Mathematical Sciences: Finite Groups and Complex Oriented   Theories in Homotopy
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Group theoretic constructions have traditionally been part of              the arsenal of an algebraic topologist .  On the most basic level,              tools from (finite) group theory have been used to construct                    topological tools:  Steenrod squares, characteristic classes,                   spaces with A(n)-free cohomology, etc.  On a more subtle level,                 many basic topological structures seem to be determined by group                theoretic structures - for example, infinite loop space structures              are determined by the combinatorics of the symmetric groups, and                the theorem of Atiyah that calculates K(BG), says that group theory             determines this ring.                                                                Nicholas J. Kuhn's recent and current research continues this              tradition of using group theory to guide one's study of                         topological questions.  In particular, in his ongoing                           collaboration with M. Hopkins and D. Ravenel, a combination of                  group theory, equivariant homotopy, and number theory is being                  used to gain greater understanding of the "geometry" underlying                 complex oriented cohomology theories.  These include, in                        particular, the Morava K-theories - known by the work of E.                     Devinatz, Hopkins, and J. Smith to be of central importance in                  stable homotopy theory.
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{'Code': '126700', 'Text': 'TOPOLOGY'}
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