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Hypergeometric systems after Gelfand-Kapranov-Zelevinsky and their cohomological properties
Hypergeometric functions have been studied at least since the times of Gauss. An old idea is to investigate the differential equations that are satisfied by the hypergeometric functions since the combinatorial structure allows to obtain recursion formulae for series representations.
In this talk we consider systems of partial differential equations H_A(b) that arise through the action of a torus on complex n-space via a matrix A, and depend on a complex parameter b. These systems were introduced by Gelfand, Kapranov and Zelevinsky in the 1980's and give rise to A-hypergeometric functions as their solutions.
Varying b can be seen as deforming the solutions of H_A(b) on a Zariski open set in the parameter space. However, for special b the number of local solutions for H_A(b) changes with b. An old question is the determination of the set of exceptional parameters b that cause H_A(b) to have "unusually" many solutions. We discuss a conjecture of Matusevich and Miller that relates the set of exceptional parameters to the local cohomology groups of the toric variety defined by A, as well as recent progress.
Tuesday, January 13, 2004, 245 Altgeld Hall, 2:00 p.m.
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