Moduli Spaces of Holomorphic Curves: Properties and Applications

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

String theory is a model that represents elementary particles by vibrating strings with the aim of unifying the four fundamental forces of nature. While string theory is one of the main paradigms in physics today, it has yet to make experimentally testable predictions. However, it has generated many mathematical predictions that have led to fundamental developments in algebraic geometry and symplectic topology, especially in relation to (pseudo-) holomorphic curves. This project aims to further test string theory mathematically, while deepening the mathematical understanding of such curves with an eye toward applications to more classical problems in geometry. Some of the projects in this work will be pursued by graduate students and other junior researchers in collaboration with the investigator. This project has four distinct directions at the juncture of algebraic geometry, symplectic topology, and string theory. It will explore connections between the rigidity of pseudo-holomorphic curves in symplectic topology and birational algebraic geometry. It will study the local structure of moduli spaces of stable morphisms of genus 2 and higher, with the aim of later applications in mirror symmetry and enumerative geometry. The PI will also apply his method for computing genus 1 Gromov-Witten invariants to more targets, with the aims of verifying predictions of string theory predictions in additional cases. The fourth direction aims to develop positive-genus real Gromov-Witten theory and its relations with open string theory and real enumerative geometry. This award is jointly funded by the Algebra and Number Theory and Geometric Analysis programs.