New Problems in Homotopy, K-Theory, and Representation Theory

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9802493Kuhn Professor Kuhn has a long record of developing both homotopy- theoretic and representation-theoretic methods to solve problems in both areas. More recently, some of these methods have led him also in algebraic K-theoretic directions, and his research and techniques are being used extensively by others. The largest part of this project is directed towards connecting classic loopspace theory with the more recent developments of `S-module' and `Goodwillie calculus' technology, with the focus of these efforts aimed towards establishing new conjectures concerning well known and much studied topological objects: Eilenberg-MacLane spaces, Hopf invariants, and combinatorial function space models. A second part of the work is part of the rapid development being made now in algebraic K-theory and generic representation theory over finite fields by Professor Kuhn, K-theorists Eric Friedlander and Andrei Suslin, and their collaborators. Finally, Professor Kuhn and his students are continuing work on topological realization problems, using both new homotopy theoretic and algebraic techniques, following the direction of earlier work by Professor Kuhn and Lionel Schwartz of Paris. Homotopy theory, K-theory, and representation theory aremathematical subjects in which one is trying to discover, andultimately classify, fundamental ``building blocks'' of various sortsof mathematical structure. (This is quite analogous to a chemist'sstudying simple molecular configurations, and how these can beassembled in more complex ways.) Homotopy is concerned withdeformations of geometric objects such as higher dimensional surfaces.Representation theory is concerned with the algebraic symmetries ofmore rigid and discrete objects such as configurations of lines andplanes. Finally K-theory is a hybrid of the two. Professor Kuhn isstudying intriguing new connections relating these subjects, using avariety of state-of-the-art algebraic and homotopy theoretic tools,many developed by himself.***