Algebraic geometry of moduli spaces

Princeton University, Princeton NJ

Investigators

Abstract

Gromov-Witten theory is a rapidly expanding field with basic connections to many central areas of current research in mathematics and physics. The project proposed here is a wide ranging study of Gromov- Witten theory based on the techniques and discoveries of the last few years. The main topics covered are: the definition and exact evaluations of integrals on the moduli space of curves with boundaries, the proof of the universal Virasoro constraints, the establishment of the Gromov-Witten/ Donaldson-Thomas/Pairs correspondences, and the study of tautological classes. These topics point in several different directions: topological string theory, integrable hierarchies, and classical algebraic geometry. Each topic is central to progress in the field, and each will be addressed with a new point of view. Algebraic varieties, defined by the zeros of polynomial equations, are basic objects in both classical and modern mathematics. Algebraic geometry is the study of algebraic varieties. Ideas from symplectic geometry and string theoretic physics have opened new fields in algebraic geometry: the study of algebraic varieties via the Gromov-Witten theory of their curves and the Donaldson-Thomas theory of their sheaves. Since the topic has basic connections in several directions, progress will have a direct impact on the neighboring fields. Topological string theory is the most obvious connection and the two fields are in frequent contact. But also, for example, topics varying from the Fukaya category to random 3-dimensional partitions will be affected.