Mathematics. A large amount of technical work can surely still be done along this paradigm, as reductions to least purely formal frames are often very informative; it is the underlying philosophy that must be overcome. Many theorems tell us that this failed, yet the philosophical prejudice, the unscientific myth of "absolute knowledge" or the reference to certainty as mechanical deduction only or the idea that the foundational issues of Mathematics may be treated only mathematically, still resists, along the lines of Hilbert's relevant, but too rigid stratification, not proper to the human-mathematical experience. The major merit of the formalist-mechanicist approach was to set the philosophical and mathematical basis for inventing, in the thirties, the (mathematics of) computing machinery: Boole-Frege "laws of thought", independent of human thinking, are finally implemented in meaningless strings of symbols, pushed by machines according to meaningless rules. The idea of transferring human rationality in machines, to refer to computers as the paradigm for "logic and rigor", is probably the main fall-out (indeed, a relevant one) of this "splitting" proposal: splitting man from one of his major forms of knowledge, Mathematics. Yet, as a consequence of some of the incompleteness theorems mentioned above, the concept of infinity turns out to be essential to prove the consistency of formal arithmetic; let alone of those theories of infinity, the theories of sets whose consistency depends, at each infinitary level, by the use of even more infinitary constructions. Now, infinity is a robust human conceptual construction, the object of a lively debate for centuries, which stabilised with Cantor into a operational, mathematical notion. As suggested in [Longo, 1999a], we have, so far, no better "foundation" for the infinite than the reference to its historically meaningful specification as a mathematical concept, i.e., the analysis of its "progressive conceptualisation", to put it in F. Enriques' terms. The set theoretic specification of this concept, a technically remarkable clarification and "stabilisation" of the notion, is not a "foundation", since it transfers to larger infinities, by the consistency proofs, the foundation of each level of infinity. The foundation of mathematical infinity lies in the analysis of its conceptual genesis, of the knowledge process which brought it to stabilise as a mathematical invariant (see [Longo, 2001]). The rooting of this path in our relation to the world, its constitutive role for our (mathematical) interpretation and re-construction of it and, thus, its meaning, are the reasons of its mathematical effectiveness, as a conceptual tool of analysis. In particular, the use of the concept of infinity is robust and effective because it is co-defined with our ways of organising the world by Mathematics: from the early "theological" debates, to the structuring of trajectories and lines, by tangents and limits (Newton), rich of physical meaning (speed, acceleration ...). These notions organise, for us humans, the movements of finite objects around us by actual infinity and, once distilled in a rigorous mathematical practice (since Cantor), they are as effective as no other human construction in describing physical reality.

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1.3 Mathematics as a "three dimensional manifold" In a very schematic way, I tried to summarise the main approaches to the foundations of Mathematics by a very incomplete reference to three major scientific personalities: Riemann, Frege and Hilbert. Each one of their intended foundational ways stresses some crucial aspects of the conceptual construction of Mathematics. As a matter of fact, there is no doubt that Mathematics is grounded in logic, those "if ... then ... else ..." and much more that unfold along proofs (Frege). Similarly, one uses formal computations, in an essential way: purely algebraic reasoning pervade proofs and equations follow from equations following abstract rules, whose meaning is irrelevant to deductions (Hilbert). But symmetries also or other regularities of space (connectivity, say) contribute in singling our structures and their relations (Riemann). These have no logical meaning, yet they appear in theory building and in proofs (see for this the novel insight which originated in [Girard,1987]). This richness of Mathematics is lost in the logicist and formalist approaches: only logic or finitist formalism (and possibly not both!) found Mathematics. In particular, our relation to space is only a matter of "ad hoc extensions", largely conventional for the formalist, to be found on the concept of "ratio as number" for the logicist [Frege, 1884; pp. 56-57 and §.14]3. One may instead synthesise the variety of grounding components of Mathematics by looking at it as to a "three dimensional manifold" (a generalized "three dimensional space"). Mathematics is found on, and uses in its developements: • logic, • formal computations, • geometric construction principles. The generative nature of Mathematics is due exactly to the blend of these "three dimensions". For example, once some key regularities of space are singled out (by suitable linguistic descriptions), one uses logical or formal principles (which belong to language) to derive new properties of space (to be given in language); similarly, but entirely within language, formal computations unfold consequences of logical principles. In either case, invariance preserving transformations are applied, as both the rule of logic and of formal computations preserve meaning as invariant or stable conceptual structures: this is what they have been spelled out for! (technically: they preserve "validity w.r.t. the intended interpretation"). In particular, they preserve meaning in space. That is, if a statement about space, say, is realized in some structuring of it, then logic and computations preserve its validity via deductions and lead us to new realizable statements about the world. Thus, we add the generative power of (logical or formal) reasoning, a key linguistic tool for organising/understanding the world, a "meaning preserving" tool, and transform basic invariants of space, say, into novel meaningful invariants. Sometimes this is done by taking major detours and, then, remote techniques for algebra turn out to be useful in Geometry or in differential calculi or alike. But one may also go from (formal) language to space. For example, by symmetries and dualities one "understands", in the Cartesian plane, the "meaningless" i = \ - l and

-* This is essentially insufficient: only Euclidean geometry is closed under homotheties - or only its group of automorphisms includes ratio preserving maps; even in the '20s Frege will keep referring to Euclidean geometry only, in his attempt to broaden the logicist foundation of Mathematics.

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even -i (by a key generative simmetry of the Cartesian plane: the addition of the negative coordinates). And then the complex numbers, the result of the formal/mechanical solutions of certain equations, suddenly become particularly relevant for Physics, that very Physics which describes space and action in space: formal matrices, say, represent generalised vectorial computations in multidimensional spaces ... (a crucial step in Quantum Mechanics). Then one derives properties of complex numbers, and of functions on them, by a blend of spatial properties (symmetries ...) and language transformations, i.e. by geometry, logic and algebra. There is no miracle here, but the relevance of a conceptual construction, Mathematics, whose aim is to focus on key invariances: of space, of reasoning and of formal deduction. Geometry makes space intelligible by singling out some key regularities of it and turning them into invariant properties w.r.t. to the intended transformations that action in space suggest to us. Similarly, logic evidentiate some invariants of language: the logical principles "pass through proofs" or are present on all proofs and do not depend on contextual constructions. Some may be detached from meaning, even logical meaning, and become purely formal computations, to be applied mechanically: the formal rules then impose the computational invariants. The blend of these three conceptual dimensions makes Mathematics generative and effective. This form of generativity is the reason of the "extraordinary" effectiveness of mathematics. In summary, starting with some basic regularities (invariants) of space, say, Mathematics "generates" further invariant properties by using logical trasformations and invariants (regularities) of language (that very language we use to "organise" space) and so on so forth in all possible combinations of its "three dimensional" nature (in Girard's logical systems, for example, one uses geometric principles - such as symmetries, connectivity, ... the unfolding of knots in "proofs nets" - to carry on deductions). This is effective, as its strength is in the maximal (not absolute!) invariance and stability of each dimension of the conceptual constructions and on its "interpretation preserving transformations". It may be surprising, because, say, the unexpected spatial interpretation of the formal symbol i embeds it into a novel meaningful frame, the locus for deductions grounded on very different principles, space, yet compatible as given in another conceptual dimension. 2. Meaning and Intuition The foundational project, besides the formalist analysis that can still provide relevant information on least deductive frames (and suggest ways to implement on computers as much Mathematics as possible, an interactive help to the proof), should now be extended to an analysis of "meaning" and "intuition", this betrayed notion by the logicist and formalist tradition and that many, in contrast, mentioned so often (Riemann, Poincare', Weyl ...). The point is that Mathematics is effective also because it is meaningful and because it is grounded in intuition. Mathematical knowledge is constructed by an interaction with human intuition, a notion to be discussed below, which is dynamically modified along the genesis of the discipline. As for MEANING, before proposing a further specification, in connection to intentionality and life (in §. 3), let's more closely describe meaning as given by a network of (sufficiently stable or invariant) practical and conceptual experiences, grounded on mathematical but also on other (conceptual) praxis. For example, meaning

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is added to an analytic equation of geometry, when it is understood as a line, a plane, a n-dimensional surface ... V-l aquires a meaning on the cartesian plane, as a different conceptual constructions, "per se". Thus, meaning is first in "setting bridges", displaying metaphors, elucidating rigourously references, by mutual explanations. The very fact that this embedding and bridging is made in many and different active experiences allows us to extract the meaningful mathematical invariants, the stable "conceptual contours" common to many of them. Then, INTUOTON may be analysed as a direct, yet pre-conceptual, reference to a variety of meanings, whose interconnections motivate and provide robustness to the mathematical construction. It is a largely pre-linguistic experience; often an "insight" or "mental seeing" of (part of) the mathematical structures, as mental constructions, whose network is meaningful to us. The mathematical intuition is the ability to insert a more or less formal expression, either a "hint" or a symbolic notation, into a "network" of meanings. Seing is its main organ, as the trained mathematician can (re-)construct an image from formulae, similarly as a trained musician may "hear" music when reading a piano score (the mental reconstructions of images from verbal descriptions is a very common experience). Thus, intuition precedes and follows language. It follows it, in the sense that it is "seing" the result of a conceptual, even formal, construction, also or entirely developped in language; it precedes it, as this seing, usually, needs to be later specified in language, as it may yield a vision of a novel structure, a combination of previously inexisting ones, to be fully determined and communicated by language. Mathematical intuition is far from being static and "pure". Training is an essential part of it: it is ever changing and rich of the impurities of the subjective experience, yet it is brought to be shared and "objective" by the common cognitive roots and by intersubjective exchange. It is a crucial part of the good teaching of mathematics, it really makes the difference w.r.t. the bad teaching, as teaching to use mechanically "compulsory", meaningless formulae: the good, passionate mathematician teaches "to see" and to conjecture (seing before the proof), before and jointly to teaching how to prove. Intuition is mostly "local", as pointed out in [Piazza, 2000], even if it may be very broad (yet, "if everything were intuitable, nothing would realy be so" [Piazza, 2000]). The mathematician "understands" by using well established or original references to meanings and structures: he then "sees" the meaning as a structure (geometric, algebraic) or in a structure, this is his intuition. Intuition integrates different conceptual experiences, it allows a blend of methods without caring of the details of a formal proof: one "sees" unrelated structures toghether, proposes unexpeted bridges, by joining, in a new meaningful structure, long lasting work in different areas. This constituting of meaning should not be understood in a shallow way: its analysis must span from the earliest and deepest relation of our "being" in the world and relating to others, even in pre-human phases, up to the human, historical and most complex endeavour towards knowledge. The project then is to single out the objective elements of this formation of meaning and how it underlies intuition, since they step in the proof, as we said. The aim is to analyse in a scientific fashion what has been explicitly "hidden under the carpet" along the XX century: the role of intuition and meaning, even in proofs. Memory, for example, is one of the ways by which we constitute "meaningful" invariants, by selecting, comparing, unifying, by providing analogies. Forgetting is one

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of the major tasks of human memory, in contrast to digital data base: a goal directed or "intentional" choice, yet largely unconscious, of what is relatively relevant, a way to focus or single our what is stable or what is worth considering as an invariant. And by memory I both refer to individual and collective memory, that one which stabilises and enriches through history or through shared, intersubjective experiences. Or even to phylogenetic memory, as it seems that some pre-conceptual invariants, such as small numbers, are part of our inherited experience (see [Dehaene,1998], [Longo, 1999]) By this cognitive and historical formation of sense, meaning, as reference to space, or action, or to structures of language, that we later called "logical", or to other forms of knowledge, is at the core of the certainty, objectiveness and effectiveness of Mathematics; these are the final result of this constitutive processes towards conceptual stability. In other words, Mathematics is based on the constitution of conceptual invariants, grounded of a variety of "acts of experience", distilled in praxis, by our action in the world, from movement in space to memory and displayed by language in intersubjective communication; these invariants are "what remains" once the "details" are erased; they are the common conceptual structure which explicitly expresses our relation to the phenomenal world. The permanent reference, while theory building or problem solving, to the networks of constituting meanings is the reason for their certainty, effectiveness and objectivity. The mathematician's intuition is the grounding of understanding into this network of meanings. Its analysis is an integral part of a foundational project and, as such, it cannot be only a (meta-)mathematical problem: it is a cognitive issue, which spans from Biology to History. 3. Meaning and Intentionality, in Space and Time "Geometry ... is engendered in our space of humanity, beginning with a human activity " [E. Husserl, 1936]. The first locus for meaning are space and time. Well before any explicit or conscious representation, the first goal directed action is that of the living cell that moves in a direction, in order to maintain or improve its metabolism. And this a "meaningful action" and its meaning is at the core of life: it is meaningful, at the most elementary level, as it is part of a goal, of an intention (it is intentional4). In order to understand this approach to meaning, we need to analyse the difficult entangling of finalism and contingency that is central to life. In contrast to inanimate objects, a living being needs to "interpret" the world, in order to live in it. This may be at the most elementary bio-chemical level of the cellular reflex or at the incredibly complex level of our brain. At each moment we need to interpret the environment relatively to our main aim: survival. It is the "finalistic contingency" of life that forces 4

I am broadening by this the notion of intentionality in Husserl, as the prevailing husserlian tradition restricts intentionality to a conscious activity: intentionality is the (conscious) "aiming at an object (of consciousness)" and this object is meaningful exactly because it is "aimed at", consciously (see, for example, [Lanfredini, 1994] or the many papers on this in [Petitot et al, 1999]). In view of the remarks below, I dare to expand Husserl's clear and robust notion, in a compatible way, I hope.

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us to give meaning relatively to an aim (finalism), our unavoidable aim, but it does not need to be there and strictly depends on, but cannot be reduced to, the context (its contingency): each individual life or even that of a species, life itself is contingent. This paradigm is finalistic, but it stresses contingency, as no species, no living being would be alive if it did not have this aim at each instant of its life; yet, life and its specific realisations are not a necessary consequence of the "previous state of affairs" (they are contingent). Thus, by contingency I also mean "context dependence"; and the two meanings are related, as "depending on the context" is entangled with the lack of general necessity. The relation to finalism interests us, since giving meaning to an incoming information is "inserting/contextualising that input" into or w.r.t. an aim, a goal, this is the main thesis here (a similar idea is briefly hinted also in [Bailly, 1991], a remarkable essay in philosophy of science). We have at each instant both inputs and aims, at least a major one, life. The living being, beginning with the most elementary form of life, interprets inputs by comparing them against its main aim: preserving or improving its metabolism. Thus, "meaning" is first given by how much the inputs helps or diverts from it. In particular, this is where begins our relation to space and time, as living beings, well before any symbolic notation can be detached from them. Human geometry is effective, because it begins with the action in space of the amoeba, or with the squid choosing the shortest path to hide behind a suitably large rock (see [Prochaintz,1997] and Longo's review, downloadable). These are the very early steps towards an attempt to organise space, up to the human proposal to make space intelligible: geometry. A proposal among others: we called "mathematical" the one focusing on invariants and conceptual stability. Of course, these are just the very remote origin of our relation to the world: it is like a little stone in the enormous mountain that evolution and, later, human history have been adding on top of it. Between the "meaning" of a chemical stimulus interfering with its metabolism, for an amoeba, and the meaning, for us, of a ... mathematical proof, there is an abyss, two billions years wide. However, there is also the continuity of life, a "non differentiable" continuum perhaps, with sudden turns, as life is necessary to meaning: this is constructed, first of all, as interpretation w.r.t. life's implicit finalism, then as a network of mutual references, of "explanations", up to the reflective equilibrium of our (scientific) theories. In summary, meaning is at the core of effectiveness, in Mathematics and in other forms of knowledge as well. That is, Mathematics is effective since it is meaningful, if one understands meaning as relation to "our active presence in the world", from the simplest action of the living cell up to our endeavour towards complex relational life and knowledge. Or the rich blend of them, as one cannot clear cut between the "meaning" for the individual cell and that for the human individual, made out of cells. In our brain, neurons react as cells to stimuli, but they do it as part of assemblies of neurones ([Edelman,1992]), which in turn are influenced by our unity as living beings and, thus, by our "external" action and intersubjective exchange. These levels are not "stratified", one on top of the other, the one below specified independently of the above, but they interact in a self supporting way (in an "impredicative" fashion, to put it in logical terms). Moreover, senses are far from being "input channels", their computational parody, but they are a dynamical interactive systems, where the "form" of the input cannot be detached from action and aims: sensorial inputs are always actively

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selected and structured, according to an ongoing interpretation, for the purposes of action. This yields the complexity and richness of meaning for humans, as the non additive sum of "meanings" from the living cell up to intersubjectivity in history. 3.1 From Euclid's "aithemata" to Riemann's manifolds I pause here to sketch again some historical and philosophical remarks at the base of the author's contribution to a project, a scientific analysis in progress, as this analysis of human meaning in Mathematics is a long term goal, even when "restricted" to phenomenal space and time (see [Longo et al, 1999]). First, we react to space as living beings: distance is measured by movement, muscular thresholds, vestibular memory of rotations, [Berthoz,1997J. Time is related to it by action. But how do we go from these pre-conceptual experiences to the mathematical concepts? The interlocking of space and time contributes to give meaning to both of them: explicit metaphors for time refer to space and conversely, or we understand one in terms of the other and of our presence in them (see the "metaphors" in [Lackoff and Nunez,2000]). We perceive the symmetries of Physics (the reflections of light, crystals ...) and we give them a major relevance: they are "meaningful" to us, as symmetries shape our bodies and our lives, they underlie our actions and aims, towards pursuing life. Our geometric proposals are shaped along symmetries, as well as along lines of least action, geodetics, a further relevant regularity of space. This is so, say, for Euclid's "aithemata" (requests), which are five practical constructions with least tools (ruler and compass), grounded on least paths and symmetries: they are surely not "axioms" in the formalist sense, as they are rich of meaning, as action or constructions in space. The first axiom, for example, that is "draw a straight line from any point to any point", describes an action along a Euclidean geodetic, and so on so forth with three more. The fifth axiom describes the most symmetric situation when drawing a line on a plane, by a point distinct from a given line. More precisely, following Euclid's statement, consider, on a plane, a straight line d cutting two straight lines b and c. Then b and c meet exactly on the side where they form, with d, two angles of sum less than 180°. Or, they are parallel exactly when, on both sides of d, the sum of the angles is 180°. Why this geometric assumption should be "the most convenient" for understanding the space of every-day life, as many claim, Poincare' in particular, but not fully general for a physico-mathematical analysis, as we know since relativity theory (and as Riemann and Poincare' himself had conjectured, see [Boi,1995]) ? There are here two phenomenal levels, which stress the internal generativity of Mathematics, once its conceptual tools are well established: the local and the global analysis of space. First, the local analysis of neighbouring distances, as the space of movement and local perception, and its extension to the Euclidean plane. At this level, Euclid's fifth axiom describes the most symmetric situation: if one assumes convergence of the two lines on both sides (Riemann), or many lines that would not converge on either side, even when the internal angles are different form 180° (Lobachevskij), then many symmetries, on the Euclidean plane, are lost. That is, for the local-Euclidean analysis, the two non-Euclidean cases lose all symmetry axes orthogonal to the two lines, except one, as well as the parallel (central) axis of symmetry. In equivalent terms, if one draws, in a point, exactly one parallel to a given line on a Euclidean plane, then

