Editing John Milnor (section)

==Research==
One of Milnor's best-known works is his proof in 1956 of the existence of [[Seven-dimensional space|7-dimensional]] [[Hypersphere|spheres]] with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term [[exotic sphere]], referring to any ''n''-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres by [[Kervaire–Milnor group]]s, showing in particular that the 7-sphere has 15 distinct [[differentiable structure]]s (28 if one considers orientation).

[[Egbert Brieskorn]] found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a [[Singular point of a curve|singular point]] is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on the [[topology]] of isolated [[Singular point of a curve|singular points]] of complex hypersurfaces in general, developing the theory of the [[Milnor fibration]] whose fiber has the [[homotopy]] type of a bouquet of ''μ'' spheres where ''μ'' is known as the [[Milnor number]]. Milnor's 1968 book on his theory, ''Singular Points of Complex Hypersurfaces'', inspired the growth of a huge and rich research area that continues to mature to this day.

In 1961 Milnor disproved the [[Hauptvermutung]] by illustrating two [[simplicial complex]]es that are [[Homeomorphism|homeomorphic]] but [[Combinatorial topology|combinatorially]] distinct, using the concept of [[analytic torsion|Reidemeister torsion]].<ref>{{cite book
 | last = Ranicki | first = A. A.
 | editor1-last = Ranicki | editor1-first = A. A.
 | editor2-last = Casson | editor2-first = A. J.
 | editor3-last = Sullivan | editor3-first = D. P.
 | editor4-last = Armstrong | editor4-first = M. A.
 | editor5-last = Rourke | editor5-first = C. P.
 | editor6-last = Cooke | editor6-first = G. E.
 | contribution = On the Hauptvermutung
 | doi = 10.1007/978-94-017-3343-4_1
 | isbn = 0-7923-4174-0
 | mr = 1434101
 | pages = 3–31
 | publisher = Kluwer Academic Publishers, Dordrecht
 | series = {{mvar|K}}-Monographs in Mathematics
 | title = The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds
 | volume = 1
 | year = 1996}} See pp. 3-4</ref>

In 1984 Milnor introduced a definition of [[attractor]].<ref>{{Cite journal|last=Milnor|first=John|date=1985|title=On the concept of attractor|url=https://projecteuclid.org/euclid.cmp/1103942677|journal=Communications in Mathematical Physics|language=EN|volume=99|issue=2|pages=177–195|issn=0010-3616|doi=10.1007/BF01212280|bibcode=1985CMaPh..99..177M|s2cid=120688149}}</ref>  The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors.

Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of ''Topological Methods in Modern Mathematics'':

<blockquote>It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with [[William Thurston|Thurston]]. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with [[Henri Poincaré|Poincaré's]] work on [[Diffeomorphism#Topology|circle diffeomorphisms]], which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.<ref name="John Milnor's Sixtieth Birthday Symposium">{{cite book |author-last=Lyubich | author-first=Mikhail |chapter=Back to the origin: Milnor’s program in dynamics | editor-last=Goldberg | editor-first=Lisa R. | editor-last2=Phillips | editor-first2=Anthony Valiant | title=Topological Methods in Modern Mathematics: A Symposium in Honor of John Milnor's Sixtieth Birthday | publisher=Publish or Perish | date=1993 | isbn=0-914098-26-8 | pages=85–92}}</ref> </blockquote>

His other significant contributions include [[microbundle]]s, influencing the usage of [[Hopf algebra]]s, theory of [[quadratic forms]] and the related area of [[bilinear form|symmetric bilinear forms]], higher [[algebraic K-theory]], [[game theory]], and three-dimensional [[Lie group]]s.