Problems in Homotopy, K-theory, and Representation Theory

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DMS-0100710Nicholas J. KuhnProfessor Kuhn has a long record of developing methods to solve interesting problems in homotopy theory, K-theory, and representation theory. The goal of the largest part of this project is to connect new 'polynomial' resolutions of classic function spaces to other things: model categories of structured spectra, equivariant stable homotopy, classial loop space theory, and periodic homotopy. These connections will be used to calculate previously inaccessible cohomological invariants of both such function spaces and infinite loop spaces. A second part of the work is a continuation of the study of the generic representation theory of finite fields. In particular, from his homotopical/K-theoretic point of view, he is studying his newly discovered lattices of generalized Schur algebras, with an eye towards gaining insight about the modular representations of the finite permutation groups, and the finite general linear matrix groups.Homotopy theory, K-theory, and representation theory are mathematical subjects in which one is trying to discover, and ultimately classify, fundamental building blocks of various sorts of mathematical structure. (This is quite analogous to a chemist studying simple molecular configurations, and how these can be assembled in more complex ways.) Homotopy is concerned with deformations of geometric objects such as higher dimensional surfaces. A sample hard problem is to understand the `shape' of all continuous functions from a sphere (surface of a ball) to itself. Representation theory is concerned with the algebraic symmetries of more rigid and discrete objects such as configurations of lines and planes. A sample hard problem is to understand all the basic ways in which the set of n by n matrices of 0's and 1's can act on strings of 0's and 1's. Finally K-theory is a sophisticated hybrid of the two. Professor Kuhn is studying new connections relating these subjects, by developing and using a variety of state-of-the-art algebraic and homotopy theoretic tools.