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one obtains more symmetries than when following the two "possible negations" of the Euclid's fifth axiom, naively represented in the Euclidean plane. Euclid's geometry constructively proposes an organisation of physical space grounded on (planar) symmetries (and straight lines understood as light rays, according to [Heath, 1908]) Yet, there is another phenomenal level. This one focuses on the locality of the Euclidean manifold and suggests also a global look to space. This understanding of the world is constructed in a difficult intersubjective practice, through history: it is the passage from the Greek geometry of figures to geometry as a science of space, from Descartes to Gauss and Riemann. Along this path, man had to get used to manipulate actual infinity, a difficult historical achievement, far away from Euclid's understanding (see the fuzzy use of "apeiron", indefinite, in the fifth axiom and in the definition of parallel: "eis apeiron", "in apeiron" ..., a truly indefinite concept, in contrast to the perfect rigor of the other notions). One major step towards this achievement, consisted in conceiving the convergence point of two parallel lines, "out there", into actual infinity, a necessarily global look to space. This is projective geometry: it provided an early, implicit, distinction between the local level of the figures of Euclidean geometries and a global level of a geometric space, which includes the point at infinity. Projective geometry is still compatible with the Euclidean approach, yet it is a relevant extension of the latter, largely due to the pictorial experience of the prospective in the Italian renaissance (this is when projective geometry was actually invented). The mathematical proposal grew out from the interaction with painting, a remarkable example of this singling out of mathematical concepts from our attempts to describe the world, even for very different purposes. By this, also, it became possible to conceive the phenomenal level of actually infinite planes, where one may have that remote convergence point of parallel lines. In summary, the geometrical experience is enriched by the proposal of a second phenomenal level, the global level of actual infinite spaces, beyond, but compatible with the local level of Greek geometry of figures. The infinite and absolute of Newton's spaces is a further development of this new conception. Gauss and Riemann's analysis of curved surfaces is a dramatic change of view point. The idea is that the global geometric properties could differ from the local ones: ratio of distances could vary when enlarging figures. As a matter of fact, Euclidean geometry is the only one whose transformation groups (whose automorphisms) contain the homotheties (i.e., its local properties, such as ratio of distances or angles, are invariant w.r.t. arbitrary enlargements and their inverse): homotheties are not automorphisms in non-Euclidean geometries. In Gauss and Riemann's differential geometry, distance can be defined locally in an Euclidean fashion, by generalizing Pythagora's theorem to differentials (in a bi-dimensional manifold one may set ds2 = £gjj dx'dy) and this determines the local structure - the metric and curvature). As for the global structure, it is topology that matters, as the Cartesian dimension is a topological invariant (dimension is preserved exactly under topological isomorphisms), while (relative) distance is a local property (the metric structure is a local property). As already mentioned, Riemann, in 1854, conjectured that the presence of physical bodies could be related to the local properties of distance, the metric: a remarkable insight towards relativity theory, as acknowledged by H. Weyl in 1921 (see [Boi, 1995]). Riemann (but Gauss and Lobachevskij as well) was explicitly working at a geometry of physical spaces. This stresses the relevance of his revolutionary proposal

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towards a way by which we organise space and understand the world (see [Bottazzini, Tazzioli, 1995]). A proposal grounded on "acts of experience", as Weyl would say, i.e. on a progressive formation of sense, through history (see [Weyl, 1927]). This change of perspective largely influenced our understanding of Physics, as we all know, since it modified the key "phenomenal veil", in the sense of Husserl, as the interface between us and physical space, and thus it modified the geometric intelligibility of it. Non-Euclidean geometries, or, more generally, the treatment of space as a (riemannian) manifold, proposed a novel phenomenal level, as locus of the interaction between us and space. A new Physics is constructed over it: the effectiveness of the proposal is due to the fact that it is the very (mathematical) language for a new conception of space and time. It organises the world and generates the new objects of the physical reality; by this, it defines an understanding and helps to predict. There is no pre-organised reality that we perfectly describe, by miracle, by our independent tools (either formal or platonic Mathematics): these very tools are proposed while organising reality, in space and time, and trying to make sense of it, from Euclid to Riemann. 3.2 More on symmetries and meaning As already mentioned, in Euclidean geometry, the local properties are extended for free, by homotheties, to the entire space; in particular, the local notion of parallelism is extended "indefinitely", by the (apparently) more symmetric situation (i.e., w.r.t. the naive Euclidean interpretation of the two negations of the fifth axiom). And symmetries are meaningful for us, as living beings: our own body is organised according to symmetries; we detect them very easily and use them regularly in action and pattern recognition ([Berthoz,1997], [Ninio,1991] and many others, for example in gestaltist approaches to vision). But how could physicists give up, in this century, such a meaningful and relevant physico-mathematical property as symmetry and prefer, in some contexts, nonEuclidean geometries? The point is that the newly constructed phenomenal level, the one which allows to conceive different "global" geometries, has still enough symmetries, from an algebraic point of view. Yet, how to count symmetries, as they are infinitely many? A sound way is to analyse the group of isometries. Now, this group, on the plane, is generated by the symmetries (as reflections) and it can be proved that there are isomorphisms (of different sorts: algebraic, topological) between these groups in the various geometries. Thus, from the new, subsequently generated mathematical point of view, that of algebraic geometry, one has that symmetries, in the different geometries, have a "similar" algebraic expressiveness. And then, physicists, when working at the abstract level of formal representations of space, may indifferently find more suitable, in order to describe space, one geometry or the other, according to empirical evidence, when possible. That is, as far as symmetries are concerned, there are no general-algebraic reasons to prefer one geometry to the other, at least not grounded on symmetries, while there may be empirical ones (the curvature of light in astrophysics, typically); yet, the Euclidean approach is the most obvious (convenient?) extension, by homotheties, of our local space of senses, with all its symmetries. These are the "evidences" behind Euclid's axioms. Note instead that the mathematical descriptions of space in microphysics and in astrophysics are not closed under homotheties, so far: the geometry of quantum

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mechanics, the Euclidean geometry of "medium sized objects" and the spaces of relativity have properties that cannot be transferred one from the other by homotheties. There is no unified geometry (so far; yet progresses are being made) for these three phenomenal spaces, a geometry invariant under homotheties; but there are good reasons, there are evidences which underlie each geometric proposals. As a matter of fact, the evidences for the non-Euclidean negations of Euclid's fifth axiom, are based on a peculiar "path through history", which we sketched above and which enriched our mathematical constructions: by this we could propose new physical experiences and describe a different understanding of geodetics, say (the light rays of Relativity). As a matter of fact, the concept of infinity, symmetries and their interplay in the distinction between local and global phenomenal levels, in the analysis of space, gave us a language by which could propose a new understanding of (the notion of) rigid body and light ray. As for the notion of "evidence" above, the point we are trying to develop is beautifully hinted by Husserl: "the primary evidence should not be interchanged with the evidence of the "axioms"; since the axioms are mostly the result already of an original formation of meaning {Sinnbildung) and they already have this formation itself always behind them" [Husserl, 1936; p. 193]. Axioms, then, even the "meaningful constructions" by Euclid's, are not the bottom line of the foundational analysis: geodetics or symmetries, as meaningful aspect of our manifolded relation to the world, are "behind them", in Husserl's sense. And these properties of space and of our relation to space, do not depend on the specific geometry, but, in different forms, they are also "behind" the axioms of non Euclidean geometries, jointly to the other properties that Riemann, Poincare', Weyl and a few others begun to analyse: isotropy, continuity, connectivity (see [Boi, 1995]). Thus, the formal, unintepreted axioms, Hilbert's style, are far away from founding Mathematics. Instead, they, in turn, are grounded on meaning, often to be made explicit, often necessary to the proof, as mentioned in the section 1.2, even for the most mechanisable of our mathematical theories, the arithmetic of integer numbers. These meanings relate Mathematics to the world, ground its constructions in it and, by this, turn Mathematics into a certain, objective, effective science. Intuition is the bridge that provides foundation, by "understanding", that is by embedding formal notations in a network of meanings; these are "behind" the constituting of conceptual invariants, as intentional selection of common elements, of bridges and analogies, interpreted for the purposes of aims, such as life, actions and human search for knowledge. 4. Contours and Stability There is no split between mathematical constructions and the world, as we draw Mathematics, its "geometric or conceptual contours", on the "phenomenal veil", that is on the interface between us and the world. We ground it by this in regularities of the world, while singling out these very regularities and defining our own "self. The construction of objects, our singling them out, and of concepts, derives and gives meaning from and to the world. By these reasons, by this rooting of knowledge in the meaningful presence of our biological and historical life, by this cognitive presence of ours in the world, Mathematics is so effective for its purposes. Mathematics is, by definition, the collection of the maximally stable concepts that we can draw on the phenomenal veil, the invariants that we may transfer in many other forms of description of phenomena, exactly because of their stability and invariance or strong contextual

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independence. And we always need to single-out an invariant in order to constitute a (physical) object. Moreover, we simultaneously design and force a formalism from and onto reality: Mathematics is normative and generative, as it proposes rules and methods for deriving new concepts, new structures. These may then suggest a new understanding of the world, a new tool for "reading" it. Mathematics, thus, is effectively applicable to different contexts of knowledge: it forms the underlying texture of our representations of the world, by its very construction. The point is that we actively single out objects and propose concepts. And we extend this "action" into autonomous mathematical constructions: the (more or less formal) mathematical developments are a purely mental extension, made in language and intersubjectivity, of the human praxis of generating stable (geometric or conceptual) contours. Of course, Mathematics also departs, by its internal methods, from its direct rooting in our understanding of the world, but these very methods derive their sense and are made possible exactly by our active and creative intuition, in the husserlian sense (see [Boi, 1998] for a clear understanding of Husserl vs Kant on intuition). In other words, while determining our "spaces of humanity", we simultaneously draw the borders of "objects", we single out relevant contours, we understand while interpreting and naming. Our ego is constructed at the same time as the world of phenomena around us. Of course, there is a reality "outside there", which oppose "resistance and frictions" to our action, but phenomena are constituted in the interface between us and this unorganised "reality". Mathematics plays a major role in this process, at least in Physics, or as soon as our action tends towards scientific or sufficiently general knowledge. Mathematics "singles out" contours of objects, by the drawings of geometry and, more generally, by conceptual shaping of images and ideas. Mathematics is the drawing of contours which do not need to be there. Human vision is a good paradigm for this, as we do not "swallow" images passively, but they are (re-)constructed by active interpretations. In vision, some areas of our primary cortex reacts only to contours. But contours are not objects, they are singularities, in the mathematical sense, at the edge of bunches of wave length. We perceive these singularities and use them to isolate one object from the other, by deciding where and how to "cut" or delimit those inexisting lines. This is done in a continual interplay between incoming messages and interpretation: visual illusions tell us the permanent role of interpretation in vision, on the grounds of memory, interpolation, tridimensional reconstructions. Consider, say, the names of colours, so history dependent, yet not arbitrary. It is a completely human and historical choice that of categorising colours, like separating blue from green, by giving, with a name first of all, "individuality" to this or that colour, marking borders in the "continuum" of wavelengths, between say "burnt siena" and "red amaranth" or even "blue" and "green" (Euclid, as all Greeks, had the same name for blue and green). Yet wavelengths are there, as well as our retinal receptors, which have "pigments" sensitive to three primary colours (they have excitement peaks corresponding to the wavelengths of red, blue and green): these colours, as parameters in a three-dimensional space (an example of a three dimensional manifold, mentioned by Riemann), allow the reconstruction of all possible wavelengths, but many other triples would do as well.

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It would be interesting to know more about the role played, in the history of language, by those three primary colours that evolution has given us to reconstruct the others. They are like "pivots" upon which we build our mental categorisations and that probably drive our choices favouring some lengths over others, thus making nonarbitrary our categorisations. Because this is the point: scientific (and mathematical) reconstructions of the world are possible proposals and yet they are not arbitrary. Thus, the foundational issue is in singling out the phenomenal "pivots" on which, along history, we built up our forms of knowledge. As for geometry, and following Riemann, Poincare', Weyl, we referred to symmetries, isotropy, continuity and connectivity of space, regularities of action and movement, as "meaningful" properties. They are meaningful as they are embedded in our main intentional experience, as hinted above: life. In Mathematics, then, we have been singling out conceptual contours, grounded on these "pivots" and regularities in the world, comparable to the three primary colours in the pigments of our retina. Then, we stabilised them in "abstract" (contextual independent, yet meaningful) geometric and linguistic invariants. More precisely, the core of the mathematical work is in turning the relations between invariants into norms and, then, using these norms to carry on further constructions (and proofs of further relations between constructions). This is the normative character of Mathematics: by Mathematics, we structure scattered phenomena by norms. Then, further mathematical structures extend the conceptual construction to more complex forms, built one on top of the other, interrelated by morphisms, which preserve the intended invariant (continuity say, for topological morphisms, operations for algebraic ones) ... and this gives "categories of objects and morphisms"; then categories are related by "functors", which transform objects into objects and morphisms into morphisms. By continuing the category-theoretic metaphor, "natural transformations" relate functors and categories and so on so forth. Mathematics acquires then that typical autonomy from the world which singles it out from other forms of knowledge: it is grounded on a few cognitive and historical pivots and, once some invariants are stabilised by drawing and language, we use a variety of conceptual tools, a blend of many experiences (logical, formal, spatio-temporal ...), to constitute norms and derive new invariants, often far away from the ones we originally derived by our active, interpretative presence in the world. 5. Microphysics and Dynamics Physics has always been the privileged discipline of application for Mathematics. Indeed, Mathematics itself owes most of its own constructions to attempted descriptions of the physical world, beginning with Greek geometry, a dialogue with the world and with Gods at the same time. By this, geometry was detached by them from the "measure of the ground", an early process of "singling out" perfectly stable and invariant figures. Up to the infinitesimal calculus and Gauss-Riemann differential geometry, explicit attempts to describe movement and physical space. And further on, till the Mathematics of Quantum Mechanics, where the audacity in singling out inexisting but non-arbitrary contours reaches its highest level. "Each element ... must be prepared; it must be sorted; it must be offered by the mathematician. We then see to appear, in physical sciences, the opposition between descriptive and normative. The attributing of a quality to a substance was once of a descriptive nature. Reality had just to be shown. It was known as soon as it was recognised. In the new philosophy of science, we must

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understand that the attributing of a quality to a substance is always of a normative nature. Reality is always the object of a proof." [Bachelard, 1940; p.89]. As a matter of fact, in microphysics, we see nothing but some "crick-crack" or spots on measure instruments, which are far away from "phenomena". Then a mathematical theory is proposed, which gives unity to these "symptoms" by drawing a mathematical interpretation and, by this, by turning them into elements of a phenomenal description. Actually, even the physical measure instruments are constructed on the grounds of a theory, as they are just a practical explicitation of a theoretical hypothesis: we should measure this and that, in this way. Atoms, electrons, photons, gravitons ... are not there, they are "mathematical contours" which we single out by unifying a few signals. They are a way to propose non-arbitrary physical or conceptually stable borders. They are not "objects", yet they are as objective as our most robust theories of the world, since the physical world does "make resistance" and forces some signals towards us and the measure instruments we constructed. Physicists are ready to update them continually, even if, at each stage, the proposed invariants may be at the core of a relevant mathematized theory, often rich in applications. This drawings of contours, these constructions of invariants and of perfectly stable conceptual entities are at the core of the applicability of Mathematics to inanimate matter, they actually are at the origin of Mathematics itself. Or, following Boi's interpretation of Weyl, "physics is but geometry in act" ... "so that the mathematical understanding of this world cannot be separated from the understanding of reality itself, [Boi,1998] (see [Connes,1990] for a geometric insight into Quantum Mechanics). Or, to put it otherwise, we understand Physics (movement, gravitation, non-locality, say) or we have it as "phenomena", by the very act of proposing a mathematical theory of space-time. Of course, the effectiveness of the mathematical tool is relative to the interaction of an ongoing proposal and the various phenomenal levels: unifying phenomena is a major criteria for effectiveness. This proposal is an integral part of phenomena, but gaps are possible as we make "choices" while setting conceptual contours. These choices are not arbitrary, as they are grounded on incoming signals, on meanings, on accumulated, historical knowledge. The selection is made with reference, often, to other forms of knowledge, by implicit analogies and metaphors (such as the metaphor of the planetary system, say, to understand the atom). Thus, they may yield incomplete descriptions. An example is given by the dynamical systems which are sensitive to initial conditions: we cannot predict completely they quantitative evolution by our mathematical tools. Their effectiveness then is limited, as predictability is a component of effectiveness, even if they give a better understanding of phenomena, by unification or explanation. Of course, impredictability is not a matter of the world: there is no way to know whether the "physical reality" is "chaotic per se". The question only makes sense at the phenomenal level, the only accessible one, the actual interface between us, or our nonarbitrary proposals to organize the world, and "reality". As a matter of fact, God may know very well where the Earth will be in more than 100 millions year (see [Laskar,1990] for impredictability results on the solar system). In deterministic, chaotic systems, the only "fact" is given by an increase of "complexity" of some phenomena; but this understanding is already a theoretical proposal, an organized understanding. That is, impredictability shows up when something is said (dicere), or concepts are

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displayed, or they are proposed by a (living and thinking) conceptor: it is not a metaphysical truth. Impredictability requires an attempted prediction, while interacting with "physical reality" (or signals coming out of it); it concerns our mathematical attempts to organise these signals and it says that our attempts are "provably incomplete". 6. Incompleteness in Mathematics and Physics One may draw then an analogy between the incompleteness results in formal theories (see sect. 2) and the one mentioned above, concerning dynamical systems. Let us first recall that the incompleteness results in mathematical logic, in formal arithmetic in particular, evidentiate a "gap" between the formal, theoretical level and meaningful mathematical structures: one cannot "remove the machinery" from proofs, a machinery which refers to transfinite ordinals or well-orderings, as constructions in mental space and time. In these cases, the normative structuring of Mathematics extends iteration and order, beyond phenomenal time and space, towards and by the concept of infinity. This is the nature of purely mental constructions, well beyond the finite, such as transfinite orders or infinite well-ordering. In a sense then, infinitary constructions in mental space and time may be understood as the subjective traces of intersubjective extensions of the objectivity of the phenomenal world, i. e. the are the "mental marks" of the objectivity we constructed in intersubjective, historical praxis, over basic regularities. The concept of actual infinity is the result of many historical conceptual constructions (theological, based on the projective geometry of renaissance painters ...). Its objectivity is obtained as an integration of "metaphores" (see [Lakoff&Nunes, 2000]; but they are not just linguistic metaphores) and by the normative structuring of Mathematics, well beyond phenomena and leaves traces in our minds; it is extended by a blend of manifolded experiences (the metaphysics of infinity, say, played a major role in the constituting of the mathematical concept, from St. Thomas to Leibniz, see [Zellini,1980].) The finitary formal approach, which does not include these meaningful structures, such as infinitary mental constructions (the well-ordering of the set of all numbers, say), cannot completely describe their properties, as it has been proved. In the case of arithmetic, the incompleteness is due to the fact that the well-ordering of standard numbers cannot be axiomatized in a finitistic-effective way, yet it is an absolutely clear and robust structural property of integer numbers, seen as actual infinity in mental spaces: when used in (human) proofs it yields formally unprovable results (see, for example, the proof of Friedman's Finite Form of Kruskal's theorem, quoted in §. 1.2). One century before Hilbert's wrong conjecture of the completeness (and decidability) of formal theories, Laplace had formulated a similar one, in mathematical Physics: in his opinion, the systems of (differential) equations could completely describe the physical world. More precisely, if one wanted to know the state of the world in a future moment, with a given approximation, than it could suffice to know the current state of affairs up to an approximation of a comparable order of magnitude. By formally computing a solution of the intended equations, or by suitable approximations by Fourier series, one could deduce (or predict or decide) the future status, up to the expected level of approximation. Poincare', as a consequence of his famous theorem on the three bodies problem, proved that minor variations of the initial conditions could give enormous changes in the final result or, even, that the solutions could depend discontinuously on the initial

