Beyond Geometric Invariant Theory

In many branches of mathematics one encounters sets of equations with symmetries -- transformations that take any solution of the equations to another solution. In this situation, it is often very useful to classify all solutions up to the action of these symmetries, two solutions considered equivalent if they are related by a symmetry transformation. Within the field of algebraic geometry, the theory known as geometric invariant theory provides a very good answer to this classification question. There is reason to suspect that the methods of geometric invariant theory extend to a much broader context. In particle physics, the universe is described as a set of solutions of some equations up to the action of a very large set of symmetries. This project aims to broaden the methods and results of geometric invariant theory and bring them to bear on a large set of classification problems in algebraic geometry, including some of those studied in high energy physics.

This project envisions a new general approach to moduli problems in algebraic geometry. The main technical tool is a special kind of stratification on an algebraic stack called a theta stratification. Theta stratifications are a common generalization of the Kempf-Ness stratification studied in geometric invariant theory and the Harder-Narasimhan stratification of the moduli space of vector bundles on a curve. Classically these stratifications were used, in the case of smooth stacks, to study the Betti numbers of the moduli stack of semistable objects. The project will pursue several generalizations of these results to situations where the stack is not necessarily smooth, and to extracting more subtle topological information about the stack. Examples include wall-crossing formulas for integrals of tautological K-theory classes, and more generally extracting information about the derived category of coherent sheaves on the stack and the semistable locus.