Figures Associated with Stephen Smale

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1 Figures Associated with Stephen Smale This file contains brief accounts on the careers of seven mathematicians mentioned in Chapter 18 in association with Stephen Smale Robert Thrall, Raoul Bott, Arnold Shapiro, Mauricio Peixoto, Elon Lima, Norman Levinson, and John Stallings. It also provides snippets on five of Smale s students Michael Shub, Nancy Kopell, Jacob Palis, John Franks, and John Guckenheimer. Robert Thrall ( ) was a pioneer in operations research. Educated entirely in his home state of Illinois, he received a bachelor s degree from Illinois College and a Ph.D. from the University of Illinois in 1937 (at age 22) with the dissertation, Metabelian groups and trilinear forms, supervised by Henry Brihana. Upon graduation, Thrall accepted a position at the University of Michigan, where he remained until During this time, he supervised eighteen doctoral students, including Walter Feit in He was introduced to operations research during World War II, when he developed mathematical models that dealt largely with the military problems of the day including a game theoretic approach to assessing the relative effectiveness of military forces, a game-theoretic model of air warfare, and the evaluation of the effectiveness of surveillance. He continued to be involved with military OR after the war. Thrall moved to Rice University in 1969 to head the department of mathematical sciences, which had separated from the general mathematics department. He chaired the department for nine years before receiving a joint appointment in Rice s graduate school of business. Thrall became very involved with operations research for the state of Texas, where he helped establish a major, physics laboratory in the state. He retired in Overall, Thrall supervised five doctoral dissertations while at Rice. Stephen Smale s dissertation supervisor, Raoul Harry Bott ( ), was by far the most accomplished mathematician introduced in this file. A memorial article in the Notices summarized, Over a five-decade career he made many profound and fundamental contributions to geometry and topology. 1 Bott was born in Budapest but raised in Slovakia until 1939, when he fled to England (his mother was Jewish). He emigrated to Canada two years later, and entered McGill University, which he attended Upon graduation with a degree in electrical engineering, he enlisted in the Canadian army but the war ended so, just four months

2 later, he returned to McGill and wanted to pursue a doctorate in mathematics, but was advised to get an undergraduate degree in the field first: 2 Sensing his disappointment, Professor Williams of McGill then suggested Carnegie Tech (now Carnegie Mellon University) to Bott, where John Synge was just forming a new graduate program. Like Smale, Bott did not distinguish himself in school, yet was accepted into the new graduate program at Carnegie Mellon, where he quickly wrote a dissertation, Electrical network theory, supervised by Richard Duffin. Their resulting joint paper established the Bott-Duffin theorem, which, although it came to be well known among electrical engineers, it was Bott s last contact with that profession. 3 Upon receiving his Ph.D. in 1949, Bott was invited to the Institute for Advanced Study by Hermann Weyl, who had become acquainted with the Bott-Duffin paper. Raoul Bott spent and, later, , at the Institute, which introduced him to an entirely new world of mathematics, notably closed geodesics on Riemannian manifolds. From the IAS, Bott moved to Michigan in 1951, and remained there eight years, playing a fundamental role in bridging the gap between Lie groups and topology, which culminated with the periodicity theorem, published by the Proceedings of the NAS in In the meantime, he supervised his only doctoral student at Michigan Stephen Smale. Michael Atiyah observed: 4 His reputation was made, and shortly afterward (in 1960) he moved to Harvard, where he remained for the rest of his life. At Harvard, Bott wrote several notable papers with Atiyah, including contributions to index theory, as well as singular works on equivariant cohomology. His last significant paper (1989) proved Edwin Witten s rigidity theorem from two years earlier. The accompanying photograph of Bott was snapped by Paul Halmos in April 1980 at Indiana University. Like Smale, Bott attracted a cadre of talented students because of his large supply of feasible dissertation problems, but, unlike Smale, due to his warm and engaging personality. Other than Smale, he supervised thirty-four doctoral students, all at Harvard except two at MIT. Among them were Fields Medalist Daniel Quillen, noted historian of mathematics Harold Edwards, and Robert MacPherson. Arnold Samuel Shapiro ( ), whose alternative proof of Smale s sphere eversion was popularized in a Scientific American article, was born and died in Newton (MA). He