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conditions. Then, predictability, as "completeness w.r.t. the world" of a suitable set of formal equations, failed. These results, thus, and the subsequent work in dynamical systems, are the mathematico-physical predecessors and analogues of the many prooftheoretic incompleteness theorems, since Godel. They set a limit to the effectiveness of mathematical tools in Physics, but they are also at the origin of beautiful and new mathematical theories, where qualitative predictions replace quantitative ones and where the "mathematical understanding" does not need to coincide with predictability (see [Thorn, 1972]). Moreover, these theories, by forcing more geometry and topology into the prevailing analytical approaches to Physics, gave further richness and unity to Mathematics (see [Devaney, 1989] for a survey of the geometric approach to dynamical systems). In conclusion, there is no absolute effectiveness of the mathematical tools, in Physics, but the constructed objectivity of Mathematics is grounded on an interaction with the world around us (and in us) that guaranties a relative effectiveness, though remaining often incomplete. No choice of a specific level of description, such as the mathematical one, given by some linguistic constructions and some geometric contours, may yield a complete representation of the richness of the universe we are embedded in as we have several and interacting forms of representation. Moreover, language and drawings are rich of our internal finalism and interpretation and cannot perfectly coincide with any "independent" physical reality. Their are reasonably effective because meaningful, i.e. because they are grounded in our cognitive being, as an ongoing process that constructs "reality" while living and interpreting it. Yet they are incomplete, by this very same reason: their objectivity is constructed by us, with our changing limitations. unpredictability and incompleteness results are there to remind this to us, and to encourage the permanent invention of new methods and the construction of new "phenomenal levels". We have been able to do so throughout history by dramatic expansions of our tools or changes of paradigms: the birth of infinitesimal calculus and of non-Euclidean geometries are two of the most fantastic examples of this open-ended process, in Mathematics. The believe in "perfect completeness" or "absolute effectiveness" of the current mathematical tools (even of "formal" tools w.r.t. to specific mathematical structures, say), instead of the understanding of their relative completeness and reasonable effectiveness, may be misleading and may made us blind w.r.t. the growth of other form of knowledge, which may stimulate the change. 7. Some Limits to Effectiveness: The Phenomenology of Life The extension of the mathematical method to other sciences, where conceptual stability and invariance are not the main concern, is even less straightforward and sets further limits to its successes. As a matter of fact, the richness of Mathematics is grounded on its unique invariance and stability: one may even define Mathematics as the locus of the maximally invariant and stable concepts that humans could propose, in their endeavour towards knowledge. Mathematics is normative in that this invariance and stability provide the norm, they are not (passively) descriptive. Its effectiveness, in Physics, is due to the essentially constructed (and mathematical) nature of "physical objects", as this is how we "single them out" (by mathematics). But, are mathematical invariance and stability, its (fully general) norms, at the core of Biology?

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Consider, for example, the notion of neural synapses. Of course the biologist has to "define" it, as he has to communicate knowledge, propose a description. Yet, the notion of synapses does not need to be as stable as a mathematical concept, for many reasons. Synapses change in evolution and ontogenesis; they are dynamical; their behaviour is largely contextual and no fully general norm describe it, as the causal relation w.r.t. the context is less relevant that the finalistic organisation of the system of which it is part. Norms are very effective in explaining causes, much less in understanding the contingent finalisms of life. The ecosystems of life change continually the rules of the game. The underlying physico-chemical invariants are part of the phenomenon, but are not sufficient to describe it. Their analysis contributes to the explanation, but the phenomenal level of our relation to living beings is a different one. By this, Physics and Chemistry, and their invariant laws, are necessary to understand life, but they are not sufficient to derive its properties. Clearly, we may change mind, in history, as for the description of an electron, as well: experiences may bring in new facts and suggest novel interpretations. But, in Biology, it is not just knowledge that may be revised, as in Physics and all empirical sciences, but stability, full generality of the "formal" description, repeatability of the experience, are less central than in Physics. What really matters are variation, nonisotropy, diversity, behaviours in an ever changing ecosystems. Again, one needs to write about and, then, to define the synapses, but, by the reasons above, a mathematical definition of it would not have the same interest nor relevance as the unavoidable and crucial or fully explanatory, mathematical description of a particle, in microphysics (see [Jacob,1970], [Bailly,1991], [Longo,1998], among others). For example, one may mathematically define a quark and derive (some) of its properties as theorems, to be later checked by experiences; a mathematical definition of a neural synapses in no way (or in minor, very specific ways) could give properties of the biological entity as "theorems", to be formally derived from the definition. However, although "biological objects" may be hardly captured by the normative nature of the physico-mathematical description, one may consider another element of the biological phenomenon, which has no counterpart in Physics: functionality. It is possible, that the "function" of a living component, organ or being (in an ecosystem or in a compound form of life), may be more effectively described, by mathematical tools, than the "object". The problem of effectiveness then is transferred to the analysis of the "right level" of invariance to attribute to functionalities, i.e. to propose an informative and effective level at which the function can be actually abstracted form contextual dependencies, or may be given the right level of dependence on them and, thus, stabilised in a mathematical description. The point is to find "what depends on what", or how much a specific function of life may be independent, say, from the "hardware" that realises it. Indeed, this kind of problem is typical of Mathematics, in its own context. Category Theory, in Mathematics, beautifully centres it: one has to find the right category to work in, i.e. the structural properties that morphisms or isomorphisms are exactly meant to preserve. A category spells out the invariants that matter, this is the main reason for its conceptual superiority in the foundation of Mathematics w.r.t. Set Theory. Moreover functors relate categories to other categories, tell what must be added or forgotten to embed one into the other. Natural transformations relate functors and unify the various notions stabilised in and by functors and categories. But proofs as well, in Mathematics,

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require a close analysis of invariance. When proving a theorem for all real numbers, say, or an arbitrary algebraic structure, one may "use", in the proof, a generic real or a generic example. Then, at the end of the proof, the mathematician has to observe: "look, my proof only depends on the fact that this is a real or that is this kind of structure, no more no less is used". The task may be non obvious: it may happen that the proof "proves more" the statement, i.e. that less properties where required or that, under those hypothesis, more may be actually proved. Or, conversely, some implicit assumptions have been used. In both cases, the level of invariance proposed is the wrong one, too narrow or too large. The research problem then, w.r.t. the uncertain effectiveness of Mathematics outside Physics, is to single-out some truly stable invariants, in Biology, say. Or, otherwise, to "adjust" Mathematics to more plastic conceptual constructions: instead of using well established methods and structures, with their usual conceptual stability, and work on the data provided by biologists, we should perhaps reconstruct concepts by interacting also with their methods, which are very different form those in Physics (see [Longo, 2001] for more on the distinction between concepts and structures in Mathematics). As a matter of fact, it is very hard to transfer outside Mathematics the crucial theoretical praxis of the discipline, grounded on invariance. It is already very hard to apply it on the borderline of the mathematical activity, e.g. with reference to its applications, as we so often slipped into metaphysics: the relevance and stability of a proposal was confused with an absolute. So, for two thousands years, we were told that Euclidean geometry was "absolute", that it perfectly coincided with physical spaces, independently of any context or assumption, to physical space "per se". Similarly, Cantor-Dedekind's construction has been seen as "the continuum" of space and time. Closer to our times and to the problem we are discussing here, Turing Machines have been presented as the mathematically invariant definition of (human) reasoning, as discussed next. 8. Thought as a Function In 1935-36 the many formal approaches to computability (Herbrand-Godel, Church, Kleene, Turing ...) were shown to be equivalent. The everybody exclaimed: we have an absolute. We coherently defined deductions as computations, independently of the formalism, machine or ... whatever implements it. May it be a Turing Machine, a mechanical (or, later, digital) computer, a set of formal rules ... provided that they contain certain features (as described by the partly informal notion of "algorithm"), then they all compute the same class of functions, the general recursive ones (this is the so called Church Thesis). And there comes the metaphysical slip: since these functions, as deductions, describe the "act of effective reasoning" (the "human computor in the act of deducing", to put it in the words of Gandy, a student of Turing) we have got the universal notion, the invariant defining human intelligence, in a complete and effective fashion. This is the so called "functionalist" approach to human mind. We all know, since then, the many failures of Strong Artificial Intelligence, and the successes of many of its more modest "sub-programs" (the interactive expert systems and theorem provers and much more, which did not assume the full generality of the strong claim). As well as the successes of Computer Science, which is a "science" exactly because the "art of programming" is independent from the machine. Programs must be portable, this is the key motto of programmers. Programming languages must

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be "universal", i.e. compute all recursive functions and being transferable from a machine to another. Both key points are grounded on Turing's remarkable idea of distinguishing software from hardware. But even operating systems, as implementation of Universal Turing Machines, are transferable: so, when a machine dies (a new technology is born) one may take its operating system, its programs and date base and transfer it on another. This practice of "metempsychosis" is a remarkable technology and is at the core of Computer Science. Yet, some proposed for decades this model of computing, its perfect mind/body dualism, as a model for human intelligence (without necessarily being Indus). The mistake again was to believe that "intelligence" may be grounded on formalised "laws of thought", as rules independent of meaning. Meaning as reference first to the "hardware" that implements it, our living and historical brain, and which is rich of the finalistic contingency of life we mentioned above. By this, the proposed level of invariance was the wrong one. The objectivity and generality of reasoning is instead grounded on that very peculiar hardware which is our brain through history, i.e. our human brains interacting by language and action with the world and among them. Its invariance is due to the common biological and cognitive roots and, later, to intersubjective exchange, which allows to focus on what is "stable" as shared with other humans, in an enlarging communicating community. Stability of (mathematical) reasoning, though very high (indeed maximal, among our forms of knowledge), is not an absolute and it is the result of a process towards invariance. It is very hard to spell out, in mathematical terms, which are the constitutive invariants, underlying this very "function", human thought, which produces invariants. The irony is that even machines now do different things than that "absolute" proposed in the '30s. Distributed, concurrent, asynchronous computing uses open systems, working with evolving operating systems and data base, with no absolute time (a crucial physical difference w.r.t. Turing Machine). They perform very different tasks, in some cases, not even vaguely comparable with those of sequential computing with an absolute clock (see [Monist, 1999]). And we still do not have a sufficiently good mathematical description of these novel computer systems, which Physics and engineering have been giving us. Yet, the "functionalist" approach to human thought still presents that wrong level of invariance (the "laws of thought" do not depend on life, are formal and, thus, they may be implemented on a Turing Machine) as the core of cognitive analyses. This shows the difficulties in transferring the normative analysis of Mathematics from "objects" to "functions", when dealing with the phenomena of life and history. It is hard, if ever suitable, to single-out living individuals, mathematically; it is a nonobvious scientific challenge to extend the reasonable effectiveness of Mathematics to "functions", from the simplest living structure (e.g. the internal and external topology of a cell) to those which span life and history, such as human intelligence. 9. Brain Plasticity and Neural Nets A more recent mathematical approach to mental functions has been grounded on the idea of "interactive net of formal neurones". These theories are beautiful physicalmathematical models, a qualitative change w.r.t. those that rely on mathematical-logic descriptions, such as Turing Machines or alike, based on the assumption of a universal "computational logic". Following McCullogh, Pitts and Hebb, neural nets on the

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contrary are inspired by the plasticity of the brain and aim at modelling this crucial aspect of it. In contrast with the functionalist theories, rather than asserting that the "hardware" which realises the thinking is irrelevant (Turing machines, systems of balls and marbles, computers or brains "are all the same" for the functionalist), the mathematical theory of neural nets assumes as essential a biological property of that specific hardware which is "the (human) brain", plasticity. In other words, instead of starting with a logical analysis of (mathematical) deduction and defining it to be an analysis of thinking, the connectionist hypothesis (such is the name for the theoretical proposal of neural nets) stresses a fundamental property of the brain, the plasticity of its electrical connections, and turns it into Mathematics. And this is a Mathematics very rich in powerful tools: "spin-glasses" and methods from statistical Physics, based on the Mathematics of dynamical systems, one of the most modern and powerful instruments of analysis in contemporary Physics. But, at this point ... one forgets the original anri-functionalist project, the one which tries to develop and is prompted by the biological reality of the brain. One forgets that neurones are alive and have behaviours which are not always independent of their individual unity, let alone of their context. Indeed the brain plasticity itself depends on a number of causes. The first set of such causes is the spatio-temporal nature of the electrical message, its geometry. The second is the ability to prime "cascades of chemical reactions", which induce changes in synaptic structure. Third, the huge number of elements which regulate the chemical reactions within cellular and extracellular liquids. And still more causes, not yet well understood, including the tridimensional geometric structure of the proteins exchanged by the synapses. The connectionist proposal seems to mutate into the following: it does not matter how brain plasticity is achieved so long as it is plasticity; the formal nets will simulate it by their very refined Mathematics of electrical connections of continuous weight. But then as soon as we start drifting away from functianaflsm, have we immediately gone back to it? We end up simulating with electrical circuits only one function, the one implemented by neural plasticity as variation of electrical conductivity, even if the real communication, along synapses, is also bio-chemical, it uses the convection of liquids etc.. At a certain point then one does not refer any longer to the structural characteristics of the brain, or forgets many of them which might be crucial but are not considered as such (fluids, bio-chemistry, ... ); again, we are back to thinking that "machines" are interchangeable. The subject then develops, driven by its internal methodology which again bears the physical-mathematical imprint. All this is very interesting, since new questions can be asked to the biologist, and some specific situations simulated. But, mostly, formal neural nets suggest an original way of building extraordinary new machines, very different from digital computers. Still I think that physicists and mathematicians working with neural nets should not present their theory as a cut and dried "Theory of Brain": as we have said, the electrical signal along the axon is very important for understanding brain functions, but there are other factors which play a relevant role and "boundary phenomena" between membranes, Chemistry and liquids have their relevance too: the synaptic connection is far from transferring only electrical information, as it was once believed. Moreover, everything is coherently managed by that living cell, the neurone, which has its own well defined and aggressive individuality. This is killed if it is "conceptually dissected", if one focuses on a single physical phenomenon, the threshold electrical calculus.

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Moreover, this cell is immersed in contexts which are themselves alive and full of connections, which have their own unity and intentions (as aims of life), that are killed off by the "cutting plane" of the mathematical formalisation of the electric signal. As I already stressed, this is the main reason why modelling in Physics differs qualitatively from that in Biology. Today, for instance, to shape a new kind of wing for an aeroplane one can often avoid experimenting in a wind-tunnel. Computer simulation of fluid dynamics has now reached a very high degree of precision and reliability. It is, in fact, an important element for cost-cutting in aeroplane construction. The mathematical description of that physical reality, namely the decoupage of the "physical outlines", of the key parameters of the phenomenon and their electronic elaborations, are a good enough approximation for this practical issue. In this case the mathematical theory, even if at a level of approximation and formally incomplete, is an effective "theory" of the physical phenomenon. On the contrary, there is a qualitative difference when trying to give a theory made of physical and mathematical invariants, which should capture the dynamics of living brains, say; or to propose a "theory" of the biological phenomenon, an effective one for simulating and forecasting the activity of the brain, beyond some very restricted aspects (mostly related to very small numbers of neurones). This is because the few mathematisable aspects will be conceptually isolated from the living whole, and one would work on them using one's own methods of conceptual stability; yet, it is that whole that contains an interconnected network of individual and global aims and intentions. As said in §. 3, intelligent behaviour is meaningful for this. Certainly one must continue working at the mathematical modelling of biological phenomena, but keeping clearly in mind the limits of this approach. And the same to be able to talk productively to biologists (I have seen biologists and physicists with great difficulties in understanding each others on the theme of modelling and neural nets). In particular, in order to progress one must always remember that there is a qualitative difference between mathematical simulation in Physics and in Biology; a difference which I have tried to single out with these observations on "decoupage", as a mathematical practice to isolate invariance, so crucial and effective in Physics, but rather unsuitable when transferred on the unity of living beings and their ecosystem. Moreover, I wish to add that prudent researchers in this area, such as Hertz, Krogh and Palmer [Hertz, 1991], admit that these theories have taken only one or two ideas from neurobiology, and do not make any pretence of giving a "mathematical model" of the brain. It is instead possible that these studies, with their autonomous practical and mathematical developments, might one day provide us with formal "neural" machines that can be even more revolutionary than those which have already changed our life: digital computers. As already mentioned, also Turing Machines were considered by many as the "ultimate" model for human reason, an absolute: they served to a, perhaps, more significant purpose, as they gave us brand new forms of elaborating and exchanging information. Computers are used for some fast numerical computations, impossible to man, or, mostly, for word-processing and world-nets of data, fantastic achievements, that do not even vaguely resemble our mental activities, but enrich them enormously. Good science is worth pursuing, when technically deep, even if the early philosophical project is basically wrong: the indirect fall-out may be as amazing as unexpected. Subsequent philosophical reflections may help to revise it, or in inventing

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new research directions or, at least, in further understanding the world or our descriptions of it. 10. Conclusion In this essay, I tried to delineate the "cognitive" reasons for the successes of Mathematics in Physics: Mathematics is drawn, simultaneously to Physics, on the phenomenal veil, i.e. on the very locus where we apprehend and (re-)construct physical phenomena, as living and historical beings. Its internal generative character, by norms and invariants, goes often further and independently of the relation to Physics; or, even, it may give, sometimes, indirect or unexpected new tools for drawings contours to novel physical phenomena. Thus, its effectiveness is contingent, as much as life itself, since it is first grounded in our active relation, as living beings, to space and time, by language and by all forms of intersubjective communication (gestures are not irrelevant in communicating Mathematics). Mathematics then may be described as a three dimensional space, as hinted in §. 1.3, since invariants of space, of language and of formal computations interfere in a synergetic way while generating mathematical concepts and proofs. Mathematics grows along with the reasonableness of History, which made us construct models of the physical world, since Greek figures of space, while creating the key structures of Mathematics or its very language. Yet, the effectiveness of extensions of Mathematics' fruitful paradigm to Biology is largely reduced, let alone to other disciplines where relational human activity is grounded in but go well beyond biological life. When departing from the analysis of the causal and local or elementary interactions in Physics, the developments of mathematical tools must be done with a similar simultaneous attention not only to the facts and data of Biology but also to its own methodology, so indebted to finalism and global phenomena. As a matter of fact, in a two-ways interaction, the phenomenal level on which we draw Mathematics may change dramatically and it may require great changes in the mathematical methods, as least as relevant as the invention and use of actual infinity or of non-Euclidean geometries. Of course, the switch from the analysis of "objects", in Biology, to that of "functions" has been at the core of a remarkable revolution. Morphogenesis is in part a consequence of this change of perspective: the analysis of singularities and fractals provided original tools for it (see [Thom,1972] and the many writings on Mathematics in Biology), yet the underlying physico-mathematical paradigm, upon which these ideas were born, still leave many biologist unsatisfied. I have been choosing above the most complex of the "biological activities", human thinking, as an example of a "function". This is probably not fair, as the reasons for its complexity go well beyond Biology, in view of the role in it of human intersubjectivity, through history. Yet, at all levels of complexity, as soon as we examine functions of life, it seems particularly hard to isolate mathematical invariants, to find the right "category" and mathematical structures. This is largely due to the systemic unity, contingency and finalism in biological phenomena. For these reasons, in [Longo,2001], it is suggested that, in order to gain in effectiveness, the interaction with Biology should begin by the analysis of the conceptual and even pre-conceptual constructions of Mathematics, which precede or underlie the explicit mathematical proposal for a structural invariant. We shouldn't only

