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.