A3 his algebra how a boy from chicagos west side became a force in american mathematics

gin the story than o n a bright day: "In Baltimore, 22 February 1877 was a day of celebration. Bright, cloudless, uncharacteristically springlike, the birthday of America's first president was feted in fine style. Bunting h u n g from the balconies . . . . " His inaugural address, in Baltimore, suggests that Sylvester relished the c h a n c e of starting again in a country without the prejudices that he had e n d u r e d in England. He addressed his n e w l y f o u n d a u d i e n c e in 1877: "Happy the y o u n g m e n gathered u n d e r our wing, who, unfettered a n d untrammelled by any other test than that of diligence a n d attainments, have here afforded to t h e m an o p p o r t u n i t y of filling up a c o m p l e t e scheme of education." This was a pivotal point in Sylvesters life, a n d Parshall uses it as the o p e n i n g scene of the b o o k very effectively to introduce a flashback. Without enthusiasm, the b i o g r a p h e r c a n n o t succeed, a n d here a n d t h r o u g h o u t the book, her e n t h u s i a s m for writing the life matches Sylvesters o w n for mathematics. As with any of Sytvester's current researches there was now, in the year 1877, no field more worthy of study than invariant theory. It had everything--it was topical, there were routine calculations for students, and there were also challenging p r o b l e m s for himself to consider. O n e desideratum was plugging the gap in "Cayley's theorem" which purported to c o u n t the n u m b e r of linearly i n d e p e n d e n t invariants and covariants of a binary form. Both Cayley and Sylvester believed it true, and indeed, their calculations were based on its truth. But Cayley had m a d e a crucial assumption, a n d while this caused no flurry in the 1850s, the h e i g h t e n e d importance of "mathematical proof" in the 1870s suggested it was a result which definitely n e e d e d proof. It was indeed a p r o u d m o m e n t in Baltimore w h e n Sylvester p r o v e d it. and, in a n n o u n c i n g his success to the mathematical world, he blew his t r u m p e t with all the Sylvestrian puff he could muster. Less successful were his attempts to reprove Gordan's t h e o r e m in which the G e r m a n mathematician Paul G o r d a n had demonstrated that the n u m b e r of irreducible invariants a n d covariants was finite in the case of an algebraic binary form. A proof using only plain algebra w o u l d vindicate the English methods of algebra and put the G e r m a n semi-abstract calculus

in tile shade. The quest took o n a nationalist dimension because Sylvester also believed that the so-called symbolic methods of Alfred Clebsch a n d G o r d a n had b e e n appropriated from o n e of Cayley's early discoveries in the theory of hyper-determinants. These objectives were theoretical, but of equal value in invariant-theory circles of the nineteenth century were the finding a n d recording of the actual algebraic expressions for the invariants and covariants. For this Sylvester started up an ambitious calculatory program for the binary forms of the first ten orders (in modern language, for polynomials u p to degree ten). For the first [our, the task was simple and had b e e n recognised since the 1840s; the case of the binary quintic was thought to have b e e n recently comp l e t e d - t h o u g h details in that case still awaited attention. Sylvester and his b a n d of students took u p tile calculatory challenge with alacrity, and if a "truth" in the theory had not yet b e e n confirmed by mathematical proof, they p l u n g e d o n in the hasty heat of calculation. This was the inductive approach p a r excellence, a n d it is a measure of h o w Sylvester differed from mathematicians today (at least he was able to publish work b a s e d o n pure sunnise!) Sylvester did m u c h to create a research atmosphere at J o h n s Hopkins. The students were invigorated by having a mathematician with a s o u n d Eur o p e a n reputation a m o n g them, and if his teaching methods were idiosyncratic, still, he was a stimulating presence. Sylvester was instrumental in bringing the American Journal of Mathematics into being, the first successful research journal in the country. Nevertheless, after seven successful years in Baltimore, Sylvester increasingly missed his former life in England, particularly his L o n d o n social circle based at the A t h e n a e u m Club a n d at the Royal Society. The last act in Sylvester's dramatic career b e g a n with his a p p o i n t m e n t to the Savilian Chair of g e o m e t r y at Oxford in 1883. Even at the age of s e v e n t y he was not p r e p a r e d to retire to the sidelines and rest o n his laurels. He started a mathematical society in the university and set out further g r a n d theories. O n e was a theory of reciprocants, a theory akin to invariant theory (he was to discover it had b e e n s u b s t a n tially developed elsewhere). He b e c a m e