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try to use well established or autonomous mathematical tools in order to contribute to Biology, but we should rediscuss foundational issues in Mathematics with reference to biological experiences (see [Longo, 1998], [Longo et al., 1999]). In conclusion, Mathematics does not capture "the essence of the world", even not in Physics, as we only interact with physical reality at the phenomenal level, where we "draw" the structures of phenomena exactly as mathematical objects, with no pretence to understand essences. Let alone in Biology where the unity of individuals, the interactions with the ecosystem and their contingent finalism appear instead as an "essence", well independent of us. The kantian paradigm shows here how linked it was to the peculiar relation between Physics and Mathematics. In any case, though, Mathematics is as effective as it may be human language, as a tool for communicating. Human language is very effective, as we understand each other very well, but it is not unreasonably effective, nor "complete" in any sense. We need gestures, smiles, caresses, we need to hit or touch each other and make love to communicate more fully what language, in many situations, cannot express. Moreover, each human historical language is incomplete relatively to the others (and thus, it is "essentially" incomplete), as the following example may suggest. A friend of mine, a French sinologue, had to translate a classic Chinese poem. This poem described a river running through a forest. In each ideogram there was a fragment of the ideogram that evocates the notion of fluidity, of running water ... . The visual, pictorial, impression was (reportedly) fantastic and it was an essential part of the poetic communication. This was clearly lost in the oral communication and, a fortiori, by the translation in an phonetic writing. Dually, it is almost impossible to translate in Chinese our complex temporal constructions, such as the past of the future or the future of the past; yet, both Chinese and our languages are very "effective" in describing the world. The universal and complete language of all possible expressions is a wrong dream as much as the universal (and complete) system of (mathematical) signs for all sciences. And this is fortunate, as it confirms the richness of the world and of our tools to understand/organise it. This is why we invented autonomous conceptual (and linguistic) tools, w.r.t. Mathematics, for the analysis in Biology or, say, in History. Indeed, each method has some mathematical aspects, the mathematizable fragments. For example the morphology of a jelly-fish, which is shaped like a drop of milk falling into water (an old example recalled by R. Thorn), or the spots on some fours whose distribution is optimal, according to some geometric criteria (more work in morphogenesis, since Turing), are beautifully mathematisable fragments of live; indeed, they are "physical aspects" of life. Similarly the physical structure or the "geometry" of the visual cortex may require some relevant Mathematics in order to be analysed closely. Yet, the biological phenomenon is also elsewhere: how comes, say, that this forms are genetically stabilised and are reproduced in offsprings? Or, how to analyse the crucial redundancies in evolution and even in ontogenesis, which are so unrelated to the optimal path or geodetics of mathematical Physics? By which path through evolution the "double jaw" of some reptiles of 200 millions years ago is potentially the hears of birds and mammals (the "latent potentials" in Gould's analysis)? The intended structures and concepts are so dynamical in evolution, that any mathematized focus on their conceptual stability and invariance would not be the most relevant aspect, as it necessarily is in Mathematics.

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On one hand, then, it is possible that some fragments of life may be (increasingly) understood by Mathematics, but dually Mathematics owes greatly also to the other forms of description and knowledge, which do not need to be reducible to it. It contains some elements of these forms, which actually contribute to its foundation, as it is rooted also in them or, better, it possesses some common roots, the cognitive ones. And this is one of the reasons for the effectiveness of Mathematics. Once more, this is not a vicious circle, but the virtuous spiral of the open and dynamical system of our forms of knowledge, if we reconstruct its unity as a network, not as a fake ultimate system of axioms which explains or "covers completely" everything, by formal derivations (the "fake wooden frame" to which refers Weyl in the introduction to Das Kontinuum, see the English translation; Weyl's anti-formalist stand is even more strongly presented in his posthumous [Weyl, 1985]). The project then is in acknowledging first the differences in languages and methodologies, as well as their internal limitations, as for effectiveness, and then try to enrich them by interaction and, possibly, by singling out the common cognitive roots of our different conceptual constructions. Fortunately, these constructions, including Mathematics, are not God given, nor perfect and static platonic realms, but human and "plastic", as much as our interacting brains: thus we may invent better ones, as we did very often along history, and then unify them by cross explanations and mutual influences or translations. We may follow new meaningful aims, which may lead us to propose entirely novel concepts and ideal structures. References (Preliminary or revised versions of Longo's papers are downloadable http://www.di.ens.fr/users/longo).

from

Bachelard G., La Philosophic du non, Paris, PUF, 1940. Bailly F., "L'anneau des disciplines", special issue of Revue Internationale de systemique, Vol. 5, No. 3, 1991. Berthoz A., Le sens du mouvement, Paris, Odile Jacob, 1997. Boi L., Le probleme mathematique de I'espace. Une quite de I 'intelligible, Berlin and Heidelberg, Springer-Verlag, 1995. Boi L., "The role of intuition and formal thinking in Kant, Husserl and in the modern Mathematics and Physics", Mathesis, 2005. Boole G., The laws of Thought, 1854. Bottazzini U., Tazzioli R., "Naturphilosophie and its role in Riemann's mathematics", Revue d'Histoire des Mathematiques, 1 (1995), 3-38. Connes A., Geometrie non-commutative, Paris, InterEditions, 1990. Dehaene S., The Number Sense, Oxford, Oxford University Press, 1998. (Review/article downloadable from http://www.dmi.ens.fr/users/longo.) Devaney, R. L., An Introduction to chaotic dynamical systems, Addison-Wesley, 1989. Edelman G., The matter of Mind, Naw York, Basic Books, 1992. Frege G., The Foundations of Arithmetic, 1884 (english transl. Evanston, 1980.) Galileo G., Dialoghi sopra i due Massimi Sistemi, U, 1632. Girard J.-Y., "Linear Logic", Theoretical Comp. ScL, 50 (1-102), 1987. Girard J.Y., Lafont Y., Taylor P., Proofs and Types, Cambridge U. Press, 1990. Godel K., Nagel E., Newman J., Girard J.-Y., Le theoreme de Godel, Paris, Seuil, 1989.

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Harrington L. et al. (eds) H. Friedman's Research on the Foundations of Mathematics, Amsterdam, North-Holland, 1985. Heath T.L., The Thirteen Books of Euclid's Elements, Cambridge, Cambridge University Press, 1908. Hertz J., Krogh A. and Palmer R., Introduction to the Theory of Neural Computation, 1991. Husserl, H., The origin of Geometry, part of Krisis, 1936. Jacob F., La logique du vivant, Paris, Gallimard, 1970. Lakoff, G. and Nunez, R., Where Mathematics Comes From: How the Embodied Mathematics Creates Mathematics, New York: Basic Books, 2000. Lambert D., Structure et Efficacite des interactions recentes entre Mathematiques et Physique, These de Doctorat, Louvain-la-Neuve, 1996. Lanfredini R., Husserl: la teoria dell'Intenzionalita, Roma, Laterza, 1994. Laskar J., "The chaotic behaviour of the solar system", Icarus, 88: 266-291, 1990. Longo G., "Mathematics and the Biological Phenomena" in International Symposium on Foundations in Mathematics and Biology: Problems, Prospects, Interactions, Pontifical Lateran University, Vatican City, November, 1998 (proceedings, Basti ed.). Longo G. "The mathematical continuum, from intuition to logic" in Naturalizing Phenomenology: issues in comtemporary Phenomenology and Cognitive Sciences, (J. Petitot et al, eds) Stanford U.P., 1999c. Longo G., "Mathematical Intelligence, Infinity and Machines: beyond the Godelitis" Invited paper, Journal of Consciousness Studies, special issue on Cognition, vol. 6, 11-12, 1999a. Longo G., "The Constructed Objectivity of Mathematics and the Cognitive Subject" Quantum Mechanics, Mathematics and Cognition (M. Mugur-Schachter ed.), Kluwer,2001. Longo G., "Memoire et Objectvite en Mathematiques", Le Reel en Mathematique, colloque de Cerisy, 1999 (a paraitre). Longo G., Petitot J., Teissier B. Geometrie et Cognition, research project, downloadable from http://www.dmi.ens.fr/users/longo, 1999. Philosophy of Computer Science, special issue of The Monist (G. Longo ed.), vol. 82, n.l,Jan. 1999. Ninio J., L'empreinte des sens, Seuil, Paris, 1991. Piazza M., Intorno ai numeri. Oggetti, proprieta', finzioni utili, Mailand, Mondadori, 2000. Prochiantz A., Les anatomies de la pensee, Odile Jacob, 1997. (Recensione/articolo downloadable da http://www.dmi.ens.fr/users/longo.) Riemann B., "On the hypothesis which lie at the basis of geometry", 1854 (english transl. by W. Clifford, Nature, 1873; trad, italiana di R. Pettoello, Boringhieri, 1999). Shanker S., Godel's Theorem on focus, Croom Helm, 1988. Tazzioli R., Riemann, Le Scienze, Aprile 2000. Thorn R., Stabilite structurelle et Morphogenese, Benjamin, Paris, 1972. Zellini P., Breve Storia dell'Infinito, Adelphi, 1980. Waismann F., Wittgenstein und der Wiener Kreis, Frankfurt a. M., Suhrkamp, 1967 (Waismann's notes: 1929-31). (English translation: Wittgenstein and the Vienna

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circle: conversations recorded by F. Waismann, Barnes & Noble Books, New York, 1979) Weyl H., "Uber die Definitionen der mathematischen Grundbegriffe", Math, naturwisss. Blatter, 7, 1910. WeylH., Das Kontinuum, 1918. Weyl H., Raum, Zeit, Materie, 1918 Weyl H., "Comments on Hilbert's second lecture on the foundations of Mathematics." [1927] in van Heijenoort J., From Frege to Goedel, 1967. Weyl H., Philosophy of Mathematics and Natural Science, Princeton, Princeton University Press, 1949. Weyl H., Symmetry, Princeton University Press, 1952. Weyl, H. "Axiomatic Versus Constructive Procedures in Mathematics." (Edited by T. Tonietti) The Mathematical Intelligence, Vol. 7, No. 4, 1985. Wigner E., "The unreasonable effectiveness of Mathematics in the Natural Sciences", Comm. Pure Applied Math., XIII, 1-14, 1960. Wittgenstein L., Philosophical Remarks, translated into English by G.E.M. Anscombe, Barnes & Noble, New York, 1968.

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PATHWAYS OF DEDUCTION A. CARBONE Equipe de Genomique Analytique INSERM U511, Universite Pierre et Marie Curie 91, blv de I'Hopital, 75013 Paris (France)

1

Deductions, foldings and the brain

Different models of various regions of the brain have been proposed and they stimulated the discussion on the way our mind works. The essential feature of most of these models is the hierarchical structure which is underlying the organization. What we "see" is nevertheless not necessarily the basic mechanism. Recent studies in computational complexity and proof theory reveal that hierarchical organizations, even though structurally appealing, are computationally inefficient. In fact, our brain seems to be "fast" in performing certain tasks (such as perceiving the presence of an animal in the landscape, or intuitively grasping a complicated mathematical idea) and extremely "slow" in performing others (as the construction of a mathematical proof). No hierarchical structure conceived by man displays similar features. In this paper we would like to show how the usual hierarchical approach to the construction of formal mathematical proofs (introduced with the work of Prege and established through the work in logic until this last decade) is inappropriate to reveal the intricate structures underlying proofs. There are two basic operations which one finds in every complex system: replication and folding (or cancellation). (Based on these two operations one can suggest a formal definition of complexity.) Replication and folding arise in many forms, going from a more abstract to a more concrete nature. On the abstract side, it has been and remains a challenge to study the effect of replication and folding on mathematical structures. For example, the folding in the combinatorial group theory corresponds to word cancellation and it is encoded in the 2-dimensional language of the van Kampen diagrams. On the concrete side, these two operations underly many biological systems, where the perfection and the symmetry of a mathematical structure is broken. Yet "pseudo-structures" seem to be preserved and one looks for mathematical tools to handle them. 383

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Replication and folding manifest themselves in vastly different situations where they are governed by different laws. In molecular biology for instance, one observes the replication in the production of thousands of copies of the same RNA strand, followed by the physical folding of those. This kind of folding constraints the dynamics of the cell and may interfere with the rate of production of RNA (via the network of protein-DNA interaction). DNA also folds, but for different purposes. For example, the 2 meters of human DNA must fit into the cell nucleus of 10 microns. This folding is organized (in chromosomes) along very specific structural patterns serving several biological functions. One more example involves proteins. They fold in the course of translation into a compact 3-dimensional conformation where the geometry of the specific active sites on the boundary determines their bio-chemical activity. Summing up: - folding allows a large object to fit into a small space, - the complexity of the functions performed by the object depend on the complexity of the folding, and - these two properties go along, since reducing the space necessarily produces complicated foldings. Possibly the evolution first folded "primordial DNA" in order to save space and then as a bonus came up with more complicated behavior. Folding and size reduction make possible to control the object but sometimes make the manipulation of the object a slow process, in particular if a local unfolding needs to be realized (as in the transcription of RNA from DNA). Biologists are searching for rules of folding and unfolding and it remains unclear, for example, in how many ways a (natural) protein can fold. Do foldings follow specific pathways? Amazingly, this pure biological (and superficial) discussion remains meaningful in the context of formal proofs. In proofs, and as we believe in most complex systems, replication and folding play a crucial role, and the analysis of the interaction of these operations in proofs is the main object of this paper. The biology will remain in the background, to contrast what happens in formal proofs. In proofs the folding and the replication are tide up in intricate ways (the unfolding induces replication and the folding induces identification) while biological systems tend to separate folding and replication in order to operate efficiently. As a result, the dynamics of proofs turns out to be very different (at least to a casual eye) from the dynamics of biological systems.

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Dynamics in proofs underlies a process of computation. We are after the structure of this dynamics, or if one prefers, of the computation behind proofs, where we shall see that short proofs need to contain cycles and that the elimination of these cycles might enlarge the proof in such a way that no human mind could possibly embrace the details. A proof, as understood in this article, represents a finite computation despite the presence of cycles which might create an illusion of infinite computations hidden behind. (There are proofs that might describe infinite processes but we shall not consider them here.) This will be clarified in Section 6 where we shall show that infinite iterations (intrinsic, for instance, to autonomous dynamical systems) are not present in proofs. The absence of infinite iterations becomes apparent if one introduces possibilities of parallel computation in a proof (see Section 6), otherwise one necessarily has cycles which bias our perspective towards a dynamical view. Why do we look at proof theory and at the notion of formal deduction instead of considering the notion of "truth"? Here is a basic fact which might give a clear picture of the reason to the non-logician: if we extend our mathematical theory with an inconsistent axiom, we might end-up with a new theory for which there is a k > 0 such that all proofs of length (i.e. the number of formal deductive steps) smaller than k are proofs of true statements. This means that even if we work in an inconsistent setting we can still deduce interesting results. Proof theory allows us to speak about what is going on in the stretch of computational time < k and permits an analysis of the structure of the computation when resources are bounded (e.g. with respect to time and space). I would like to acknowledge the numerous conversations I have with Misha Gromov in the subjects touched in this paper.

2

A formal language to describe proofs

In the thirties, Gentzen introduced a logical system (i.e. a finite set of rules for the manipulation of logical formulas) which, nowadays, is used at large in the study of formal proofs. Its success is due to the useful combinatorial properties concerning the formulas appearing in the proofs as well as the graph-theoretical features of the structure of the proofs. This logical system allows the manipulation of sequences of formulas which, for our purposes, will be simply chains of symbols that one can combine through controlled transformations of the sub-chains as described below. The alphabet out of which the chains are constructed is the logical language containing variables, constant, function and relation symbols, as well as

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logical connectives A (and), V (or) and logical quantifiers V (for all), 3 (there exists). Some extra symbols are used as separators, that is (,) (parenthesis) and , (comma). For each relation symbol R there is a unique complement R1 that represents the logical negation of R (often written ^R). The symbols A, V are complementary to V, 3, and the complement of a chain of symbols is the chain obtained by complementing all relation symbols and logical symbols in it. For instance the complement of the formula p Viq1 A r) is p1 A (qVr1), and the complement of (A1{c)AB{d))\/3x C(x) is {A{c)V Bx (d)) AVx Cx(x). For convenience, if A is a formula then we shall denote with the symbol AL its complement. (The reader might notice the analogy with the WatsonCrick complementarity in DNA for which the complement of the sequence ACGGGGTTTCC is TGCCCCAAAGG, where the letters A, C are complementary to T, G. One might entertain herself by looking at other examples of duality and parity in mathematics and physics.) An example of sequence of formulas is J4J-(c),Vi/ B(y,d) A C(d),BL(a), where Ax(c), Vj/ B(y,d) f\C{d) and BL{a) are formulas. An example of rule is T,A

A,B

T,A,AAB to the effect that given two sequences of formulas T, A and A, B one can construct a new sequence containing both collections of formulas T and A (which might be empty) and the formula A A B. (The logical meaning of the rule should not worry the reader.) Similar rules are defined for the other logical connectives and quantifiers. Together with those rules allowing the construction of new formulas lying in a sequence, there are two extra rules: A, A, A A,A contraction

A, A1

T,A

T,A cut

The contraction rule says that if two copies of a formula A lie in the same sequence, then they can be identified and only one remains. The cut rule says that if two sequences contain complementary formulas then the two formulas cancel out. In logic, the cut rule is a generalization of the wellknown rule of modus ponens, saying that if the lemma A has been derived, then the sequence of formulas A is derivable from the sequence A, A1. The occurrences A and AL in the two sequences considered in the cut rule, are called cut-formulas or lemmas.

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Figure 1: Triangles with inscribed circles. The set of rules which we described, together with sequences of the form (called axioms), define the so-called predicate logic. If quantifiers are not allowed to occur in the formulas, the logic is called propositional. A formal proof is a manipulation of sequences of formulas by means of the rules above, which starts from axioms, creates new sequences which can be combined with any of the previously derived sequences and any of the axioms, and which ends with a sequence called theorem. An example of formal proof is illustrated on the left hand side of Fig. 2. The use of the cut rule in formal proofs allows the construction of compact deductive arguments. We give two intuitive examples of the use of cuts in proofs. They illustrate the power of lemmas in deductions.

