Editing Stephen Smale (section)

==Research==
Smale proved that the [[diffeomorphism group|oriented diffeomorphism group]] of the two-dimensional sphere has the same [[homotopy type]] as the [[special orthogonal group]] of {{math|3 × 3}} matrices.{{sfnm|1a1=Smale|1y=1959c}} Smale's theorem has been reproved and extended a few times, notably to higher dimensions in the form of the [[Smale conjecture]],<ref>{{cite journal|mr=0701256|last1=Hatcher|first1=Allen E.|author-link1=Allen Hatcher|title=A proof of the Smale conjecture, Diff(''S''<sup>3</sup>) ≃ O(4)|journal=[[Annals of Mathematics]]|series=Second Series|volume=117|year=1983|issue=3|pages=553–607|doi=10.2307/2007035|jstor=2007035 |zbl=0531.57028}}</ref> as well as to other topological types.<ref>{{cite journal|mr=0276999|last1=Earle|first1=Clifford J.|last2=Eells|first2=James|title=A fibre bundle description of Teichmüller theory|journal=[[Journal of Differential Geometry]]|volume=3|year=1969|issue=1–2|pages=19–43|doi=10.4310/jdg/1214428816|doi-access=free|author-link2=James Eells|author-link1=Clifford John Earle Jr.|zbl=0185.32901}}</ref>

In another early work, he studied the [[Immersion (mathematics)|immersion]]s of the two-dimensional sphere into Euclidean space.{{sfnm|1a1=Smale|1y=1959a}} By relating immersion theory to the [[algebraic topology]] of [[Stiefel manifold]]s, 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,{{sfnm|1a1=Smale|1y=1959b}} and his doctoral student [[Morris Hirsch]] extended his work to immersions of general [[smooth manifold]]s.<ref>{{cite journal|mr=0119214|last1=Hirsch|first1=Morris W.|author-link1=Morris Hirsch|title=Immersions of manifolds|journal=[[Transactions of the American Mathematical Society]]|volume=93|year=1959|pages=242–276|zbl=0113.17202|issue=2|doi=10.1090/S0002-9947-1959-0119214-4|doi-access=free}}</ref> Along with [[John Forbes Nash|John Nash]]'s work on [[isometric immersion]]s, the Hirsch–Smale immersion theory was highly influential in [[Mikhael Gromov (mathematician)|Mikhael Gromov]]'s early work on development of the [[h-principle]], which abstracted and applied their ideas to contexts other than that of immersions.<ref>{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete|Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge]]|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|doi=10.1007/978-3-662-02267-2|zbl=0651.53001|author-link1=Mikhael Gromov (mathematician)}}</ref>

In the study of [[dynamical system]]s, Smale introduced what is now known as a [[Morse–Smale system]].{{sfnm|1a1=Smale|1y=1960}} For these dynamical systems, Smale was able to prove [[Morse inequality|Morse inequalities]] relating the [[cohomology]] of the underlying space to the dimensions of the [[stable manifold|(un)stable manifold]]s. 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.{{sfnm|1a1=Smale|1y=1961a}} 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.<ref name="milnor">{{cite book|mr=0190942|last1=Milnor|first1=John|title=Lectures on the h-cobordism theorem|others=Notes by [[Laurent C. Siebenmann|L. Siebenmann]] and J. Sondow|author-link1=John Milnor|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1965|zbl=0161.20302|doi=10.1515/9781400878055|isbn=9781400878055 }}</ref> Using these self-indexing Morse functions as a key tool, Smale resolved the [[generalized Poincaré conjecture]] in every dimension greater than four.{{sfnm|1a1=Smale|1y=1961b}} Building on these works, he also established the more powerful [[h-cobordism|h-cobordism theorem]] the following year, together with the full classification of [[simply-connected]] smooth five-dimensional manifolds.{{sfnm|1a1=Smale|1y=1962a|2a1=Smale|2y=1962b}}<ref name="milnor" />

Smale also introduced the [[horseshoe map]], inspiring much subsequent research. He also outlined a research program carried out by many others. Smale is also known for injecting [[Morse theory]] into mathematical [[economics]], as well as recent explorations of various theories of [[computation]].

In 1998 he compiled a list of 18 problems in [[mathematics]] to be solved in the 21st century, known as [[Smale's problems]].{{sfnm|1a1=Smale|1y=1998|2a1=Smale|2y=2000}} This list was compiled in the spirit of [[David Hilbert|Hilbert]]'s [[Hilbert's problems|famous list of problems]] produced in 1900. In fact, Smale's list contains some of the original Hilbert problems, including the [[Riemann hypothesis]] and the second half of [[Hilbert's sixteenth problem]], both of which are still unsolved.  Other famous problems on his list include the [[Poincaré conjecture]] (now a theorem, proved by [[Grigori Perelman]]), the [[P = NP problem]], and the [[Navier–Stokes existence and smoothness|Navier–Stokes equations]], all of which have been designated [[Millennium Prize Problems]] by the [[Clay Mathematics Institute]].