Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics.

[6] Smale finally earned his PhD in 1957, under Raoul Bott, beginning his career as an instructor at the University 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.

In 1960, Smale received a Sloan Research Fellowship and was appointed to the Berkeley mathematics faculty, moving to a professorship at Columbia the following year.

[23] For these dynamical systems, Smale was able to prove Morse inequalities relating the cohomology 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 the gradient flow of any Morse function can be arbitrarily well approximated by a Morse–Smale system without closed orbits.

[25] Using these self-indexing Morse functions as a key tool, Smale resolved the generalized Poincaré conjecture in every dimension greater than four.

[26] Building on these works, he also established the more powerful h-cobordism theorem the following year, together with the full classification of simply-connected smooth five-dimensional manifolds.