Moduli of Pointed Curves and Relative Stable Maps

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The investigator proposes to use Kontsevich's space of "stable maps" (originally motivated by mathematical physics last decade) and related objects to tackle problems in a variety of fields. Traditionally, the space of stable maps has been studied using facts about the fundamental "moduli space of curves" defined by Deligne and Mumford. The investigator proposes to conversely study the moduli space of curves by studying maps from curves to varieties. Some of the proposed work will likely rely on Jun Li's recent extension of Kontsevich's work, the definition of a space of "relative stable maps" in the algebraic category. It has long been known that nodal algebraic curves are a powerful tool in algebraic geometry. They can be thought of as surfaces with holes (picture a ball, a donut, or a french cruller) with pairs of points "glued together". Earlier this decade, ideas from string theory in physics led to the introduction of "stable maps", parametrizing certain kinds of maps of nodal curves into another space. This development has proved to be incredibly fruitful, sparking advances in a variety of fields. The investigator's area of research is the use of these ideas in the field of algebraic geometry, in particular with applications to many other fields (such as enumerative geometry, arithmetic geometry, combinatorics, and physics).