Moduli Spaces, Tautological Rings, and Theta Functions

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The project addresses questions in algebraic geometry. Algebraic geometry concerns the study of solution sets of polynomial equations several variables. It is one of the oldest, most developed and active areas of research in mathematics. It has connections to many sub-fields within mathematics, both pure and applied, and also to computer science and theoretical physics. Moduli theory concerns the behavior of geometric shapes as the coefficients of the defining equations are allowed to vary. The focus of this project is the moduli of complex surfaces. These spaces have been studied abstractly in the past. This project will further study these spaces by understanding the set of natural loci inside the moduli space of surfaces that arise in geometric computations, as well as its structure. In particular, combining tools from classical algebraic geometry as well as theoretical physics, the project aims to develop a flexible machinery to work quantitatively with these loci, and to further apply this machinery to answer questions arising in the enumerative geometry of surfaces (that is, questions of the type: enumerate all surfaces that enjoy certain geometric properties). In addition to research, the principal investigator will be involved in high school mathematics competitions, will advise undergraduate students, graduate students, and postdoctoral associates, work to increase diversity in mathematics, and organize seminars and local conferences. Specifically, the project concerns the study of the tautological ring of the moduli of surfaces (K3 surfaces, abelian surfaces, etc.). Some of the projects here concern obtaining structural results, understanding the size of the tautological rings, and developing an explicit calculus for tautological classes, expressions for natural geometric cycles in terms of tautological classes. This also leads to projects of independent interest, e.g., generalizations of a conjecture of Lehn concerning tautological integrals over the Hilbert scheme, a higher-rank stable pair correspondence, and connections with the Verlinde formula for new classes of surfaces. Over the moduli of curves, K3 or abelian surfaces, the investigator will continue work on bundles of generalized theta functions. Questions here deal with refined Chern character calculations and connections with the tautological rings. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.