CAREER: Algebraic Geometry of Moduli Spaces

William Marsh Rice University, Houston TX

Investigators

Abstract

The investigator and his colleagues will address a number of fundamental problems about the geometry and arithmetic of moduli spaces: What is the canonical model of the moduli space of curves of genus g? Can one construct natural spaces that interpolate between the canonical model and the moduli space of stable curves? Can these be interpretted as moduli spaces in their own right? Can one count the number of rational points of moduli spaces of curves, bounded with respect to various heights? What are the natural functions (effective divisors) one might use to specify these heights? What are the naturally defined strata in the deformation space of a plane curve singularity? How can one interpret blow-ups along such strata? Algebraic geometry is the study of the solutions to systems of polynomial equations. Geometric aspects of the solution sets translate into algebraic properties of the equations, and vice versa. This approach has the advantage that computers can manipulate equations very efficiently. Promising hypotheses can thus be checked on explicit examples. Increasingly, computational approaches to geometry are transforming the education of undergraduate and graduate students, both in pure mathematics and in related fields where mathematics is applied. This emphasis on concrete examples and explicit computation creates new research opportunities for young people, especially those who have only just started to learn the technical intricacies of the subject.