e n t r a n c e d with matrix algebra, w h i c h he attacked by considering a plethora of low-dimensional c a s e s - - f o r example, his system of n o n i o n s i n v o l v i n g 3 • 3 matrices. It was again the inductive approach which p e r m e a t e d all his mathematical endeavours. Sylvester was fortunate that, like Cayley's, his attainments were recognised in his lifetime. To some extent, his huge ego was soothed. As the Oxford years ran on, his health deteriorated, his eyes gave him trouble, a n d he retired to Mayfair, London's fashionable West End. His last years were spent writing poetry and delving into problems in n u m b e r theory. In Professor Parshall, Sylvester has f o u n d a worthy chronicler of his life. T h r o u g h the exercise of a great deal of care and diligence, Parshall has overc o m e the practical difficulties of a scattered archive of t h o u s a n d s of n o t e s a n d letters, some of them almost indecipherable. Her earlier work, ,lames

Joseph Sylvester: Li/~ a n d Work in Letters (Oxford University Press, 1998), provided some of the raw materials, but writing a b i o g r a p h y is a different p r o p o sition. In this quest, n o w c o m p l e t e d , she has succeeded admirably. Sylvester is o n c e more before us, a n d rejuvenated he excites o u r e n t h u s i a s m for the great mathematical enterprise.

Tony Crilly [email protected]

A 3 & His Algebra: How a Boy from Chicago's West Side Became a Force in American Mathematics t)le Nancy E. Albert iUNIVERSE INC., LINCOLN, NEBRASKA, 2005, 3 5 2 PP., ISBN-13:978-O-595-32817-8 US $ 2 3 . 9 5 . REVIEWED BY JOSEPH A. WOLF

Jrian Albert was o n e of the most mportant American algebraists )f the twentieth centu~-y. He was