TJAJA1

Example 1 We take an "isoscele" triangle and we inscribe a circle in it as illustrated in Fig. 1. Repeatedly, we inscribe a smaller circle on the top of the previous one and so on. We shall obtain an infinite number of such circles, one on the top of the other. To compute the sum of the radiuses of such circles (see Fig. 1 on the left), we can calculate the radius of each circle and then sum up the values, but this sum is infinite. This approach produces an explicit computation in the sense that each radius is explicitly considered in the reasoning and contributes to the sum. A finite but implicit solution, is illustrated in the picture on the right of Fig. 1, where one can see that the sum of the radiuses coincides with half of the height of the triangle. (Infinity is present in this solution but suppressed by the implicitness.) The short cut we used to pass from an argument involving an infinite computation to a finite argument corresponds to the presence of lemmas in a formal proof. (For all apparent simplicity, no known automatic deduction system is able to come up with this short cut.) Example 2 To avoid the explicit calculation of 20 +19 +18 + . . . + 3 + 2 + 1 , at the age of seven, Gauss observed that if we add by columns the following

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additions 20 + 1 +

19 + 2 +

18 + 3 +

... + ... +

2 + 19 +

1 20

the result will be 10 times 21. By using the lemma n + ra = p—>(n — 1) + (m + 1) = p one can even avoid the addition by columns: one computes 20+1, and applies 20 times the lemma. The lemma allows to codify the explicit calculation and to deduce the solution faster. The reader might have a sense now of the shortening induced by certain mathematical arguments contrapposed to the length of the constructions that one would generate if lemmas were not allowed to be used. These are toy examples, but in the next section we discuss some concrete mathematical ones.

3

Unfolding

A precise statement concerning explicit and implicit constructions, that is constructions allowing or not the use of lemmas, was proved by Gentzen Theorem 3 (The Cut Elimination Theorem - Gentzen) If U is a formal proof (with cuts) of a statement S, then there is an effective way to transform II into a formal proof II' of S which does not make use of the cut rule. This holds for both propositional and predicate logic. The transformation is possible by means of local manipulations which unfold the proof II into the proof II', usually much larger than II. The price to pay for the elimination of cuts may be exponential for propositional logic, and multi-exponential (i.e. a tower of 2's) for predicate logic. From these values, it is clear that there are situations where one might find a small proof with cuts, even though one might never be able to look at the cut-free form because too huge. But why would we like to look at the cut-free form of a proof? Roughly speaking this form corresponds to the combinatorial version of the original proof. In a proof free of cuts, the construction of an "object" in the proof is done "piece by piece" and we might be interested to know how large this object is, or in how many steps we can build it, etc. (We shall comment on a concrete example of such construction in Section 5.) Since the complexity of the object and the length of the proof are related, one hopes to extract bounds

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from formalized proofs by analysing cut elimination. This is a direction of research proposed by Kreisel. In mathematical practice proofs with "large" cuts are regarded as less elementary than those with "smaller" ones. For instance, the Prime Number Theorem was originally proven with the cutting edge of the Riemann zeta function while the elementary proof was found much later and with a great effort. On the other hand, it took a long time to furnish a short non-elementary proof of the van der Waerden Theorem on arithmetic progressions. The beauty of the van der Waerden Theorem resides in the simplicity of its formulation given a finite set of points in a finitely colored (partitioned) plane, there is a parallel translation followed by a dilation which moves the set into a monochromatic position. Amazingly, there is still no logically simple proof of the result. In 1987, Girard showed that one can formally recover (by cut elimination) the elementary combinatorial argument used by van der Waerden from the dynamical system proof of Furstenberg and Weiss (where the key transcendental ingredient is the fact that the intersection of a decreasing sequence of non-empty compact sets is non-empty). This transformation consists of purely local combinatorial manipulations of formulas (together with some finitarisation of compactness). Thus the proof based on dynamical systems contains, in some codified form, all information needed for the combinatorial proof. Girard's theorem is a single result of this kind, and in most significant cases in number theory and algebraic geometry there is no elementary counterpart to analytical proofs. In no single case there is a formal derivation of one kind of proofs from the other.

4

A geometrical view of t h e unfolding

This section and the following one represent a condensed survey of the subject and the reader should not be surprised if they turn out to be hard to follow in detail. A formal proof is usually visualized as a "finite tree" of rules (growing downwards), whose root is the theorem, the leaves are axioms, and the internal nodes are intermediate sequences of formulas derived from one or two sequences (which label the antecedents of the node in the tree) through the formal rules. An example is illustrated in Fig. 2. There is another kind of graph, not necessarily a tree anymore, that can be associated to a formal proof. It represents the "flow of formulas" in the proof, and it is called the logical flow graph. This graph carries more information

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C ,

p^, p P

X

,CAP

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C

C^,C^, C /v P, P^/N C C\

CAP,

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C"", (Os P) v- (P"1" A C ) Figure 2: A formal proof and its associated tree of derivation. Each node of the tree corresponds to a sequence of formulas in the proof. There are three axioms and three leaves of the tree, three intermediate sequences and three internal nodes of the tree. The theorem corresponds to the root.

&~, c

cr, c

Figure 3: Logical paths between formula occurrences in formal proofs. The proof on the left displays a splitting point of the graph correspondent to the contraction of the formula C; the figure on the right displays a proof whose logical flow graph contains an oriented cycle. (In both pictures, only parts of the logical flow graphs are traced.)

than the tree derivation (for instance, it distinguishes proofs with cuts from those without cuts) and the process of cut elimination can be adequately expressed in terms of combinatorial operations over this graph. Two examples of logical flow graphs are illustrated in Fig. 3. They show the occurrences of the formula C logically linked within the proof. The left hand side diagram in Fig. 3, displays a proof with links for P as well as for C. By looking at the pictures one observes that contractions correspond to branching points in the graphs. We distinguish the branching points where a directed edge forks in two edges by calling them defocusing points, while vertices where two directed edges come together and are followed by a single edge are called focusing points (see Fig. 4).

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focusing

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defocusing

Figure 4: Branching points. When lying in a logical flow graph, they correspond to the use of contractions.

Figure 5: A defocusing point follows a focusing one. Between the two branching points there is a path linking them. This configuration corresponds to the presence in the proof of two contractions, and of a cut in between them.

A logical flow graph might be arbitrarily complicated. In fact, every (non-oriented) graph with degree of the vertices < 3 can be topologically embedded into the logical flow graph of some proof. The more complicated the graph is, the more logical information the graph carries. A good combinatorial way to understand this point is to count the number of different paths encoded in a logical flow graph. Let us consider for instance the graph in Fig. 5. It is easy to count 4 different oriented paths coded in it. Each one of the paths shares with the others, some part which might be constituted by a very long chain of nodes. It is the sharing of nodes that reduces the complexity of the representation. In Fig. 7 we can see how an exponential number of paths can be coded in a small graph. During the process of elimination of cuts, the logical flow graph of a proof undergoes significant topological changes and with this, its size blows up. The key idea is to transform a graph which contains paths starting from a focusing branching point and arriving to a defocusing one (see Fig 5) into a graph free of such configurations. The branching points where the graph splits or recombines are locally disrupted and this induces global effects in the topology of the graph (for instance cycles might be disrupted as illustrated in the second diagram on the right of Fig. 9).

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Figure 6: A subgraph H of the logical flow graph G is duplicated. The branching points below the nodes b, c and above the nodes e, / disappear with the duplication. Some new branching point is introduced: one below the node a and the other above the node d.

Figure 7: A graph where several directed paths share common parts (left) and its unfolding (right). The number of paths is exponential in the number of diamonds in the graph. Observe that the folding plays the role of the universal covering in the category of oriented graphs.

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Figure 8: Duplication of a path. This kind of duplication might give two different connections between the nodes a, b and c, d. (This is reminiscent to how the enzyme recombinase alters DNA strands by switching one of the connections above (on the right) into the other.)

d

c

d

c

d

Figure 9: The duplication of a path might induce the splitting of a cycle.

The combinatorial idea which underlies the cut elimination procedure is the following. Given the logical graph of the proof, the procedure chooses a subgraph of it and resolves some of the focusing or defocusing points by duplicating the subgraph, as illustrated in Fig. 6. By doing this, some of the branching points are eliminated but some new ones might be introduced. All in all, one moves around branching points and pushes them towards the boundary of the graph, until they are absorbed by the boundary and eliminated. Gentzen Cut Elimination Theorem ensures that this can be always done with a finite number of duplications. Among the effects that the operation of duplication might generate, one can see Fig. 8 and Fig. 9.

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Cyclic structures

Proofs without cuts are acyclic as well as proofs with cuts and no contractions. This means that cycles are built through the interaction between cuts and contractions. As a side effect of the rules for the manipulation of sequences of formulas, cycles in a proof organize following an interesting geometric pattern. In fact, any cyclic structure (i.e. any group of cycles nested together, as illustrated in Fig. 10) is linked to the rest of the graph in such a way that - there is a point in the graph, which is external to the cyclic structure, and from which it starts a path that arrives to any point lying in the cyclic structure, and - there is another point in the graph, which is again external to the cyclic structure, and to which it arrives a path that starts from any point lying in the cyclic structure. This means that for each cyclic structure, there is (at least) a path going in the structure and a path going out from it (as in Fig. 10). In other words, cyclic structures in proofs behave essentially as "open systems". The existence of ways in and of ways out in a cyclic structure can be proved by observing that if no way in or no way out was present in a cyclic structure, then the operation of duplication illustrated in Fig. 6 could not split (some) cycles and therefore reduce the logical flow graph of the proof to a cycle-free graph. Since we know that cycles can always be eliminated by applying the procedure of cut elimination, the above hypothesis leads to a contradiction. The elimination of cycles through duplication is done at great expense of the proof size. Hence, it is useful to ask, whether cycles are relevant for a proof to be short.

Figure 10: A cyclic structure lying in the logical flow graph of a proof, together with a path going in and a path going out the cyclic structure.

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When one considers a logic with quantifiers the answer is positive. This can be shown by proving in a few steps of deduction, that large integers exist. It turns out that not only one can prove in a very few steps that n, 2 n , 22" exist, but that also 222 and e(2, n) (i.e. a tower of n 2's) can be constructed with essentially n steps of formal deduction. To construct 22 and e(2, n) with such a small complexity of derivation, one needs cycles, and it can be shown that any cycle-free proof of size n can construct only objects of size much smaller than 22 . These short proofs codify in a small space the complete description of the object of large size. This codification exploits the high symmetry of the object and this latter is reflected in the argument of the proof by a logical description of a repeated substitution of logical "terms" (in essence, given a chain of symbols, some of these symbols are substituted with the chain itself, and this operation is allowed to repeat). In the logical flow graph, this repeated substitution corresponds to the presence of cycles. Are cycles necessary to short cuts in propositional logic? Surprisingly the answer is negative. In fact, there is a fine procedure for the manipulation of lemmas in proofs that allows to transform locally the structure of the proof until cycles are eliminated (but cut-formulas will still be present). The resulting proof is of polynomial size in the size of the original proof, and the degree of the polynomial is the number of cycles of the original proof. We do not know whether all true propositional formulas have polynomial size proof in the logical system described in Section 2. If this were the case then NP would be equal to co-NP. This problem is a brother to the famous P = NP question. We believe that an understanding of the folding and unfolding of propositional proofs might turn out to be crucial for complexity issues and possibly shade some light on P =? NP.

6

Cycles and spirals

Cycles cannot be eliminated silently when quantifiers are involved in the deduction (as seen in Section 5) since an explosive blow up in the size of the proof might occur. Also, cycles suggest a codification of an infinite process while we know, by the Theorem of Cut Elimination, that this codification represents only a finite number of iterations. Therefore the "intuition" of infinity is an illusion, and one wonders why cycles appear at all in our formal representation of proofs. There is a set of rules for the manipulation of chains of formulas that creates different geometric structures in proofs, where cycles do not appear but spirals replace them instead. A comparison of the two deductive systems shows that cycles are projections of spirals, as illustrated in Fig. 11.

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Figure 11: The projection of the spiral back down to the circle. Observe that the spiral is the universal covering of the circle and that the map represented in the figure is the canonical projection. Spirals properly represents the dynamics within a proof but the complexity of this representation is large: if the size of a proof with cycles equals n, then the size of the corresponding proof where spirals replace cycles is roughly 22 . Thus the proof with spirals is too big to be handled, and one is obliged to work in the "projected space", where spirals turn into cycles. One might speculate that computational complexity forces the brain to follow cyclic pathways of deduction, being aware, at the same time, that the "real" dimension of the space of reasoning is bigger than the "descriptive" dimension. The interweaving of these two modes of human reasoning underly the difficulty of our attempts to understand how our mind works. Taking this view, one starts doubting how often what we call "natural" corresponds to "reality".

7

Conclusions

We have seen how proof theory provides a procedure for unfolding a proof with cuts, e.g. turning the trascendental proof of the van der Waerden theorem into the elementary one. Yet there is no procedure for compactifying a proof by folding. Not every proof can be in principle folded. For this, a proof needs to contain many repetitive patterns, some kind of hidden symmetry. There are at least two situations where such folding is desirable: the first concerns real mathematical proofs such as the above van der Waerden theorem where one has an intuitive feeling for the symmetries but yet the actual process of folding is by no means automatic. The second concerns automatic proofs

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generated by computers where the symmetry, even if present, might be hard to detect. Moreover, once it is detected, the symmetry does not immediately yield the pattern of folding. Transforming an automated formal proof into one acceptable by a human mind remains an open problem. The difficulty resides in the fact t h a t local relations between formulas may be lost during the unfolding (this is due to duplication) and in order to fold one has to "guess" where to insert these relations. As a comparison, look at the following geometric picture: start with the universal (infinite) covering of a compact space and consider a bounded domain in this covering which projects onto the underlying space. The problem is how to reconstruct this map of the domain together with the underlying space where it goes, by using the information encoded in the (partial) local transformation of the domain which comes from the Deck transformation group of the covering. One should be aware that the combinatorics of proofs is by far more complicated than t h a t of the coverings.

8

Bibliographical guide

An introduction to the combinatorics and complexity of cut elimination can be found in [CS97a]. An exhaustive source (but more technical) is [Gir87b]. For a survey on propositional proof systems and their relations with complexity theory see [Pud96, Kra96]. Gentzen's original paper on the cut elimination theorem appeared in [Gen34] and, more recently, in [Sza69]. The bounds on the complexity of cut elimination are due to Statman, Orevkov and Tseitin [Sta78, Ore79, Tse68]. Some references for the work of Georg Kreisel on the importance of extracting bounds from proofs using cut elimination as the main tool are [Kre77, Kre81a, Kre81b]. The original paper of van der Waerden with the combinatorial proof of his theorem on arithmetic progressions is [VdW27]. The proof by Furstenberg and Weiss appeared in [FW78]. Girard's proof of the transformation (through cut elimination) between the combinatorial proof and the proof in dynamical systems is included in [Gir87b]. Other examples of formal proofs analysed through cut elimination (and Kreisel's counterexample interpretation) can be found in [Bel90, Kol93a, Kol93b, Kol94]. The notion of a logical flow graph is introduced in [Bus91] and an analysis of its properties appears in [Car97, Car99a]. The constructions of numbers as 2 2 , e(2, n) together with the cyclic structure underlying these constructions are studied in [Car98a], and the relation of these cyclic structures to group presentations is developped in [Car99b]; cyclic structures with their ways in

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and ways out are analysed in [Car99a, Car97a]; the evolution of the logical flow graph during cut elimination is studied in [Car97a, CS97b]; cycles and their spiral representations are the object of study in [Car98b]. The cost of the elimination of cycles for predicate logic is computed in [Car98a], and for propositional logic in [Car97b]. The notion of symmetry and quasi-symmetry in proofs and mathematical structures in general is the main theme of [CS97b, CS96a].

References [Bel90]

G. Bellin. Ramsey interpreted: a parametric version of Ramsey's Theorem. Contemporary Mathematics, 106:17-37. AMS Publications, 1990.

[Bus91]

S. Buss. The undecidability of fc-provability. Annals of Pure and Applied Logic, 53:72-102, 1991.

[Car97]

A. Carbone. Interpolants, cut elimination and flow graphs for the propositional calculus. Annals of Pure and Applied Logic, 83:249299, 1997.

[Car97a] A. Carbone. Duplication of directed graphs and exponential blow up of proofs. Annals of Pure and Applied Logic, 100:1-76, 1999. [Car97b] A. Carbone. The cost of a cycle is a square. Symbolic Logic, 2000. To appear.

The Journal of

[Car98a] A. Carbone. Cycling in proofs and feasibility. Transactions of the American Mathematical Society, 5:2049-2075, 2000. [Car98b] A. Carbone. Turning cycles into spirals. Annals of Pure and Applied Logic, 96:57-73, 1999. [Car99a] A. Carbone. Streams and strings of formal proofs. Theoretical Computer Science, 2000. To appear. [Car99b] A. Carbone. Asymptotic cyclic expansion and bridge groups of formal proofs. Journal of Algebra, 2000. To appear. [CS96a]

A. Carbone and S. Semmes. Looking from the inside and the outside. Synthese, 2000. To appear.

[CS97a]

A. Carbone and S. Semmes. Making proofs without modus ponens: An introduction to the combinatorics and complexity of

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cut elimination. Bulletin of the American Mathematical Society, 34:131-159, 1997. [CS97b]

A. Carbone and S. Semmes. A Graphic Apology for Symmetry and Implicitness. Mathematical Monographs, Oxford University Press, 2000.

[FW78]

H. Furstenberg and B. Weiss. Topological dynamics and combinatorial number theory. Journal d'Analyse Mathematique, 34:61-85, 1978.

[Gen34]

G. Gentzen. Untersuchungen iiber das logische Schliefien I—II. Math. Z., 39:176-210, 405-431, 1934.

[Gir87b]

J-Y. Girard. Proof Theory and Logical Complexity, volume 1 of Studies in Proof Theory, Monographs - Bibliopolis, Napoli, Italy, 1987.

[Kol93a] U. Kohlenbach. Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallee Poussin's proof for Chebycheff approximation. Annals of Pure and Applied Logic, 64:27-94, 1993. [Kol93b] U. Kohlenbach. New effective moduli of uniqueness and uniform a priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory. Numer. Fund. Anal, and Optimiz., 14(5&6):581-606, 1993. [Kol94]

U. Kohlenbach. Analyzing proofs in analysis. Proceedings of the Logic Colloquium 93 - Keele, 1994.

[Kra96]

J. Kraji'cek. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1996.

[Kre77]

G. Kreisel. From foundations to science: justifying and unwinding proofs. Symposium: Set theory. Foundations of mathematics, dans Recueil des travaux de l'lnstitut Mathematique, Nouvelle serie (Belgrade) 2(10):63-72, 1977.

[Kre81b] G. Kreisel. Extraction of bounds: interpreting some tricks on the trade. In P. Suppes, editor, University-level computer assisted instruction at Stanford: 1968-1980, number tome 2 (10) in Institute for Mathematical Studies in the Social Sciences, 1981.

