Algebraic and Combinatorial Aspects of Generalized Hypergeometric Functions

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This project is centered around the study of A-hypergeometricfunctions. These have been introduced in the 80's by Gel'fandand his coworkers as solutions of a system of partialdifferential equations encoding combinatorial data of polytopes. Specific problems to be addressed in this project include: theclassification of rational hypergeometric functions and theirrelationship with toric residues; bounds for the holonomic rankand rational rank; hypergeometric functions arising as periods ofCalabi-Yau manifolds. Potential applications of this workinclude a deeper understanding of the properties of theA-discriminant and new algorithms for the computation of totalresidues -a rational expression on the coefficients of a systemof polynomial equations which is of considerable interest incomputational algebraic geometry- and, ultimately, thedevelopment of algorithms for solving polynomial equations. The study of polynomial equations is of fundamental importance onalmost all branches of science and technology. In the last fewyears it has become clear that the best approaches to theirsolution are those that combine numerical and symbolic methods. The work on this project attempts to answer some basic questionson the symbolic study of a class of partial differentialequations closely related to systems of polynomial equations withthe goal of developing new algorithms. This same system ofdifferential equations arise also in Mirror Symmetry, afundamental theory in high-energy physics, with remarkableconsequences in pure mathematics. The investigator hopes to beable to relate the symbolic approach to the classical algebraicgeometric (Hodge Theory) approach.