Thesis Research
Summary: In 2020, I completed a PhD in pure mathematics at Princeton University. In that work I made contributions to the foundations of the field of Symplectic Field Theory. Geometric objects in high dimensions can be difficult to understand. To study these objects, mathematicians generally resort to attaching simpler proxies called ‘invariants’—like numbers, such as the volume, or algebraic objects--to these complicated geometric objects. Mathematicians’ ability to study high dimensional geometric objects often reduces to the clarifying power of these ‘invariants’. My thesis developed the technical foundations for a richer set of algebraic invariants to further elucidate the study of high dimensional symplectic manifolds.
Title: Polyfolds of Lagrangian Floer Theory in All Genera
Abstract: Polyfold theory was introduced by Hofer, Wysocki, and Zehnder as a framework to study compactness, smoothness, and transversality problems in solution spaces of families of non-linear Fredholm operators. My thesis work establishes a polyfold approach to the construction of an integration theory on moduli spaces of pseudo-holomorphic curves with Lagrangian boundary conditions as a technical foundation for a theory of Lagrangian Floer theory in all genera and open Gromov-Witten theory. Similarly to the construction of Lagrangian Floer theory (in genus 0) established by the seminal work of Fukaya, Oh, Ohta, and Ono, the construction of Lagrangian Floer theory in all genera separates into two pieces:
1. The construction of an integration theory on the moduli spaces of pseudo-holomorphic maps with Lagrangian boundary conditions
2. An ordered perturbation scheme on these moduli spaces which relates geometry (in particular integration) along the boundary of a given moduli space in terms of geometry of moduli spaces of lower order
My thesis treats the fi rst of these pieces, while the second piece is left to subsequent work.