Moduli Spaces in Algebraic Geometry

Stanford University, Stanford CA

Investigators

Abstract

Algebraic geometry provides a powerful theory with which to define moduli spaces (spaces of solutions of geometric problems) for interesting mathematical objects. Once they are defined, there are natural compelling questions about their geometry and topology. What do they look like? Are they irreducible, connected, of what dimension? Are they smooth? if not, what singularities arise? What structure is exhibited by their cohomology rings, and why should it be geometrically expected? What are their equations? The investigator intends to address many of these fundamental problems in a number of cases. The investigator has a track record of sustained and serious effort both in outreach to students at all levels (high school, undergraduate, and graduate), and in building institutions in which algebraic geometry can grow. The investigator will continue to attract graduate students into algebraic geometry and continue to nurture the careers of graduate students, post-docs, and young researchers. The investigator will also continue to work with large numbers of students at the secondary and undergraduate levels, attracting students into the mathematical sciences. The investigator works in algebraic geometry, although his interests connect to other areas of mathematics, including topology, combinatorics, physics (string theory), number theory, and symplectic and differential geometry. This proposal, continuing various strands of the investigator's work, deals with moduli spaces and related notions in a variety of settings. In particular, the proposal deals with a number of fundamental questions regarding the foundations of "tropical" geometry, the stabilization of moduli spaces in the Grothendieck ring, the study of K3 surfaces through elliptic fibrations, and the topology of various moduli spaces of curves.