Geometry of Moduli Spaces, Arithmetic Quotients and Theta Divisors

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This project addresses the geometry of moduli spaces that parameterize algebraic and geometric objects. The focus will be on spaces where there are several natural approaches to the moduli problem, particularly via geometric invariant theory, moduli of pairs, and Hodge theory. The former two approaches provide a method where the geometry of the objects parameterized plays a central role. The latter approach can allow for the use of arithmetic methods, including modular forms. Specific problems to be considered include giving modular interpretations to boundary loci of arithmetic quotients, describing log-canonical models of the moduli space of curves, and investigating the local structure of compactified Jacobians. Abelian varieties and theta divisors will also be of special interest.Algebraic geometry is the field of mathematics that focuses on the solution sets of polynomial equations. Inasmuch as polynomial equations are ubiquitous, the subject sits at the crossroads of many different fields including complex geometry, number theory and theoretical physics. A motivating question for algebraic geometers has been to classify solution sets by their invariants. For instance, one could try to classify those complex valued solution sets that can be identified with a torus (the surface of a donut); this would correspond to fixing the invariants known as the (complex) dimension and genus equal to the number one. Often the collection of solution sets with fixed invariants can itself be viewed naturally as a solution set of polynomial equations. These are known as moduli spaces, and their algebraic and geometric properties yield a tremendous amount of information about the original solution sets of interest. The PI intends to study a number of such spaces. The proposed research will have a significant impact on a field that plays a central role in mathematics and interacts with numerous other fields.