3 attended the University of Michigan, where the thesis he wrote for his master s degree was published in the Annals. He wrote: 5 This paper represents work done for a Master s thesis at the University of Michigan under Prof. N. E. Steenrod, whose guidance and encouragement were invaluable Shapiro then enrolled in the graduate program at Chicago, where he earned his Ph.D. in 1950 based on the dissertation, Cohomology relations in fiber bundles, supervised by André Weil. Due to Weil s influence, the major results were announced in French in the Comptes Rendus, but other than that, the dissertation was never published. 6 Arnold Shapiro spent at the Institute for Advanced Study along with Smale s advisor, Raoul Bott, which might account for Shapiro s early knowledge of the famous sphere inversion. Bott later referred to Shapiro as his personal remedial tutor. 7 Shapiro did not publish extensively, but his work was generally received positively. A Bulletin paper written with Henry Whitehead formulated an extension of Dehn s lemma based on a method of Christos Papakyriakopoulos, introduced above with Milnor. 8 In 1960, Shapiro contributed a paper to the Bourbaki Seminar, Algèbres de Clifford et periodicité des groupes πk(bo). Four years later, three after he had died of leukemia at age 40, Shapiro was named as a coauthor of a paper in the fledgling journal Topology in which the other two authors, wrote: 9 This paper developed in part from an earlier version by [Bott and Shapiro]. It is presented here, in its revised form, by [Atiyah and Bott] in memory of their friend and collaborator ARNOLD SHAPIRO. Atiyah later recalled, This [paper] has since proved useful in index theory. 10 Mauricio Matos Peixoto (b. 1921) was born in Brazil, and graduated from the National School of Engineering at the University of Brazil in 1943 with a degree in Civil Engineering. Upon graduation, he joined the faculty of his alma mater, where his research turned to structural stability. He and his wife, Marilea Chavez Peixoto, also a mathematician, began conducting mathematical seminars, which were unusual at engineering schools in Brazil at that time. Mauricio Peixoto visited the U.S. for the first time in 1949, when he spent eighteen months at the University of Chicago. In 1953, he and Brazilian mathematician Leopoldo Nachbin ( ) founded the Instituto de Mathemática Pura e Applicada (IMPA) in Rio de Janeiro. Peixoto next came to the U.S. when he spent the year studying with Solomon Lefschetz at Princeton. He overlapped with Smale for the summer of 1958, when Smale arrived at the Institute, and introduced him to structural stability of dynamical systems.

4 After the year at Princeton, Peixoto returned to IMPA, where he remained until he held professorships at Brown ( ) and the University of Sao Paulo ( ). Other than these two positions, he stayed at IMPA until retiring in Altogether he supervised six doctoral students, half at IMPA and the other half at Brown. A 2001 article on Peixoto by his former IMPA student Jorje Sotomayor (Ph.D. 1964) ended: 11 One could inquire how from the modest and rough formal mathematical training of engineers offered in Brazil at his time. Mauricio developed the geometric vision, penetration, sophistication, and esthetic touch that is the mark of his mathematical work. The fact is that he belongs to the generation of self-taught Brazilian scientists the leaders, a class of talented special spirits, living ahead of their time, who themselves defined their training, opening their own paths and pointing to new directions. Elon Lages Lima ( ) was a Brazilian mathematician who became an influential figure in the development of mathematics in Brazil. He received a bachelor s degree in mathematics from the University of Brazil in He then enrolled in the graduate program at the University of Chicago, where he earned a Ph.D. five years later based on the dissertation, Duality and Postnikov invariants, supervised by the topologist Edwin Spanier. From Chicago, he spent a year at the Institute for Advanced Study on a Guggenheim fellowship. After that, he returned to Brazil, where he spent a lifetime of research in differential topology, algebraic topology, and differential geometry. He wrote over thirty books on mathematics, most aimed for secondary teachers. Norman Levinson ( ) was responsible for informing Stephen Smale of an error in one of his papers that inspired Smale to discover the horseshoe. Called the department godfather 12 at MIT, Levinson spent almost his entire career at MIT. As an undergraduate student , he elected about twenty graduate courses even though he majored in electrical, the field he earned his BS and MS degrees in that field. However, Norbert Wiener took Levinson under his wing during his graduate course Fourier series and integrals, and gave him a copy of the unpublished manuscript, Fourier Transform in the Complex Domain. Then a dramatic event dramatically changed the arc of Levinson s career: 13 When Levinson found a gap in a proof and was able to prove a lemma that corrected it, Wiener typed the proof, affixed Levinson s name to the paper, and submitted it to a journal for him. Wiener and department head Henry Phillips then arranged for a one-year fellowship to study at Cambridge with G.H. Hardy. Although Levinson s lifetime habit of working all night and sleeping in daytime prevented him from attending Hardy s lectures, he obtained class notes from

5 a fellow student and communicated with the Cambridge don in that manner. Moreover, Levinson submitted two papers and had begun two others during his year abroad. He had been assured beforehand that he would be granted a Ph.D. when he returned and that indeed did occur, based on the dissertation On the non-vanishing of a function, supervised by Wiener. Norman Levinson was then awarded a two-year NRC fellowship to study at the Institute for Advanced Study with John von Neuman. However, he was released early from the fellowship in February to assume the courses at MIT taught by Jesse Douglas, whose nervous breakdown caused him to be hospitalized. Initially, MIT provost Vannevar Bush resisted appointing another Jew to the department, but Levinson s wife recounted the following dialog between Bush and Hardy, who was visiting the U.S. at the time: Hardy: What a marvelous theological institution this is! Bush: It s not a theological institution! Hardy: Then why don t you hire Levinson? Hardy s ploy worked, and Normal Levinson served on the MIT faculty from 1937 until his death thirty-eight years later. Over this time, Levinson contributed to Paley-Wiener theory, probability, quantum mechanics, complex programming, and number theory. He supervised thirty-four doctoral students, including my former colleague Seymour Haber Chapter 18 noted a contentious priority dispute between Stephen Smale and John Robert Stallings, Jr. ( ) over the proof of the generalized Poincaré Conjecture of dimension at least seven. Stallings was born and raised in Arkansas, graduating from the University of Arkansas in He then enrolled in the graduate program at Princeton, where he received his Ph.D. in 1959 for the dissertation Some topological proofs and extensions of Grushko s Theorem, supervised by Ralph Fox. After a one-year postdoctoral fellowship, Stallings returned to Princeton, this time on the faculty. He moved to Berkeley I 1967, based on his proof of the Poincaré Conjecture in dimension 5 for piecewise linear manifolds. He remained at Berkeley until taking early retirement in 1994, the same time as Smale. Overall, Stallings supervised twenty doctoral students, all but four at Berkeley. Michael Ira Shub (b. 1943) earned his Berkeley Ph.D. in 1967 with the dissertation, Endomorphisms of compact differentiable manifolds, supervised by Stephen Smale. Shub taught at Brandeis, California at Santa Cruz, and Queens College From then until 2004, he worked at IBM s Thomas J. Watson Research Center. He then taught at Toronto 2004-

