Mathematically, this research project falls into the broad category of algebraic geometry, one of the most varied areas of today's mathematics. Fundamentally, algebraic geometry is the study and classification of geometric objects described by algebraic equations through manipulation of the input data using a 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, which are points that are unusual when compared to their neighbors. Examples of singularities include cusps such as the tip of a funnel cloud, or self-intersections such as the center in a figure-of-eight; they indicate states in which a given physical system becomes anomalous. This project also contributes to the training of the next generation of researchers by engaging graduate as well as undergraduate students in research.
The main focus of this project is the study of singularities through cohomological methods. To a singularity defined by a set of polynomial equations 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" (that is, get larger in some sense), and such jumps are often accompanied by an appropriate nonzero local cohomology group. The singularities where jumps occur typically exhibit worse behavior than their neighbors. On the other side, near typical members of the family, the invariants often deform according to so-called "hypergeometric" differential equations. One component of this project investigates, using local cohomology and combinatorial methods, jumps and solutions of the appearing hypergeometric differential equations. The other part of the project is concerned with the study of specific invariants in families 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.
