This proposal supports the research in algebraic geometry of Professor Uli Walther of Purdue University. The main focus of this proposal is the study of singularities. To a singularity (defined by the vanishing of a set of polynomials) one may attach several invariants; these may be of discrete type (such as the number of branches of a curve meeting in a point) or of continuous nature (such as the space of all vector fields tangent to the singularity). If one considers a family of singularities, such invariants behave in interesting ways: on one side, at special members of the family they "jump" (i.e., get larger in some sense), while on the other side, near typical members of the family, they deform according to so-called "hypergeometric" differential equations. One component of this project investigates, using homological and combinatorial methods, jumps and solutions of the hypergeometric differential equations. The other part of the proposal is concerned with the study of specific invariants of singularities derived through either calculus (the Gauss--Manin connection and Bernstein--Sato polynomial), counting techniques (the Igusa zeta function), or deformations (cohomology of the Milnor fiber), and their interplay.
This research falls into the broad category of algebraic geometry, one of the most variegated areas of today's mathematics. Fundamentally, algebraic geometry is the study of geometric objects described by algebraic data through artful manipulation of the input data using an incredibly wide array of mathematical tools. Because of its diversity, algebraic geometry permeates such different branches of science as robotics, cosmology, and computer encryption. The origins of algebraic geometry can be traced to the works of Euclid and Pythagoras. In its modern form, the focus of algebraic geometry is on singularities. These include cusps, foldings, and self-intersections, and are generally comprised of points that are unusual when compared to their neighbors.