6 2010, and has been at the University of Buenos Aires and CUNY since then. He has supervised nine doctoral students. Nancy Jane Kopell (b. 1942) earned her Berkeley Ph.D. in 1967 with the dissertation, Commuting diffeomorphisms, supervised by Stephen Smale. Born in the Bronx, she graduated from Cornell in 1963 before being awarded a fellowship to attend Berkeley, where she struggled with one dissertation advisor before switching to Smale. Upon graduation, Kopell taught at MIT for two years before moving to Northeastern University, where she remained She then moved to Boston University, where she met her husband Gabriel Stolzenberg. She worked on various topics in applied mathematics after leaving graduate school. She has supervised nine doctoral students at Boston University The Brazilian mathematician Jacob Palis, Jr. (b. 1940) earned his Berkeley Ph.D. in 1968 with the dissertation, On Morse-Smale diffeomorphisms, supervised by Stephen Smale. Born to immigrant parents from Lebanon and Syria, Palis graduated from the University of Brazil (now UFRJ) in He enrolled at Berkeley two years later. Upon receiving his Ph.D., he returned to Brail as a researcher at the Instituto Nacional de Mathemática Pura e Aplicada (IMPA), and has remained there since then, supervising forty-two doctoral students from 1972 to John Moore Franks (b. 19**) earned his Berkeley Ph.D. in 1968 with the dissertation, Anosov diffeomorphisms, supervised by Stephen Smale. Two years after graduation, Franks joined the faculty at Northwestern, where he has remained ever since, publishing over eighty papers. He has supervised twenty-six doctoral students from 1975 through 2013, including Steven Batterson in John Mark Guckenheimer (b. 1945), graduated from Harvard in 1966 before enrolling at Berkeley, where he earned his Ph.D. in 1970 with the dissertation, Endomorphisms of the Riemann sphere, supervised by Stephen Smale. Upon graduation, he taught at IMPA, the University of Warwick, and MIT, sandwiched around a two-year as a visitor at the Institute for Advanced Study, before joining the faculty California-Santa Cruz, where he remained for twelve years, He then moved to Cornell, where he has taught ever since. Guckenheimer has published papers in neuroscience, algorithms for periodic orbits, and dynamics in systems with multiple time scales. He was a co-recipient of the 2013 AMS Steele Prize for mathematical exposition for the book Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector

7 Fields (1983; revised 1990., and served as president of SIAM Guckenheimer has supervised twenty-four doctoral students, all but two at Cornell. Endnotes: 1 On p. 398 of Loring W. Tu (coord. ed.), Remembering Raoul Bott ( ), Notices Amer. Math. Soc. 60 (2013), On p. 556 of Loring W. Tu, The life and works of Raoul Bott, Notices Amer. Math. Soc. 53 (2006), On p. 4 of Sir Michael Atiyah, Raoul Harry Bott: September 24, December 20, 2005, Biographical Memoirs Nat. Acad. Sci. Available online at 4 Ibid., p Footnote on p. 581 of Arnold Shapiro, Group extensions of compact Lie groups, Annals Math. (2) 50 (1949), Arnold Shapiro, Cohomologie dans les espaces fibrés, Comptes Rendus Acad. Sci. Paris 231 (1950), On p. 532 of Raoul Bott, Topological constraints on analysis, in Duren Century II, pp Arnold Shapiro and J.H.C. Whitehead, A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), On p. 3 of M.F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964, suppl. 1), On p. 7 of Atiyah, Raoul Harry Bott. [Endnote 2.] 11 J. Sotomayor, Introduction: A few words about Mauricio M. Peixoto on his 80 th birthday, available online at 12 On p. ix of Joel Segel, Introduction, in Joel Segel (ed.), Recountings: Conversations with MIT Mathematicians, Natick (MA): A K Peters, 2009, pp. ix-xiii. 13 On p. 545 of Enders A. Robinson, Levinson, Norman, in Charles Coulston Gillispie (ed.-in-chief), Dictionary of Scientific Biography, Vol. 18, 1980, pp