Moduli Spaces of Pseudoholomorphic Maps

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Mathematical objects known as symplectic manifolds are of fundamental importance in the study of modern geometry and physics. This award supports research that lies at the intersection of symplectic geometry, which provides the framework for classical mechanics, and string theory, which developed as a potential candidate for unifying general relativity and particle physics, enabling the study of models of space-time and phase-spaces inaccessible by other means. The problems considered in this project are of foundational nature, and answers to these are expected to lead to new techniques and interactions among different fields of mathematics and have potential applications in string theory. There are several interesting applications of this work, and promising directions for continuing this line of research, some of them involving graduate students. The research and outreach activities of the principal investigator will have an impact on the education of next generation of mathematicians, engaging them in cutting-edge research early in their graduate career.The theme of this research involves symplectic manifolds and the structure of the invariants associated to them. Specifically, one of the projects aims to understand the geometric reasons behind a general Gopakumar-Vafa type structure theorem for many flavors of Gromov-Witten invariants. These types of structures were conjectured using string theory and have spurred a lot of interest in mathematics and many attempts to understand the geometry behind them. Another project aims to understand the properties of the Gromov-Witten virtual fundamental cycle in symplectic geometry and explore its functorial aspects. It includes a proposed set of axioms that would uniquely determine it, inspired by the Eilenberg-Steenrod axioms for homology theories. There are many different constructions of the Gromov-Witten virtual fundamental class of closed symplectic manifolds, each one with its own strengths and advantages, so knowing that most of these constructions carry the same information is useful, allowing one to choose whatever construction may be best suited for the particular problem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.