Mathematical Sciences: Algebraic, Geometric and Combinatorial Structures Related to Multivariate Hypergeometric Functions

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Zelevinsky This award provides funding for a project that will mainly be conducted in the following two areas: (A) Multivariate discriminants, resultants, and hyperdeterminants. (B) Representations of quantum groups and piecewise- linear combinatorics. Both of these subjects have deep relations with the theory of multivariate hypergeometric functions. Another unifying feature is that in both areas an essential role is played by new kinds of combinatorics whose main objects of study are lattice points in convex cones and polytopes and their piecewise-linear transformations. The principal investigator is going to continue his work on the following topics: (1) explicit polynomial formulas for multivariate discriminants and resultants, (2) singular loci of discriminant varieties, (3) algebraic and combinatorial properties of canonical bases in representations of quantum groups, (4) geometry of totally positive matrices, (5) Chow varieties of 0-cycles and symmetric functions in several vector variables, (6) secondary polytopes and maximal minor polytopes. This research explores connections between Combinatorics, Algebra, and Geometry. One of the goals of Combinatorics is to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.