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Stephen Smale: The Mathematician Who Broke the Dimension Barrier

Steve Batterson
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Shirley B. Gray
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This biography of Stephen Smale will bother many readers. For those of us with a memory of the 60s, the reaction is certain to be both pain and nostalgia. For mathematicians in particular there are multiple layers of messages. We are asked to review the launch of a stellar career in research mathematics projected upon the planes of McCarthy era communism, the Free Speech Movement, and the push to include more women mathematicians amongst our inner circle. We are asked to factor in the nature of our institutions, including the National Academy of Sciences, the National Science Foundation, the administration of the University of California at Berkeley, the legends surrounding the absence of a Nobel Prize in mathematics, and the Fields Medal selection process.

Batterson has done an outstanding job of collecting factual information on Smale's youth and education. He even attended his Grand Blanc Class of 1948 reunion. This information should be read by the rank and file of all undergraduate and graduate students in mathematics. Batterson uses Smale's youth to debunk the popular myth that one must demonstrate outstanding early talent to pursue mathematics research. Consider the following evidence:

  • "In retrospect, neither Smale nor anyone else was the mathematical genius."
  • "During Smale's high school experience, chess provided the only hint of future success."
  • "Smale failed to impress his peers with his precociousness."

However, as a Freshman at the University of Michigan, Smale had the luck to be put, after a placement exam, into a small class with an exceptional young instructor, Bob Thrall. Smale describes him as his "first memorable teacher." Still, as an undergrad in mathematics, his interest in his major was eclipsed by other interests, especially campus politics and travel. Even as late as the second semester of his graduate work, Smale dropped two courses and received a C in another. The chair, T. H. Hildebrandt, felt it necessary to notify Smale that to remain a graduate student he would have to improve his grades. Later, for Smale's first job search, Hildebrandt wrote that Smale was a "marginal, underachieving graduate student." Very few graduate students or faculty would accept the proposition that a truly outstanding research career could be built on such a tenuous mathematical background.

Although his thesis advisor, Raoul Bott, was not yet well known, he did provide Smale with an excellent problem: to classify, up to regular homotopy, regular closed curves in an arbitrary manifold. Smale was Bott's very first graduate student. Either through wisdom or innocence, but certainly not from experience, Bott fed him an accessible problem that laid the foundation for very creative work. In his first year as an Instructor at the University of Chicago, Smale proved the counter-intuitively famous theorem that one can turn a 2-sphere in R3 inside out. The whole area of immersion theory flourished. Topology was front and center stage in mathematics.

Very few in mathematics, science or engineering find open doors when seeking to migrate to peripheral subdisciplines. Smale opened his own doors. From topology he moved laterally to dynamical systems where he discovered his now famous "horseshoe" having stability amid chaos. A few months later, he attacked the classic Poincaré Conjecture, where he "broke the dimension barrier" by announcing his proof for dimensions greater than 4. But he also entered into a still contested battle over credit and recognition with another in the field, John Stallings. And in San Diego, Mike Freedman solved the 4-dimensional version of the problem. The original Poincaré Conjecture that every simply connected compact 3-manifold is homeomorphic to S3 is still unproved, and has replaced Fermat's Last Theorem as one of our Olympian quests.

Batterson provides the reader interested in knowing more about Smale's and others' work the helpful option of reviewing the mathematics isolated by topics. He presents four clearly written appendices: Smale's Thesis, Everting the Sphere, Chaos and Horseshoe, and The Higher Dimensional Poincaré Conjecture. This is most helpful to those not in the field or not wanting to delve into the extensive literature. A report or presentation based on any one of these sections would be an excellent project for an undergraduate student.

Smale received many honors for this work. Readers should wonder if there is any significance in his being awarded the National Medal of Science by Clinton, and not by earlier presidents. After all, the National Medal was conferred fully 30 years after the Fields. I would have enjoyed seeing a picture of the Fields ceremony in Moscow in 1966, but there is none. There is a photo of Smale at the White House with Clinton.

There are several gaps in the biographical material. Flint, Michigan, Smale's home, was a "Waspish world of autoworkers and farmers." Batterson asks throughout the biography how a future mathematician from Flint could successively become so involved in undergraduate politics at the University of Michigan, the House Committee on Un-American Activities, extensive worldwide travel, organizing the Vietnam Day Committee during the Free Speech Movement at Berkeley and challenging both the NAS and the NSF? Batterson should scrutinize his own sentence. The clues are all there. Farmers take major risks with every planting season. To see the American West — Yellowstone, Little Big Horn, the Pacific — was the dream of every young man of Smale's father's generation. Union activity and rebellion against the establishment (in the form of both management and government) following World War Two (more interest in travel) were intense in Michigan. And, as to his mathematics, Smale found a few good teachers and accessible problems as a young man that permitted him to make the giant leap to Fields Medal research.

Another disquieting gap emerges in the discussion of the Jenny Harrison case. Batterson, a Smale academic "grandson," and the AMS reviewer Rob Kirby, a Berkeley professor, essentially avoid these chapters in the history of Smale's life, the life of the Berkeley Department of Mathematics, and the history of the women's movement. This woman reviewer asks if it is balanced to write 110 pages on the Free Speech Movement and conflicts with the NSF and less than six pages on issues and problems that have been vigorously discussed in every mathematics department in the world. It is ever so much easier to criticize external institutions than to examine our own internal structure and flaws. Yet one loud message of the Free Speech Movement was that we should have the courage to address our own weaknesses.

The final gap in the narrative is a consequence of the publication date. Smale took early retirement at Berkeley and is now at the City University of Hong Kong. He continues to be quite active. I had the pleasure of a short visit with him this past October. His large office is located amid an ultramodern grid complex of corridors and hallways stacked into a campus array that shares the new Hong Kong subway with a luxurious shopping mall. This backdrop of many malls, The City University, the airport, suspension bridges, tunnels and hundreds of other construction projects stands in sharp contrast to his surroundings in Berkeley, or especially at the Institute for Advanced Study. These surroundings suggest "Think Future."

We discussed his presentation in Paris to the International Mathematical Union of his problems for the 21st century. [See Smale, Steve (1998). Mathematical Problems for the Next Century, Mathematical Intelligencer, 20, 7-15.] The organizers of the conference tried very hard to identify the exact room in Paris where Hilbert delivered his now famous address in 1900. Smale was not sure if the room was the same, but he was certain that he had a larger audience than Hilbert. Smale placed the 3-dimensional Poincaré Conjecture as Problem #2 behind the Riemann Hypothesis. I chided him for selecting only 19 problems, when Hilbert had chosen 23, Euler had calculated pi to 23 places, and the oldest editions of Euclid had 23 definitions.

In addition to hosting a 70th birthday celebration, Smale is now editing a new journal in yet another field, computational mathematics.

In closing, one notes Batterson repeats information at several junctures in the narrative. This has the disquieting effect of making a reader question if Batterson forgot what he had written in earlier sections. Perhaps he was, like a good teacher, repeating for emphasis. He also makes a jump from a detailed example selected from earliest algebra to a topological equivalence, i.e., homeomophism. There are so many missing links, this example and others probably should not have been used.

Batterson deserves thanks for his efforts. Read his book. You are certain to react. Batterson's insight on Smale's personality is a wonderful suggestion for all in research: "He possessed a quiet confidence in his ability and a willingness to accept carefully chosen risks."

Shirley B. Gray ( teaches History of Mathematics at California State University, Los Angeles. She is married to a National Medal of Science winner, the chemist Harry B. Gray of Caltech.
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