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[Kre81a] G. Kreisel. Neglected possibilities of processing assertions and proofs mechanically: choice of problems and data. In P. Suppes, editor, University-level computer assisted instruction at Stanford: 1968-1980, number tome 2 (10) in Institute for Mathematical Studies in the Social Sciences, 1981. [Ore79]

V.P. Orevkov. Lower bounds for increasing complexity of derivations after cut elimination. Journal of Soviet Mathematics, 20(4), 1982.

[Pud96]

P. Pudlak. The lengths of proofs. In S. Buss, editor, Handbook of Proof Theory. North Holland, 1996.

[Sta78]

R. Statman. Bounds for proof-search and speed-up in predicate calculus. Ann. Math. Logic, 15:225-287, 1978.

[Sza69]

M.E. Szabo (Editor). The Collected Papers of Gerhard Gentzen. North Holland, Amsterdam, 1969.

[Tse68]

G.S. Tseitin. Complexity of a derivation in the propositional calculus. Zap. Nauchn. Sem. Leningrad Otd. Mat. Inst. Akad. Nauk SSSR, 8:234-259, 1968.

[VdW27] B. L. van der Waerden. Beweis einer baudetschen vermutung. Nieuw Archiv Wiskunde, 212-216, 1927.

PHENOMENOLOGIE ET THEORIE DES CATEGORIES FREDERIC PATRAS Laboratoire J. A. Dieudonne CNRS-UMR6621 Universite de Nice - Sophia Antipolis Pare Valrose, 06108 NICE cedex 2 patras@math. unice.fr

Abstract: The mathematical community has become aware that 20-th century epistemologies such as Bourbaki's structuralism are not satisfactory. In particular, it appears that mathematicians should take signification problems, which have been long neglected, systematically into account. This is obvious for what concerns educational questions, but the recent writings of such an eminent and influential mathematician as Alexandre Grothendieck emphasize that dealing with significations, that is not only with formal or computational tools, is necessary even for research and for the internal progress of science. Husserl's phenomenology is, among other things, a theory of knowledge that deals with such problems. We show in the present article that modern mathematical theories, such as category theory, combine remarkably with Husserl's philosophy to create simultaneously a new image of mathematics and new approaches to classical philosophical concepts and distinctions such as transcendental subjectivity vs. objectivity.

1. Les apories du structuralisme L'art precede les savoirs theoriques : l'edification d'une grammaire serait inconcevable sans une maitrise prealable de la langue et l'idee d'une geometrie n'aurait guere de sens en dehors d'un contexte de vecus spatio-temporels, sauf a jouer assez artificiellement avec les mots en les degageant du terreau de significations et de pratiques sur lequel ils se sont formes. En regie generate, l'essence meme de la temporalite veut qu'il ne soit jamais possible de nommer et, a fortiori, de decrire que des contenus conceptuels deja constitues. Bien entendu les concepts, une fois acquis et sedimentes dans des pratiques langagieres ou scientifiques, s'affranchissent peu a peu des determinations liees a leur mode d'apparition historique. Aussi l'idee d'une grammaire universelle, si elle peut etre contestable pour telle ou telle raison tenant aux structures de la langue et de la pensee, est-elle cependant tout a fait legitime a titre de projet, de meme que la notion d'espace ou de geometrie peut etre encore employee dans des situations limites ou seules de vagues intuitions justifient l'emploi des termes sur le fondement de criteres formels et sans veritables contreparties intuitives1.

1

II en va ainsi pour les espaces de la topologie et la geometrie algebrique modemes. 401

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Dans l'univers mathematique, cette transition des contenus de pensee a des structures formelles susceptibles d'etre depouillees de toute emprise intuitive est codifiee par la methode axiomatique. L'idee de la methode est tout sauf neuve : le projet d'une organisation analytique du discours scientifique a partir d'un ensemble de propositions admises comme vraies (les postulats ou axiomes de la theorie) et de regies de deduction est tout d'origine grecque et, plus precisement, aristotelicienne. L'originalite de son usage, a la fin du dix-neuvieme siecle et surtout au vingtieme, tient, outre a son caractere systematique, a une certaine distanciation quant aux axiomes. Ceux-ci sont susceptibles de varier, donnant ainsi naissance a autant de theories qu'il y a de choix possibles de systemes d'axiomes fondamentaux. Parmi les systemes d'axiomes, il faut surtout retenir les systemes dits ouverts. Leur role a ete decisif pour la philosophie et la pratique mathematique a partir des annees 1920-30, c'est a dire des travaux de l'ecole d'Emmy Noether et la naissance d'un corpus d'"algebre moderne" avec des ouvrages comme le celebre traite de van der Waerden2. Les axiomatiques ouvertes codifient des concepts generaux comme ceux d'espace vectoriel, de groupe ou de topologie, et permettent d'unifier au sein d'un meme formalisme des objets ou des phenomenes d'essences apparemment distinctes. Au cours du vingtieme siecle, la terminologie des structures3 mathematiques, plus familiere sans doute au grand public du fait de son invocation par les differents courants structuralistes des sciences humaines, s'est graduellement substitute a celle des axiomatiques ouvertes, entre autres sous 1'influence du groupe Bourbaki4 qui a fait des structures le cceur de sa conception des mathematiques5. Le concept de structure (qui n'est d'ailleurs pas defini de maniere univoque dans la litterature et a une fonction tres souvent programmatique) coincide, dans son usage courant, avec celui d'axiomatique ouverte, et la substitution d'une terminologie a l'autre aurait peu d'importance si elle n'etait l'indice d'une mutation de sens et de valeurs. Dans la pensee hilbertienne qui, des Grundlagen der Geometrie de 1899 jusqu'aux recherches formalistes6, est au coeur de la problematique axiomatique dans toute sa dimension philosophique, la pensee axiomatique a une vocation universelle. En un sens, elle est l'heritiere de la mathesis universalis leibnizienne et d'un grand projet de legislation des sciences de la nature. C'est a ce projet que les mathematiciens de la seconde moitie du vingtieme siecle ont longtemps tourne le dos, entraines par un mouvement de constitution internaliste de leur discipline. Historiquement, 1'abandon du

2

B. van der Waerden. Moderne Algebra. 2 vols. Berlin, Springer, 1930. P. Carter. Structures. Dictionnaire d'histoire et philosophie des sciences. Ed. D. Lecourt. Paris, P.U.F., 1999. 4 P. Cartier. Bourbaki. Dictionnaire d'histoire et philosophie des sciences. Ed. D. Lecourt. Paris, P.U.F., 1999. 5 N. Bourbaki. VArchitecture des mathematiques. in Les grands Courants de la pensee mathematique. Paris, Hermann, 1948. 6 Nous renvoyons a la biographie de Hilbert: C. Reid. Hilbert. Berlin, Springer, 1970. 3

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programme axiomatique general et le desinterdt pour les grandes questions epistemologiques attenantes coincident a peu pres avec l'essor du courant "structuraliste". En France, une responsabilite importante dans cette disaffection resolue d'une part importante de la communaute des mathematiciens pour «le reel et les grandes categories de la pensee »7 incombe a Bourbaki. Les limites de la strategic de l'autruche philosophique, revendiquee explicitement dans les quelques textes programmatiques du groupe et dans presque toute l'ceuvre de son porte-parole le plus influent, Jean Dieudonne (faire abstraction des problemes logiques et philosophiques de signification ainsi que des enjeux ontologiques du discours mathematique) sont aujourd'hui evidentes. Deux exemples suffiront a en convaincre. La physique mathematique, largement ignoree par Bourbaki, est indubitablement l'un des moteurs les plus actifs du developpement du corpus mathematique et contribue de maniere decisive au renouvellement de ses problematiques. Des travaux recents, comme ceux d'Alain Connes8, montrent a quel point la question de la structuration du sens des theories physiques, celle de la renormalisation par exemple, est essentielle. Elle deborde d'un cadre strictement mathematique puisqu'il y va vraiment, au travers d'un formalisme algebrique, de la structure de l'univers physique. La pedagogie mathematique est, de son cote, a la croisee des chemins. Le structuralisme, s'il propose d'organiser le corpus, ne dit rien sur les mecanismes d'apprentissage et de creation des savoirs. La science dont il rend compte est une science figee. Les idees de Claude Chevalley, membre du groupe Bourbaki, decrites par sa fille Catherine, malgre qu'elles ne pretendent pas refleter une vision partagee par les autres membres, decrivent bien le type de mathematiques auxquelles Bourbaki visait a aboutir : « Le manque de rigueur donnait a mon pere l'impression d'une demonstration ou Ton marcherait dans la boue, ou Ton souleverait des sortes d'immondices pour avancer. Quand on les avait ecartes, on pouvait acceder a l'objet mathematique, une sorte de corps cristallise dont l'essentiel est la structure. Quand cette structure etait construite, il disait etre interesse par cet objet pour le regarder, l'admirer, presque le faire tourner, mais certainement pas pour le transformer... Si Ton observe la facon dont mon pere travaillait, il semble que c'etait d'avantage cela qui comptait, cette production d'un objet qui, alors, devenait inerte, mort en somme. »9 En d'autres termes, 1'edification de structures ou plus generalement d'axiomatiques est un processus tardif, qui ne vient qu'apres le travail createur celui-la meme dont la pedagogie moderne voudrait apprendre a rendre compte, quitte a deconstruire les belles architectures de la science constituee. 7

N. Bourbaki. Op. Cit. A. Connes et D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199, 203-242 (1998). 9 Extrait de Nicolas Bourbaki, faits et legendes. M. Chouchan. Editions du Choix, Argenteuil, 1995.

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Les alternatives a la cecite epistemologique sont nombreuses, comme le sont les directions de recherche : mathematisation et modelisation du reel physique et social, genealogie voire archeologie au sens foucaldien des savoirs scientifiques, etudes morphologiques dans une perspective 'a la Thorn'... On s'interessera ici aux problemes de fondement, au sens le plus radical du mot, a savoir en deca des questions posees par la logique mathematique ou la m£tamathematique. Plus precisement, il s'agira de dire toute l'actualite mathematique de la pensee husserlienne et des grandes distinctions qu'elle a introduites au sein de la theorie de la connaissance : ontologie formelle/ ontologie materielle, logique formelle/ logique transcendantale, noese/ noeme. II devrait apparaitre que chacune d'elles continue de faire sens pour l'epistemologie contemporaine. 2. Deconstruire le terrain des evidences naturelles L'axiomatique, on l'a dit, est un moment tardif de la pensee, mais c'est egalement le cas de tous les grands edifices de la connaissance. L'evidence de la geometrie euclidienne, de 1'analyse moderne et de toutes ces theories dans lesquelles les mathematiciens se meuvent avec autant sinon plus d'aisance que dans le "monde de la vie" ne doivent pas dissimuler qu'il s'agit la de savoirs complexes. Le terrain d'evidences originaires sur lequel ils sont d'abord parus nous est masque par l'oubli des apprentissages effectues et par la puissance du langage institue qui fait comme si des concepts comme ceux de droite, de forme geometrique ou de nombre allaient d'emblee de soi. Ces categorisations deja effectuees dans le langage ou les theories scientifiques sont evidemment indispensables : sans elles, pas de science a proprement parler. Pour autant, leur pouvoir est egalement coercitif: si 1' "espace" est celui de la mathematisation galileenne de la nature, comment accepter ensuite les espaces courbes de la relativite generale et des geometries riemanniennes ? Dire les mecanismes primaires de constitution du savoir n'est pas aller contre les interets bien compris d'une science en progres et oublieuse de ses racines ; c'est egalement menager un espace de liberte ou il sera possible, lorsque necessaire, de transvaluer les axiomatiques depassees. On sait par exemple 1'importance que Grothendieck10 attachait a la quete de noyaux de sens, quitte a devoir chambouler les edifices trop figes de la geometrie, de la topologie ou de l'arithmetique ! Pour lui, la science semblait pouvoir progresser vers une adequation parfaite de ses concepts et de ses techniques a nos intuitions les plus fondamentales. Husserl s'attache a un projet analogue, mais ses outils theoriques sont herites de la grande tradition philosophique : Descartes, Kant, Bolzano..., et ses ambitions sont resolument epistemologiques (logiques, au sens classique du mot). Les textes husserliens qui traitent du probleme de l'origine des savoirs scientifiques sont nombreux. Notre ambition ne sera pas ici d'en expliciter le contenu mais, sans 10 A. Grothendieck. Recoltes et semailles. Reflexions et temoignage sur un passe de mathematicien. Manuscrit. Grothendieck est l'un des quelques mathematiciens du vingtieme siecle dont les idees ont bouleverse l'edifice mathematique.

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veritable souci de fidelite, de penser a la maniere husserlienne et de degager d'une oeuvre complexe et fascinante quelques materiaux particulierement significatifs pour le developpement contemporain des mathematiques. L'Origine de la geometrie11 est un texte commode pour aborder la pensee husserlienne. Manuscrit bref date de 1936, il explore, a l'image de La Crise des sciences europeennes et la phenomenologie transcendantale12, le processus de constitution des objets de la geometrie a partir d'un processus d'idealisation dont les sources vives sont les materiaux intuitifs concrets du monde ambiant. Texte peu technique, il permet de degager la necessite et la saveur epistemologique de l'approche phenomenologique. Suivons-le. Un ecueil important dans la recherche des origines du sens mathematique tient au statut de l'historicite des concepts. C'est sans doute l'un des points les plus delicats dans la deconstruction methodique des evidences logiques qui forment le terreau d'une pratique normale (mais pas necessairement satisfaisante) de la science. II n'est par exemple pas indifferent que Bourbaki ou Dieudonne aient attache autant d'importance a l'histoire des sciences : une histoire figee, heroi'que et factuelle fait le jeu d'un discours teleologique et normatif. Aussi faut-il se garder de 1'historicisme, en sciences comme ailleurs. Foucault est la pour nous le rappeler, mais son archeologie du savoir hesite devant les mathematiques : «A ne reconnattre dans la science que le cumul lineaire des verites ou l'orthogenese de la raison, a ne pas reconnattre en elle une pratique discursive qui a ses niveaux, ses seuils, ses ruptures diverses, on ne peut decrire qu'un seul partage historique dont on reconduit sans cesse le modele tout au long du temps, et pour n'importe quelle forme de savoir: le partage entre ce qui n'est pas encore scientifique et ce qui Test definitivement... II n'y a sans doute qu'une science pour laquelle on ne puisse distinguer ces differents seuils ni decrire entre eux un pareil ensemble de decalages : les mathematiques, seule pratique discursive qui ait franchi d'un coup le seuil de la positivite, le seuil de l'epistemologisation, celui de la scientificite et celui de la formalisation. »13 Le renvoi par Foucault du langage mathematique a l'image qu'en a donne le structuralisme, a savoir l'idee d'une cloture formelle et d'une certaine autonomie epistemologique est significatif. II temoigne des difficultes qu'il y a a interroger les dimensions du discours mathematique qui echappent a cette cloture et forment pourtant le cceur du mouvement vivant de constitution du savoir. Les intuitions, les programmes de recherche, les deficits de sens informent et donnent corps a la pratique mathematique, mais ces composantes du discours echappent pour l'essentiel a la diffusion publique et ecrite. " E. Husserl. L'Origine de la geometrie. Trad. J. Derrida. Paris, P.U.F., 1962. Cite VOrigine dans la suite. 1 E. Husserl. La Crise des sciences europeennes et la phenomenologie transcendantale. Trad. G. Granel. Paris, Gallimard, 1976. Cite La Crise dans la suite. 13 M. Foucault. L'Archeologie du savoir. Paris, Gallimard, 1969.

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Une autre composante de 1'attitude de Foucault, tout aussi influente dans ses jugements sur les mathematiques, est une mefiance affirmee a I'egard du transcendantal, qui se retrouve d'ailleurs dans les diverses epistemologies affranchies de toute reference au kantisme. De meme qu'il faut rejeter l'illusion d'une cloture epistemologique du discours mathematique, il faut aller contre de telles prises de position, puisque vouloir interroger les origines du savoir sans se cantonner a la surface des phenomenes ou vouloir questionner les objets mathematiques dans leur surgissement passe par la prise en compte d'un moment transcendantal. Refus de toute autonomie du discours scientifique et transcendantalisme sont deux aspects decisifs de la phenomenologie husserlienne. Le type d'historicite revendique pour les mathematiques par VOrigine est, a sa maniere, tout aussi archeologique que chez Foucault. Mais l'archeologie dont il s'agit est une archeologie du sens. La sedimentation de sens qui s'est accomplie avec le developpement des methodes formelles, et ce des la naissance de la geometrie grecque, doit etre etudiee comme telle et chaque strate n'indique pas seulement telle ou telle etape d'un processus historique, mais bien un moment eidetique incontournable et irreductible. Faire revivre chacun de ces moments, non pas seulement comme percee heroi'que accomplie par un Descartes, un Gauss ou un Galois, mais egalement dans sa plenitude de sens, aussi problematique qu'elle puisse etre : telle est l'ambition d'une histoire phenomenologique. « Nous pouvons dire aussi: l'histoire n'est d'entree de jeu rien d'autre que le mouvement vivant de la solidarite et de l'implication mutuelle de la formation du sens et de la sedimentation du sens originaires. »'4 3. Ontologies formelles et materielles Le projet husserlien ne releve pas seulement de l'analyse des significations ou d'une epistemologie ancree dans la metaphysique occidentale. Ses implications sont concretes et concerneraient aussi bien la pedagogie ou l'ecriture mathematique que l'histoire des sciences. La phenomenologie ne se reduit pas a une theorie de la science et se manifeste par des methodes d'investigation concretes des contenus conceptuels. Hermann Weyl ne s'y est pas trompe, qui a reconnu 1'influence exercee par Husserl jusque sur son style mathematique et, plus generalement, son style de pensee («ce fut Husserl qui, me degageant du positivisme, m'ouvrit l'esprit a une conception plus libre du monde »)15. Pour autant, la realisation des ambitions husserliennes -reconcilier la science et le "monde de la vie", et en particulier « effectuer une reactivation integrate et authentique jusqu'a la pleine originarite, par une recursion vers les archievidences, dans les grands edifices de la geometrie et des sciences dites 14 Origine, p. 203 [380] (la reference a l'edition originate est donnee systematiquement entre crochets, apres la reference a la pagination de la traduction francaise). 15 Cite dans Hermann Weyl (1885-1955), C. Chevalley et A. Weil. L'Enseignement mathematique. Ill, fasc. 3, 1957.

