Algebraic and Combinatorial Aspects of Generalized Hypergeometric Functions

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

This project is centered around the study of A-hypergeometric functions. These have been introduced in the 80's by Gel'fand and his coworkers as solutions of a system of partial differential equations encoding combinatorial data of polytopes. Specific problems to be addressed in this project include: the classification of rational hypergeometric functions and their relationship with toric residues; bounds for the holonomic rank and rational rank; hypergeometric functions arising as periods of Calabi-Yau manifolds. Potential applications of this work include a deeper understanding of the properties of the A-discriminant and new algorithms for the computation of total residues -a rational expression on the coefficients of a system of polynomial equations which is of considerable interest in computational algebraic geometry- and, ultimately, the development of algorithms for solving polynomial equations. The study of polynomial equations is of fundamental importance on almost all branches of science and technology. In the last few years it has become clear that the best approaches to their solution are those that combine numerical and symbolic methods. The work on this project attempts to answer some basic questions on the symbolic study of a class of partial differential equations closely related to systems of polynomial equations with the goal of developing new algorithms. This same system of differential equations arise also in Mirror Symmetry, a fundamental theory in high-energy physics, with remarkable consequences in pure mathematics. The investigator hopes to be able to relate the symbolic approach to the classical algebraic geometric (Hodge Theory) approach.