Smale was born inFlint, Michigan and entered theUniversity of Michigan in 1948.[4][5] Initially, he was a good student, placing into an honorscalculus sequence taught byBob Thrall and earning himself A's. However, hissophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F innuclear physics. Smale obtained his Bachelor of Science degree in 1952. Despite his grades, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a graduate student. When the department chair,Hildebrandt, threatened to kick Smale out, he began to take his studies more seriously.[6] Smale finally earned hisPhD in 1957, underRaoul Bott, beginning his career as an instructor at theUniversity of Chicago.
Early in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher-dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio."[7][8] He has been politically active in various movements in the past, such as theFree Speech movement and member of theFair Play for Cuba Committee.[9] In 1966, having travelled to Moscow under anNSF grant to accept the Fields Medal, he held a press conference there to denounce theAmerican position in Vietnam,Soviet intervention in Hungary and Soviet maltreatment of intellectuals. After his return to the US, he was unable to renew the grant.[10] At one time he wassubpoenaed[11] by theHouse Un-American Activities Committee.
In 1960, Smale received aSloan Research Fellowship and was appointed to theBerkeley mathematics faculty, moving to a professorship atColumbia the following year. In 1964 he returned to a professorship at Berkeley, where he has spent the main part of his career. He became a professor emeritus at Berkeley in 1995 and took up a post as professor at theCity University of Hong Kong. He also amassed over the years one of the finest private mineral collections in existence. Many of Smale's mineral specimens can be seen in the bookThe Smale Collection: Beauty in Natural Crystals.[12]
In another early work, he studied theimmersions of the two-dimensional sphere into Euclidean space.[19] By relating immersion theory to thealgebraic topology ofStiefel manifolds, he was able to fully clarify when two immersions can be deformed into one another through a family of immersions. Directly from his results it followed that the standard immersion of the sphere into three-dimensional space can be deformed (through immersions) into its negation, which is now known assphere eversion. He also extended his results to higher-dimensional spheres,[20] and his doctoral studentMorris Hirsch extended his work to immersions of generalsmooth manifolds.[21] Along withJohn Nash's work onisometric immersions, the Hirsch–Smale immersion theory was highly influential inMikhael Gromov's early work on development of theh-principle, which abstracted and applied their ideas to contexts other than that of immersions.[22]
In the study ofdynamical systems, Smale introduced what is now known as aMorse–Smale system.[23] For these dynamical systems, Smale was able to proveMorse inequalities relating thecohomology of the underlying space to the dimensions of the(un)stable manifolds. Part of the significance of these results is from Smale's theorem asserting that thegradient flow of anyMorse function can be arbitrarily well approximated by a Morse–Smale system without closed orbits.[24] Using these tools, Smale was able to constructself-indexing Morse functions, where the value of the function equals itsMorse index at any critical point.[25] Using these self-indexing Morse functions as a key tool, Smale resolved thegeneralized Poincaré conjecture in every dimension greater than four.[26] Building on these works, he also established the more powerfulh-cobordism theorem the following year, together with the full classification ofsimply-connected smooth five-dimensional manifolds.[27][25]
Smale also introduced thehorseshoe map, inspiring much subsequent research. He also outlined a research program carried out by many others. Smale is also known for injectingMorse theory into mathematicaleconomics, as well as recent explorations of various theories ofcomputation.