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"deductives" »- passe par l'edification d'une theorie rigoureuse qui permette de concilier vecus intuitifs et pures formes logiques. Pour cela, il faut apprendre a ne pas eradiquer du discours scientifique les residus intuitifs et leur part d'incertitude. C'est aussi la voie a suivre pour edifier sur des fondements universellement valables le projet d'une "axiomatisation" des sciences de la vie, de l'homme et de la nature. Pour mieux en comprendre les enjeux, rappelons ici les conclusions du structuralisme sous sa forme bourbakiste : « en fin de compte, cette intime fusion dont on nous faisait admirer Fharmonieuse necessite [celle des sciences experimentales et des mathematiques] n'apparait plus que comme un contact fortuit de deux disciplines »16 ! L'enormite du propos et sa desuetude ne doivent pas dissimuler une difficulte authentique. Affirmer ainsi que la mafhematisation de la physique est un phenomene "fortuit" est evidemment un tour de passe-passe tout aussi douteux qu'inacceptable. Pour autant, l'idee d'une rigueur apriorique des sciences experimentales est difficile a soutenir. Aussi la phenomenologie, sans renoncer a tenir un discours rationnel sur les sciences experimentales et leurs fondements (y compris mathematiques), ne remet-elle pas en cause leur inaptitude acreerde l'eidos : «Une science eidetique se refuse par principe a incorporer les resultats theoriques des sciences empiriques. Les positions de realite qui s'introduisent dans les constatations immediates de ces sciences se transmettent de proche en proche a toutes les constatations mediates. Des faits ne peuvent resulter que des faits. » Pour autant, « si toute science eidetique est par principe independante de toute science de fait, c'est l'inverse par contre qui est vrai pour les sciences de fait. II n'en est aucune qui, ayant atteint son plein developpement de science, puisse rester pure de toute connaissance eidetique et done independante des sciences eidetiques formelles ou materielles [...] C'est ainsi qu'elle entre en rapport avec le groupe de disciplines qui constituent la « mathesis universalis » formelle'7 ». Pour reprendre un exemple deja evoque, qui fera office de pierre de touche, les methodes mathematiques connues sous le nom de methodes de renormalisation en theorie quantique des champs18 se situent bien, comme toute la physique mathematique, dans ce domaine eidetique intermediate entre le formel et le materiel. Les objets de la theorie, etres purement mathematiques susceptibles d'un traitement symbolique-formel, ont tout de meme une teneur materielle incontoumable des lors qu'ils ont vocation a decrire I'univers phenomenal. Que la theorie de la renormalisation ne soit pas encore vraiment bien comprise a l'heure actuelle ou le fait que ses objets mathematiques, comme les diagrammes de Feynman, correspondent ou non a une "realite" quelconque importe peu ici : il 16

N. Bourbaki, Op. Cit. E. Husserl. Idees directrices pour une phenomenologie. trad. P. Ricoeur. Paris, Gallimard, 1950. p. 33 [18], cite Ideen dans la suite. 18 Voir par exemple C. Itzykson et J.-B. Zuber. Quantum Field theory. Singapour, McGraw-Hill, 1980. 17

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suffit pour nous qu'ils fassent reference a un mode precis d'idealisation des phenomenes pour ne plus relever seulement de la mathematique, mais egalement de cette science eidetique materielle dont Husserl fait etat dans les Ideen'9. Expliquons-nous plus en detail sur un autre exemple. La physique newtonienne a ete mise en defaut en tant que systeme pr^dictif par la physique einsteinienne et la physique quantique. Pour autant, ce qui se joue au niveau des ontologies materielles et de leurs fondements formels (la theorie mathematique correspondante) dans la transition d'une physique a l'autre n'est pas du seul ressort d'une logique naive (celle de la validation et de la refutation), mais a un moment eidetique irreductible vis a vis duquel une notion comme celle de refutation n'est que tres partiellement pertinente. Aussi convient-il de penser le depassement quantique ou relativiste de la physique newtonienne egalement en termes d'a priori synthetiques materiels (les concepts classiques d'espace, de position, d'energie..., et leur transvaluations au sein des nouvelles theories physiques, quelque problematique que soit par ailleurs leur aprioricite eventuelle). A ce stade, et compte tenu de 1'extreme ambigui'te du concept de science eidetique materielle, l'epistemologue pourrait etre tente de subordonner les ontologies materielles a l'ontologie formelle comme, dans l'epistemologie traditionnelle depuis Aristote, les differentes sciences se subordonnent a la logique generale. Cela reviendrait de fait a restaurer la position dominante de la mathesis universalis dans le pantheon des sciences et a faire par exemple de la composante materielle des concepts de la physique mathematique un simple supplement de sens au regard de leur structure formelle. A cette subordination, qui est aveugle au primat du phenomenal dans la constitution des idealites physiques, geometriques, etc., il conviendra plutot de substituer une juxtaposition des relations de dependance logique et des relations de dependance genetique, qui vont du formel au materiel et temoignent de l'ancrage de toute pensee dans des vecus (spatiotemporels, de significations,...). Concretement, il y va des modes de constitution des objets mathematiques a partir des eidetiques materielles et des eidos correspondants : espace, temps, denombrement, collectivisation... C'est cette dimension-la du savoir qu'il nous faut reapprendre aujourd'hui a legitimer. 4. La genese des objets mathematiques A 1'etendue de la tache, on devine que les voies d'enquete qui s'ouvrent aujourd'hui encore a la phenomenologie sont nombreuses. D'autant que le developpement recent des mathematiques et de la logique mathematique avec, pour cette derniere, 1'emergence de phenomenes lies a des residus incontournables

19 Ce projet d'une science eidetique materielle est intimement lie a la possibility d'un a priori synthetique materiel -extremement problematique. Nous renvoyons pour une etude detaillee du projet husserlien et de ses racines historiques et philosophiques a J. Benoist, Z/a priori conceptuel. Bolzano, Husserl, Schlick. Paris, Vrin, 1999.

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de sens dans les preuves des theoremes fondamentaux , fournissent de nouveaux outils d'analyse. La suite de cet article privilegiera un probleme precis, qui aura fonction d' illustration pour les ressorts et les potentialites de la methode phenomenologique husserlienne dans la construction, aux cotes d'approches plus internalistes21, d'une epistemologie mathematique post-structuraliste. II s'agira de la logique du mode de constitution des idealites mathematiques dans la conscience, logique etant entendu ici au sens d'un ensemble de regies apodictiques structurant un champ d'actes intentionnels et conceptuels. Nous chercherons surtout a montrer que certains phenomenes, par exemple la formation de certaines axiomatiques ouvertes (i.e. de structures mathematiques) peuvent etre reconduits a des phenomenes frequents en theorie des categories (passage a la limite, existence d'objets initiaux, problemes universels, adjoints...). II ne s'agit pas par la de pretendre que la theorie des categories est un nee plus ultra epistemologique, mais c'est sans doute a ce jour la theorie mathematique la plus a meme de permettre la description rigoureuse de certains phenomenes eidetiques, des lors qu'elle codifie et organise au sein d'un symbolisme ces deux notions fondamentales pour la pensee mathematique que sont celles d'objet et de relation . Pour en revenir a la phenomenologie, elle repose, au moins pour le type de questions considerees ici, sur l'idee que les mecanismes de creation eidetiques (en termes plus classiques, grossierement equivalents, que les actes de synthese) sont organises et relevent d'une logique specifique. Des notions comme celles d'horizon ou de variation eidetique permettent de decrire cette organisation. Un des moments cruciaux de cette description est l'edification prealable d'un modele a un niveau ontologiquement pur, e'est-a-dire independamment de toute determination psychologique ou empirique, de la distinction operee classiquement entre la conscience et son objet. Revenons sur l'exemple des diagrammes de Feynman : il s'agit de diagrammes utilises par les physiciens pour etudier formellement les contradictions auxquelles conduisent certains calculs d'integrales en theorie quantique des champs. Les mathematiciens, qui deplorent depuis longtemps les lacunes mathematiques du calcul de Feynman, le concoivent quant a eux selon les normes ontologiques de leur discipline : de maniere formelle. L'epistemologue hesitera sans doute pour sa part sur le statut a donner a ces objets et calculs dont 1'introduction, justifiee par son efficacite, reste assez problematique pour ce qui est de l'elucidation mathematique des techniques de renormalisation. II s'agit done d'objets multiformes, qui relevent a la fois d'un type d'ontologie materielle ", de par leur signification physique, et de l'ontologie formelle (la

G. Longo. Colloque de Cerisy "Mathematiques et psychanalyse", Septembre 1999. C'est a dire relevant du seul domaine mathematique ou d'interactions disciplinaires. 22 Voir aussi notre article Categories et foncteurs. Dictionnaire d'histoire et philosophie des sciences. Ed. D. Lecourt. Paris, P.U.F., 1999. 23 Celle des particules elementaires. 21

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mathesis universalis) de par leur existence mathematique (par exemple lorsqu'on les considere comme des graphes independamment de toute autre interpretation). Si Ton se place maintenant du point de vue des structures de la conscience pure, le premier facteur a prendre en compte dans 1'etude d'objets comme les diagrammes de Feynman est leur existence comme objets "dans la conscience". D'une certaine facon la conscience preexiste toujours a son objet et une theorie de la science doit rendre compte de cette anteriorite. En fait, en regie generale, « ce qui informe la matiere pour en faire un vecu intentionnel, ce qui introduit l'element specifique de l'intentionnalite, c'est cela meme qui donne a l'expression de conscience son sens specifique et fait que la conscience precisement indique ipso facto quelque chose dont elle est la conscience24 ». Le terme husserlien de moment noetique ou de noese designe ce moment et cette structuration de la conscience. Sa prise en compte dans les debats sur le statut ontologique d'objets comme les diagrammes de Feynman, en restaurant la dimension intentionnelle du savoir, devrait permettre de renouveler des problematiques par trop exclusivement centrees sur la coherence mathematique du formalisme. Pour ce qui est du savoir mathematique lui-meme, les moments noetiques purs, a supposer qu'ils puissent etre isoles dans l'analyse, sont les moments decisifs. Les moments de la conscience pure : regard porte sur l'objet, variation ideale de ses caracteristiques (comme lorsqu'en topologie on deforme continument une surface), intuition d'une analogie de structure (comme l'analogie celebre entre corps de fonctions et corps de nombres); tous ces moments, en tant qu'ils sont susceptibles de descriptions "rationnelles" doivent etre abordes par l'epistemologue au travers d'analyses noetiques25. Plus radicalement, c'est l'analyse noetique qui permet de comprendre les regies de formation d'idealites : « Les problemes les plus vastes de tous sont les problemes fonctionnels portant sur la « constitution des objectivites de conscience ». lis concernent la facon dont les noeses, par exemple dans le cas de la nature, en animant la matiere et en se combinant en systemes continus et en syntheses unificatrices du divers, instituent la conscience de quelque chose »

u

Ideen,p. 174 [291] Husserl aborde le theme des variations de structure des espaces topologiques dans les Recherches logiques (trad. H. Elie, A. Kelkel, R. Scherer, Paris, Presses Universitaires de France, 1969): « C'est en 6tudiant le mode selon lequel on passe d'un genre de multiplicity spatiale a un autre par variation du degre de courbure que le philosophe ayant appris les rudiments de la theorie de RiemannHelmholtz peut se faire une certaine idee de la maniere dont les formes pures de theories, de type nettement distinctes, sont reliees entre elles par le lien d'une loi ». On trouvera une £tude des rapports entre le concept topologique de variete et la perception spatiale, ainsi qu'une reflexion critique sur la portee de la phenomenologie descriptive dans L. Boi, Questions regarding husserlian geometry and phenomenology. A study of the concept of manifold and spatial perception. Publication n. 191 du CAMS, Paris, EHESS, Avril 2000. 25

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«Le point de vue de la fonction est le point de vue central de la phenomenologie [...] Au lieu de se borner a l'analyse et a la comparaison, a la description et a la classification attachees aux vecus pris isolement, on considere les vecus isoles du point de vue «teleologique » de leur fonction, qui est de rendre possible une « unite synthetique ». On considere des divers de conscience qui sont pour ainsi dire presents par essence dans les vecus eux-memes, dans leur donation de sens, dans leurs noeses en general, et qui sont pour ainsi dire prets a en etre extraits.26 » Un certain idealisme, souvent qualifie abusivement de platonisme, est frequent chez les mathematiciens et s'exprime selon des termes vaguement analogues : il y a une necessite a l'agir mathematique ; la decouverte mathematique est la vision d'essences, etc. L'analyse eidetique revele des structures prescrites et ontologiquement pures de la conscience et releve d'un idealisme transcendantal, mais il faut bien entendu se garder de lui donner un tel contenu naif. 5. Categorisation et synthese Si le programme de la phenomenologie husserlienne est clair, sa realisation achoppe a de nombreuses difficultes. Par nature, les lois pures de la pensee sont delicates a apprehender en dehors d'un contexte analytique, et il y va chez Husserl de quelque chose comme le synthetique a priori kantien sous des formes renouvelees par les techniques intentionnelles post-brentaniennes27 et, plus radicalement, par le projet meme de la phenomenologie tel qu'il est expose dans les Ideen. Notre these sera ici que les developpements mathematiques amorces dans les annees 50 (mais dont la cristallisation en termes de fondements methodologiques et 1'acceptation par la communaute scientifique est plus recente28) permettent de donner corps au projet husserlien, au moins pour ce qui a trait aux origines du savoir. Un exemple fondamental est celui des operations de la logique elementaire et de la logique des quantificateurs. L'algebre de la logique a la Boole est un modele tres interessant mais, comme Frege le remarquait deja, il manque a une telle approche de la logique une justification a priori des lors qu'elle a vocation a s'appliquer a des contenus et des jugements effectifs, et pas exclusivement a des objets mathematiques. Si algebrisation des regies de pensee il doit y avoir, elle doit etre etayee par une etude des structures de la conscience pure, sans quoi 1'algebrisation operera une disjonction de fait entre les formes symboliques et l'activite de jugement ordinaire.

26

Ideen. p. 294 [176], Nous renvoyons a l'ouvrage de J. Benoist, ou ces questions sont traitees systematiquement: Phenomenologie, semantique, ontologie. Husserl et la tradition logique autrichienne. Paris, P.U.F., 1997. 28 La theorie des categories est encore aujourd'hui a peu pres absente des cursus mathematiques universitaires, ce qui est tout aussi aberrant que la tres faible presence de la logique. 27

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L'approche categorielle , due pour l'essentiel a Lawvere indique que la logique des connecteurs propositionnels ou des quantificateurs est fondee sur des processus universels qui ne sont pas specifiques a la logique et apparaissent egalement en geometrie, en topologie ou en algebre, pourvu que les fondements de ces theories et leurs methodes soient decrits en termes categoriels. Ces resultats suggerent qu'il existe en amont de la mathematique formelle ou de la logique mathematisee un arriere-fond de regies aprioriques de la pensee qui transcendent les compartimentations sectorielles usuelles. C'est d'ailleurs la these defendue par Lawvere, qui voit dans des techniques de theorie des categories comme Fadjonction'1 1'archetype de constructions universelles susceptibles de rendre compte de distinctions comme : etre et devenir, qualitatif et quantitatif.. .32. L'exemple de la construction categorielle de l'operateur logique "et" (&) est eclairant car elle indique le cheminement de 1'esprit dans la creation des connecteurs propositionnels. En d'autres termes, cette construction ne se contente pas de justifier ou de decrire, a l'instar d'une modelisation d'essence calculatoire ou axiomatique : elle designe au travers de processus mathematiques des structures pures de la pensee et/ou de la conscience. Nous verrons plus avant que ces processus sont assez proches de ceux qui legiferent la constitution des objets transcendantaux dans les philosophies kantiennes et husserliennes. Pour en revenir a l'operateur &, sa regie de formation est la suivante. Supposons fixe un univers de pensee (une categorie) structure en propositions (objets) et regies d'inference (relations satisfaisant les regies usuelles Nous passons volontairement sous silence les definitions precises d'une categorie ou des concepts fondamentaux de la theorie (foncteurs, adjoints...), en nous contentant d'indications sur la signification des objets considered, et de renvois a des articles ou traites pour des precisions. II y faudrait trop de developpements pour que I'insertion d'arguments techniques precis ajoute a l'intelligibilite de cet article pour un lecteur n'etant pas au fait de la theorie. Nous l'encourageons done a consulter des manuels introductifs comme le Categories for the working mathematician de S. MacLane, Berlin, New York, 1971. Pour fixer les idees, la donnee d'une categorie est la donnee d'une collection d'objets et de transformations (dites aussi morphismes ou relations) entre ces objets. Ces demieres doivent satisfaire des conditions naturelles : la composed de deux transformations est une transformation, la composition est associative... Les groupes, les espaces vectoriels, les ensembles finis, mais egalement les espaces topologiques forment des categories. 30 Voir S. MacLane et 1. Moerdijk. Sheaves in geometry and logic. New York, Springer, 1992. 31 On peut passer d'une categorie A donnee a une autre categorie B par un procede analogue aux applications ensemblistes : il faut associer a tout objet et toute relation de A un objet et une relation de B de maniere a preserver les regies de composition des relations. Les transformations correspondantes portent le nom de foncteurs. II y a adjonction entre deux categories lorsqu'elles sont reliees entre elles par deux foncteurs en sens inverse l'un de l'autre, satisfaisant en outre une condition d'isomorphisme. En pratique, la donnee d'une adjonction entre deux categories implique l'existence de relations profondes entre elles et entre les foncteurs correspondants. De nombreuses notions elementaires, comme les proprietes des bases d'un espace vectoriel ou des generateurs d'une algebre de polynomes se reformulent elegamment en termes d'adjonction. Nous allons voir que c'est egalement le cas de certaines notions logiques. 32 F.W. Lawvere. Categories of space and of quantity. Philosophical, epistemological and historical explorations, Ed. J. Echeverria, Berlin, Walter de Gruyter, 1992.

