be seen in the book—The Smale Collection: Beauty in Natural Crystals. From 2003 to 2012, Smale was a professor at the Toyota Technological Institute at Chicago; starting August 1, 2009, he became a Distinguished University Professor at the City University of Hong Kong. In 1988, Smale was the recipient of the Chauvenet Prize of the MAA. In 2007, Smale was awarded the Wolf Prize in mathematics. Research Smale proved that the oriented diffeomorphism group of the two-dimensional sphere has the same homotopy type as the special orthogonal group of matrices. Smale's theorem has been reproved and extended a few times, notably to higher dimensions in the form of the Smale conjecture, as well as to other topological types. In another early work, he studied the immersions of the two-dimensional sphere into Euclidean space. By relating immersion theory to the algebraic topology of Stiefel 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 as sphere eversion. He also extended his results to higher-dimensional spheres, and his doctoral student Morris Hirsch extended his work to immersions of general smooth manifolds. Along with John Nash's work on isometric immersions, the Hirsch–Smale immersion theory was highly influential in Mikhael Gromov's early work on development of the h-principle, which abstracted and applied their ideas to contexts other than that of immersions. In the study of dynamical systems, Smale introduced what is now known as a Morse–Smale system. 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. Using these tools, Smale was able to construct self-indexing Morse functions, where the value of the function equals its Morse index at any critical point. Using these self-indexing Morse functions as a key tool, Smale resolved the generalized Poincaré conjecture in every dimension greater than four. 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.