Breaking through the obstacles of dimensions, he made great progress in the high-dimensional Poincare conjecture, and put forward the forerunner of chaos concept-Smale horseshoe. 1998, Smale listed 18 mathematical problems of 2 1 century, and followed the famous Hilbert mathematical problems of 1900 in spirit. Part of Smale's problem also comes from Hilbert's math problem. Problems include unresolved Jacobian conjecture and Riemann conjecture.

Another famous theorem of Smale is to prove the Smale paradox that a sphere can be everted. The paradox is that if the sphere passes through itself, the inner wall of the sphere can be turned outwards without wrinkling.