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d'associativite), notees <. Etant donnees deux propositions B et C, il s'agit de construire la proposition B & C. A priori il n'est pas clair que B & C existe : il y faut des conditions sur I'univers de pensee33. En tous les cas, si la proposition B&C existe, elle doit necessairement satisfaire a la condition : pour tout A, si A < B et A < C, alors A < B & C, B & C < B et B & C < C. En fait, B & C doit etre "universel" parmi toutes les propositions satisfaisant a ces conditions. Toute proposition D verifiant (pour tout A) les conditions imposees a la proposition B&C est equivalente a B&C, au sens ou on a simultanement : D < B & C e t B & C < D . Cette construction de l'operateur & serait simplement anecdotique si elle n'etait le prototype d'un mode de construction valant pour tous les connecteurs propositionnels et pour les quantificateurs. En termes categoriels, elle s'enonce: l'operateur & est un produit (un adjoint a droite au foncteur diagonal). A son tour, l'adjoint a droite de l'operation &C est l'implication "C => ...", etc.35 Le noyau de ces arguments est la notion categorielle de limite -limite sur des families d'objets et de relations dans des univers de pensee structures comme des categories. La limite d'une famille d'objets et de relations entre ces objets est un objet qui, intuitivement, approxime au mieux et de fa9on coherente ces donnees. Ainsi, la proposition qui approxime le mieux la donnee simultanee de B et C n'est autre que la proposition B & C ! Le champ d'application de la notion de limite ne se restreint evidemment pas a la logique : dans un ensemble strictement ordonne la notion de limite coi'ncidera ainsi avec celle de borne superieure. Elle a une signification generate, qui excede le domaine des mathematiques et de logique formelle : les classifications d'origine aristotelicienne en genres et especes precedent du meme principe - trouver un denominateur commun minimal a des donnees tout a la fois diverses et organisees (comme dans les classifications botaniques ou zoologiques). Elle intervient done tout a fait legitimement dans un contexte d'ontologies materielles, dont elle contribue a 1'organisation theorique. Le formalisme des limites n'y a peut-etre pas d'emblee la validite inconditionnelle que la theorie lui confere en mathematiques, mais cela ne porte pas atteinte a la valeur intrinseque de la methode, pas plus que les ambigui'tes de l'utilisation du syllogisme dans un contexte materiel n'en inferent la valeur analytique. Un exemple simple (dans l'esprit de ce que les physiciens appellent un "toy model", essentiellement a valeur d'illustration) permettra de cerner les enjeux du concept de limite dans un contexte de vecus et de phenomenes de pensee et du langage ordinaires. Considerons I'univers (structure comme une categorie) des classes d'etres vivants avec pour relations les relations de subordination entre Techniquement, l'existence de produits finis dans la categorie. On notera que l'operateur "et" a ici une fonction externe. II coordonne deux relations d'inference, contrairement a l'operateur & que Ton cherche a construire, et qui est une operation interne. On pourra consulter pour des arguments precis l'article: S. MacLane, Categories in geometry, algebra and logic. Math. Japonica. 42, (1995), 169-178 ou S. MacLane et I. Moerdijk, Sheaves in geometry and logic. Springer, New-York, 1992. 34

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classes. La relation "avoir des ailes, des plumes, deux pattes et pondre des ceufs" definit une classe d'etres, les oiseaux36, qui est la "limite" (en l'occurrence 1'intersection) des classes definies par les relations : "avoir des ailes", "avoir des plumes", etc. Le terme de limite exprime ici le fait de considerer la plus grande classe d'etres pour lesquels la relation est satisfaite. Repetons-le encore : il s'agit la d'un toy model. La pensee ordinaire ne procede d'ailleurs pas par des calculs de classes mais plutot par un calcul de "concepts" au sens fregeen37 (un concept au sens fregeen est un critere de decision permettant de dire si tel objet donne tombe ou non "sous le concept" : lorsqu'il nous faut decider si tel animal est un oiseau ou un reptile, nous examinons s'il a des plumes, deux pattes..., et certainement pas s'il appartient a une hypothetique classe des oiseaux !). Au demeurant, la notion de limite continue de faire sens dans un contexte fregeen, ainsi que notre analyse du concept d'oiseau. Cet exemple, pour rudimentaire qu'en soit le traitement qui en a ete donne, illustre bien les deux usages epistemologiques possibles du concept de limite, ainsi d'ailleurs que les difficultes qu'il y a a le mettre en ceuvre (faut-il privilegier un calcul de classes ou un « calcul de concepts » plus en adequation avec nos modes de pensee effectifs, mais beaucoup plus problematique?). II a une fonction descriptive (decrire les subordinations conceptuelles et en rendre compte a posteriori), mais egalement une fonction constitutive (le concept d'oiseau ne preexiste pas a sa creation ; il est par essence un concept limite edifie, comme tous les concepts materiels, sur le terreau d'experiences sensibles et de classifications successives). Retenons simplement que la logique generate traditionnelle, qui codifie les processus d'abstraction conceptuelle a partir des donnees sensibles en termes d'actes synthetiques pourrait, pour une bonne part, etre ainsi reinterpretee en termes categoriels, l'idee de construction de limites venant se substituer a celle de synthese ou, plus exactement, la modeliser. La portee de ces remarques est evidente. Elles indiquent un terrain epistemologique nouveau sur lequel fonder et legitimer le synthetique a priori, y compris sous ses formes materielles. Le vieux reve kantien d'une application en philosophie des constructions mathematiques reprend ainsi corps38. Accessoirement, une telle legitimation implique une devaluation de l'ideal de scientificite cartesien, a savoir « que la science universelle ait la forme d'un systeme deductif, systeme dont tout 1'edifice reposerait ordine geometrico sur un fondement axiomatique servant de base absolue a la deduction » 9. La synthese

Des lors que nous nous inteiessons ici au langage ordinaire, c'est volontairement que nous preferons une definition approximative et naive du concept d'oiseau a une definition plus scientifique comme « vertebre couvert de plumes, aux membres anterieurs transformed en ailes, etc. ». 37 G. Frege. Grundlagen der Arithmetik, Breslau, 1884. E. Kant. £55131 pour introduire en philosophie le concept de grandeur negative. Trad. R. Kempf. Paris, Vrin, 1991. 39 E. Husserl, Meditations cartesiennes, trad. G. Peiffer et E. Levinas, Paris, Vrin, 1947, p. 26 [6]. Citees Les Meditations dans la suite de l'article.

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doit reconquerir ses droits aux cotes de ce systeme deductif. En outre, tout comme pour les deductions analytiques, la validite des actes synthetiques doit etre reconduite et subordonnee aux lois pures de la pensee, qui leur conferent un mode specifique de scientificite. 6. Noeses et noemes Par categorisation, nous entendrons desormais la mise en evidence de structures typiques de la pensee et de lois de regulation pour ces structures typiques (comme les constructions de limites) et ce aux differents niveaux possibles de purete eidetique, de la conceptualisation de I'experience sensible aux formes pures de la pensee. Les types de categorisation esquissees au paragraphe precedent ne pretendent pas etre plus que l'indication de modalites nouvelles de developpement de la phenomenologie husserlienne. Pour autant, meme a titre de simples esquisses, ils permettent de degager des problemes specifiques aussi bien que de nouveaux modes d'entente de problemes classiques en phenomenologie. Dans un univers de pensee suffisamment riche (semantiquement, historiquement, syntaxiquement...), l'existence de "limites" est portee par le langage constitue. Ainsi, dans l'utilisation ordinaire de la logique elementaire, nous raisonnons avec les differents operateurs & (et),-i (non),V (pour tout), 3 (il existe)... sans nous inquieter de leur provenance ou de leur mode de formation. II nous suffit qu'ils existent et nous soient disponibles. De meme, les concepts du langage ordinaire "vont de soi", encore que cet "aller de soi" soit beaucoup plus problematique qu'il n'y parait. Pourtant, des que Ton questionne en deca des evidences du langage ordinaire ou lorsqu'il s'agit d'enseigner des notions originales et dedicates, la question se pose du mode de formation de concepts nouveaux ou, plus simplement, de la justification de procedures conceptuelles dont la validite quotidienne se fonde sur 1'habitude plutot que sur des justifications aprioriques. Le formalisme categoriel, qui permet de degager des regies de formation d'objets comme les limites mathematiques (theoremes d'existence...), permet aussi de clarifier certaines conditions d'existence ou de formation d'objets en dehors des mathematiques. Toutefois, des lors que Ton cherche a etendre la categorisation a l'ensemble des structures de pensee, il faut au prealable bien s'entendre sur des concepts comme celui d'objet ou de relation. Nous sommes passes outre ces problemes dans un premier temps, en admettant que divers "univers de pensee" non mathematiques puissent etre structures comme des categories. II nous faut maintenant traiter du statut phenomenologique de l'objet dans une perspective categorielle. Si 1'intentionnalite est le fait premier pour la phenomenologie et la condition de possibility meme d'une activite de la conscience, son correlat theorique, la noese, herite de cette priorite. Aucun objet n'existe pour nous en dehors de la conscience que nous en avons. L'objet est, par necessite, «dans chaque moment de conscience, 1'index d'une intentionnalite noetique lui appartenant de par son sens, intentionnalite qu'on peut rechercher, et qui peut etre explicitee ». De ce fait, de

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meme qu'aux vecus intentionnels sont associes des moments noetiques, des moments noematiques ou noemes sont associes aux visees correspondantes, c'est a dire encore aux objets intentionnels40. L'objet au sens de la phenomenologie, et en particulier l'objet mathematique, n'a done pas l'immediatete que lui procure l'imagerie "platonicienne" traditionnelle et son recours a la transcendance. En particulier, aucun objet n'est concevable sans un horizon de sens, ce sens fut-il purement formel, comme avec les productions des systemes axiomatiques. L'objet mathematique est toujours insere dans les horizons de la conscience du mathematicien ; c'est ce qui rend, de fait, la creation possible. Si l'imagination etait bridee par le carcan des structures mathematiques deja constituees, la science mathematique serait indefiniment condamnee a se repeter et a n'ameliorer qu'a la marge ses contenus. En d'autres termes, c'est dans la correlation cogito-cogitatum que prend racine la pensee mathematique. L'insistance portee historiquement sur le cogitatum, a savoir sur les descriptions objectives et le corpus des mathematiques constituees, a occulte les potentialites de recherches plus originaires -la possibilite meme d'une science du sens et des lois d'essence des objets mathematiques. La theorie des categories intervient de maniere subtile dans ce debat, des lors que la structure interne des categories reflete en partie la dichotomie noeme/noese. Ainsi, la construction des connecteurs propositionnels effectuee au paragraphe 5 a une teneur noetico-noematique qu'il est possible d'expliciter. Dans ce cas, les objets sont les propositions, e'est-a-dire les correlats noematiques de visees intentionnelles d'un type particulier (assertorique). Les relations, qui sont des relations d'inference, sont, bien entendu, susceptibles d'etre egalement obtenues comme correlats noematiques de visees intentionnelles d'un autre type (deductif), mais 1'articulation de ces relations entre elles, la possibilite de les faire varier a 1'interieur d'un horizon (l'horizon de variation de la proposition A et des inferences correspondantes au paragraphe 5): tout cela renvoie aux structures noetiques de la conscience. Bien entendu, ces considerations categorielles sont loin d'epuiser la signification de la dualite noetico/noematique, mais elles donnent tout de meme de bonnes indications sur ce qu'il faut attendre, dans les textes husserliens, de notions comme celles d'horizon ou de variation eidetiques, qui ont une composante dynamique et supposent que soient inscrites dans la structure meme des objets (du monde, de l'esprit...) des lois de variation ideales. 7. Le sujet transcendantal Pour que ces interactions entre theorie des categories et phenomenologie ne demeurent pas entierement allusives et programmatiques, cet article se conclura par une analyse precise. II s'agira de (commencer a) reecrire la theorie de la

Ideen, p. 305 [182],

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subjectivite transcendantale41 dans une perspective categorielle . Le sujet, correlat indispensable de ces objets que nous souhaitons penser en termes de categories, est la donnee premiere de la phenomenologie, heritiere en cela du cogito cartesien. L'ego cogito n'est-il pas un « domaine ultime et apodictiquement certain sur lequel doit etre fondee toute philosophic radicale ? 43». Pour autant, si le sujet est phenomenologiquement premier, cela ne signifie pas que son concept aille de soi. Que signifie, typiquement, que le sujet puisse etre transcendantal ? Jusqu'ou est-il done possible d'argumenter au-dela de l'evidence cartesienne de l'ego pour essayer de fonder la subjectivite ? II n'est pas inutile d'en revenir a la premiere formulation de la subjectivite transcendantale dans la Dialectique transcendantale kantienne. N'est-il pas remarquable que cette formulation soit d'emblee presque mathematique ? « Comme fondement [d'une psychologie transcendantale], nous ne pouvons rien donner d'autre que cette simple representation, vide par elle-meme de tout contenu, moi, dont on ne peut meme pas dire qu'elle soit un concept, mais qui est une simple conscience accompagnant tous les concepts. Par ce « moi», par cet « il », ou par cette « chose qui pense », on ne se represente rien de plus qu'un sujet transcendantal des pensees = X. Et ce sujet ne peut etre connu que par les pensees, qui sont ses predicats: isolement, nous ne pouvons en avoir le moindre 44

concept .» Et Kant d'ajouter, « II doit d'abord sembler etrange que ce qui conditionne ma pensee en general, et qui n'est par consequent qu'une qualite de mon sujet, s'applique en meme temps a tout ce qui pense, et que nous puissions pretendre fonder sur une proposition qui parait empirique un jugement apodictique et universel tel que celui-ci: tout ce qui pense est constitue comme la conscience declare que je le suis moi-meme. La raison en est que nous attribuons necessairement a priori aux choses toutes les proprietes constituant les conditions qui seules nous permettent de les concevoir ». La pensee kantienne est tres explicite. II y a au fondement de l'idee de sujet et, a fortiori, de la subjectivite transcendantale, l'experience qui est la mienne et celle de ma pensee, dans toute sa specificite et sa particularite. Elle ne peut pretendre a l'universel que des lors que les structures qu'elle detecte dans les modalites de son fonctionnement sont apodictiques. Et e'est bien ce que signifie la mise en equation : sujet transcendantal des pensees = X. L'edification des concepts Telle que nous la considerons, la theorie de la subjectivite transcendantale est kantienne plutot qu'husserlienne. Cela dit, on sait la dette de la phenomenologie a l'egard de la pensee critique, particulierement evidente pour ce qui est de la subjectivite transcendantale. Aussi nos remarques s'appliqueraient-elles telles quelles a I'egologie transcendantale husserlienne et a ses diverses variantes phenomenologiques. 42 Nous suivons les descriptions du sujet transcendantal donnees par J. Benoist dans Autour de Husserl. L'ego et la raison. Paris, Vrin, 1994 et dans Kant et les limites de la synthese. Paris, Vrin, 1996. 43 Meditations p. 42 [16]. 44 E. Kant. Critique de la raison pure. Ximt ed. p. 341 [264]. Trad. J. Barni, Paris, Flammarion, 1976.

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transcendantaux, a commencer par celui de sujet, precede d'une algebre generalisee ou les concepts transcendantaux apparaissent d'abord comme les indeterminees du systeme, construits par un precede de renvois successifs. La subjectivite transcendantale est "cela" a quoi renvoient le "moi", la "chose qui pense", le "il", et tous les concepts de meme nature. C'est ce que signifie que « le sujet ne puisse etre connu que par les pensees, qui sont ses predicats : isolement, nous ne pouvons en avoir le moindre concept». Cela parce que la subjectivite transcendantale, malgre son apodicticite, n'a pas de sens en dehors du systeme de renvois qui vient d'etre evoque, et qui la constitue. De la meme fa§on, nous n'avons jamais de concept propre ou d'intuition propre de l'infini, sinon au travers de processus limites ou d'images geometriques. Au fond, la subjectivite transcendantale n'est-elle peut-etre rien d'autre que l'expression de la necessite du renvoi de toute pensee a une « chose qui pense ». Comment approximer la mise en equation « sujet transcendantal = X » par le langage des categories ? Considerons l'ensemble des objets pensables, c'est a dire le « monde » (constitue au travers des differents modes de variations concretes et ideales de la pensee). En font partie aussi bien les representations associees a des vecus concrets (cette table, cette chaise...) que les representations conceptuelles (la justice, la beaute, le nombre...), imaginaires (le centaure), les representations de mots, etc. A quoi il convient d'ajouter pour chacune de ces representations les variations possibles de ses modes d'existence (presence immediate, souvenir, possibility, anticipation...). A chaque objet primaire ainsi considere est associe une representation (et un objet) secondaire, aux contours plus ou moins bien definis, qui est celle de la conscience accompagnant la representation de l'objet primaire. Dans la plupart des cas cet objet secondaire est le moi vivant concret (je pense, je vois, concois cette table, cet homme, ce jugement), mais il peut arriver que l'objet primaire renvoie, dans sa constitution meme, a d'autres consciences que la mienne. Des propositions comme « Pierre doit me presenter a Paul » ou « un athee ne croit pas a l'existence des Dieux » font etat de tels renvois. Dans l'ensemble des objets du monde, je peux done isoler ceux qui apparaissent comme le support d'une activite de conscience : le moi actuel, passe ou futur, Pierre, les athees, l'humanite, etc. Ce sont les objets A qui sont en relation avec d'autres objets B sur le mode de la conscience (A a conscience de B). Lorsque je fais prendre virtuellement a A et B toutes les valeurs possibles selon un precede ideal il me reste le schema formel « X a conscience de Y », qui est l'archetype de toutes les occurrences possibles de jugements « A a conscience de B ». Cet X et cet Y ideaux, auxquels tous les A et B intervenant dans de tels jugements seront lies par un rapport de substitution des variables dans le schema formel « X a conscience de Y », ne sont rien d'autre que le sujet transcendantal et son correlat inevitable : « l'objet non-empirique, c'est a dire transcendantal = Y » kantien. En d'autres termes, la proposition "X a conscience de Y" (c'est a dire, le sujet transcendantal a conscience de l'objet transcendantal) est la limite sur tous les objets et les relations possibles (l'objet initial dans le langage des categories) d'une

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categorie dont les objets sont les propositions "A a conscience de B", les relations etant des relations de dependance conceptuelle et d'inference dont le prototype serait la specialisation de "X a conscience de Y" en "A a conscience de B". Les autres relations peuvent etre construites sur le modele de la specialisation de propositions comme "nous aimons la liberte et la fraternite" en "j'aime la liberte" ou "nous aimons la fraternite"45. Bien entendu, il est loisible de douter de la pertinence, et surtout de l'utilite de telles considerations. Au-dela de convictions propres aux mathematiciens ayant pu observer la portee hermeneutique du langage des categories, nous pouvons attirer 1'attention sur le fait que les analyses precedentes mettent en relief un non-dit essentiel des analyses kantiennes (et post-kantiennes) de la subjectivite et l'objectivite transcendantale. En effet, il n'est pas de sujet ni d'objet transcendantal concevable sans le "a conscience" qui unit le X et le Y dans "X a conscience de Y". En d'autres termes, il doit exister quelque chose comme un mode de conscience transcendantal reliant le sujet transcendantal = X a l'objet transcendantal = Y pour valider la construction kantienne ! A la lumiere de la construction categoriale du sujet et de l'objet transcendantal, il faut done introduire un problematique "mouvement de pensee transcendantal" de l'un a l'autre, qu'il convient d'interpreter comme l'ouverture necessaire de tout sujet a ce qui l'excede. Le sujet est transi d'exteriorite : « Ce qui constitue le sujet, comme objet singulier, en tant qu'il y va du rapport a l'objet en general, e'est la relation a ce qui n'est pas lui. Le sujet est deja toujours « dehors » »46. Le formalisme categoriel met ainsi en evidence les ressorts et racines d'arguments philosophiques delicats et longtemps debattus, en 1'occurrence les regies de constitution de la subjectivite transcendantale. Sa portee philosophique, des lors que ses conclusions rejoignent et etayent celles de travaux philosophiques recents, nous semble done aller de soi. Plus generalement, les avancees conceptuelles et techniques des dernieres decennies ont permis aux mathematiciens d'acquerir une maitrise nouvelle sur les structures de certaines de leurs intuitions (toutes celles qui ont trait a l'idee de structure mathematique, de theorie algebrique, de variation ou de transitions entre champs theoriques...). Cette maitrise peut et doit se refleter sur notre conception de l'activite mathematique, du terrain sur lequel elle se deploie, mais egalement de questions plus radicales : signification et portee de la mathesis universalis de tradition leibnizienne, conditions de possibilite et limitations intrinseques d'une axiomatisation des sciences de la nature, etc. Autant dire que la « mathematique philosophique » n'a pas dit son dernier mot !

Le choix d'autres types de relations, plus complexes, serait envisageable. J. Benoist. Autour de Husserl. Op. cit. Chap. III.

G e o m e t r i e s of N a t u r e , Living S y s t e m s and Human Cognition The collection of papers forming this volume is intended

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