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a r e m a r k a b l e p e r s o n o n m a n y accounts. author of the book u n d e r review. Roy Most obviously, he was a first-rate mathdied shortly after earning a B.A. in anematician a n d bore m u c h responsibility thropology. Alan earned a B.S. in physics, for bringing m o d e r n algebra into the worked as an engineer, and has since current mathematics curriculum. He passed away. Nancy e a r n e d a J.D. a n d was especially famous for his work o n works :is a lawyer. nonassociative algebras, o n division alI am reviewing the J u n e 2006 revised edition of this biography. With m o d e r n gebras, and on R i e m a n n matrices. Perp u b l i s h i n g methods it is easy to correct haps less k n o w n , he bore major responsibility for persuading the U.S. typographical errors and to take into ac g o v e r n m e n t to support basic research c o u n t n e w l y available source materials. in mathematics. Even before the naNancy Albert has d o n e just that. In adtional i m p o r t a n c e of science a n d mathdition, it is remarkable just h o w Nancy ematics was u n d e r l i n e d by Russia's Albert was able to transpose her legal l a u n c h of the first satellite, "sputnik," in abilities and produce such a precise and 1957, Adrian Albert played a key role accurate description of Adrian Albert's in establishing a n d increasing the mathematics, of the honors he received, a m o u n t of ONR a n d NSF research supa n d of his interactions with the federal port for active mathenmticians at all levg o v e r n m e n t and the other m e m b e r s of els of seniority. Some of this i n f l u e n c e the scientific establishment of his day. certainly must have b e e n b a s e d o n his Her description of the history of the d e f e n s e work during a n d after World University of Chicago, especially its DeWar II, in particular his effective use of partment of Mathematics, is accurate algebraic m e t h o d s in cwptology. a n d to the point. She is precise and inPerhaps k n o w n mostly to those w h o cisive w h e n she discusses the social seth a d direct personal contact with him, ting of both her father and her mother, Adrian Albert put a lot of time. thought, his c o n n e c t i o n s with the University of and effort into e n c o u r a g i n g a n d Chicago a n d the other m e m b e r s of the s m o o t h i n g the way for students a n d D e p a r t m e n t of Mathematics there, and y o u n g researchers. O n m a n y occasions, (rather delicately) the way he m a n a g e d s o m e of which are m e n t i o n e d in this despite anti-Semitic attitudes that lasted biography, he arranged financial supinto the late 1950s. She was well acport for students to e n a b l e full-time q u a i n t e d with the many mathematicians study. He e n c o u r a g e d w o m e n in the of her father's age and a bit younger, study of mathematics, a n d to the extent a n d that certainly contributed to the then possible, he facilitated their career quality of the book. Also, recently she paths. The biography contains several h a d access to some relevant AMS instances of this, and there were in fact archives. quite a few others. Adrian Albert usuThis biography has already b e e n really did this sort of thing in a very w a r m v i e w e d by Lance Small in the AMS Xohearted way: he w o u l d arrange the b e n tices ( D e c e m b e r 2005) and by Philip efit a n d then gleefully surprise the Davis in the SIAM News (June 2006). recipient with the good news. Lance Small is a n algebraist w h o started (Abraham) Adrian Albert, often nickhis graduate studies at the University of n a m e d '% Cubed," was b o r n in 1905 Chicago toward the e n d of Adrian Ala n d grew u p in Chicago, received his bert's career. I defer to his description B.S. in mathematics from the University of Adrian's mathematical results beof Chicago in 1926, a n d e a r n e d his cause I work in areas ( g e o m e t W a n d Ph.D. from Leonard Dickson at the Unianalysis) more or less orthogonal to versity of Chicago in 1928. After two Adrian's research. There is o n e n o n years as Instructor at Columbia Univer- trivial point of historical disagreement sity he r e t u r n e d to the University of b e t w e e n Lance Small's review and Chicago as Assistant Professor in 1931, Nancy Albert's book. It c o n c e r n s Adrian Full Professor in 1941, Math I ) e p a r t m e n t Albert's efforts to generate an offer from Chair 1958-1962, a n d Dean of Physical the University of Chicago to Nathan Sciences 1962-1971. He passed away in J a c o b s o n a n d the possibility that these 1972. efforts were initially i m p e d e d by antiAdrian Albert married Frieda Davis Semitic attitudes. Nancy Albert has clarin 1927. They had two sons, Alan a n d ified this situation, with better docuRoy, a n d a daughter, Nancy, w h o is the m e n t a t i o n a n d a better a r r a n g e m e n t of

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THE MATHEMATICALINTELLIGENCER

the text, in the J u n e 2006 revised edition of her book. Adrian Albert a n d my father were close friends since their u n d e r g r a d u a t e clays in the 1920s. W h e n Adrian Albert died he left a list of y o u n g mathematicians to be invited to each take some b o o k s flom his libraw, a n d I was o n the list. S o m e h o w that last gesture was typical of his generosity toward his students a n d y o u n g e r colleagues. As is clear at this point, this review is written from the v i e w p o i n t of a friend, student, a n d colleague, rather than from the v i e w p o i n t of a historian of mathematics or a mathematical critic. With that caveat, I definitely r e c o m m e n d this biography to all mathematicians interested in the interplay b e t w e e n mathematics a n d public policy, and especially those in pure or a p p l i e d algebra a n d those w h o had contact with the Dep a r t m e n t of Mathematics at the University of Chicago a n y time from the 1920s through the 1960s. Department of Mathematics University of California Berkeley, CA, 94720-3840 USA e-mail: jawolf@math, berkeley.edu

Saunders Mac Lane. A Mathematical Autobiography by Saunders M a c Lane, WELLESLEY, MASSACHUSETTS, A. K. PETERS, 2004, 358 P., $39.00, HARDCOVER, ISBN 1-56881-150-0 REVIEWED BY HENRY E. HEATHERLY

~aunders Mac Lane (1909-2005) ~was o n e of the e p o c h - m a k i n g ~: J * mathematicians of the 20th century. He knew- a n d interacted with m a n y of the o u t s t a n d i n g figures of 20th t e n tury mathematics. Add to this the wellk n o w n lucidity of his expositions, his p r o f o u n d insight into the nature of mathematics, a n d his experience with scientific organizations a n d mathematical centers of excellence, a n d one comes to the p a g e s of his autobiograp h y with high expectations. These ex-