CAREER: Holomorphic Curves in Algebraic Geometry and Symplectic Topology

Abstract Award: DMS-0846978 Principal Investigator: Aleksey Zinger The invention of pseudo-holomorphic curves techniques in the 1980s revolutionized symplectic topology, profoundly impacted algebraic geometry, and led to astounding connections with string theory. This project has three distinct directions, motivated by this interplay between different fields. The primary one is a detailed study of deformations of such curves, for arbitrary as well as generic complex structures. It aims at the fundamental understanding of properties of pseudo-holomorphic curves, including Gromov-Witten invariants (counts of such curves) and more qualititative aspects of their behavior (e.g. existence, uniruledness, etc.). The aim of another direction is to apply techniques employed in the PI's proof of the string theory prediction for the genus 1 GW-invariants of a quintic threefold in many other setting, testing additional mirror predictions and going beyond them. The third direction would further develop the PI's local excess intersection approach and its applications to classical enumerative geometry. String theory is a physical model that represents elementary particles by vibrating strings with the aim of unifying the four fundamental forces of nature. As such srings move in space, they sweep out Riemann surfaces, also called holomorphic curves. While string theory is one of the main paradigms in physics today, it has yet to make any experimentally testable predictions. However, it has generated plenty of mathematical predictions and led to fundamental developments in symplectic topology and algebraic geometry, especially in relation to (pseudo-) holomorphic curves. This proposal 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 proposal will pursued by graduate students under the PI's supervision. This award is jointly supported by the programs in Geometric Analysis and in Algebra, Number Theory, and Combinatorics.