Computational/Combinatorial Considerations In Topology, Coxeter Groups, and Representation Theory

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ABSTRACTPrincipal Investigator: Billey, Sara Proposal Number: DMS - 0800978Institution: University of WashingtonTitle: Computational/Combinatorial Considerations In Topology, Coxeter Groups, and Representation TheoryThis proposal outlines an ambitious research program attacking problems in three areas: the affine Grassmannian, fundamentals of permutations, and representation theory inspired by complexity theory. The central theme is to facilitate computation and understanding in these areas through study of very specific posets, Weyl groups, generating functions, and partitions controlling the micro level structures which we often overlook on the first glimpse of the subjects.At the heart of algebraic combinatorics is the philosophy that every aspect of mathematics can be made more precise, more concrete and more computationally feasible by identifying key combinatorial structures. Borrowing a term from analysis, this proposal is about ``micro local mathematics''. All of the work proposed will have a broad impact in several areas of pure math and theoretical computer science. All of the proposed work will have a computational focus which will further develop computer proof techniques. All of the proposed work will have a human impact component. The PI has a strong track record of mentoring students at all levels and junior faculty. This grant will greatly enhance the vertically integrated research environment in the Combinatorics Group at the University of Washington and give the group the resources needed to attack these hard problems.