Interactions Between Contact Geometry, Floer Theory and Low-Dimensional Topology

Louisiana State University, Baton Rouge LA

Investigators

Abstract

This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR). It centers on questions lying at the intersection of two fields of mathematics called contact geometry and low-dimensional topology. Contact structure are geometric objects on certain spaces known as manifolds that arise naturally in physics through differential equations, optics, and dynamics. Pioneering work of Eliashberg first showed that contact structures play an essential role in determining geometric and topological properties of three- and four-dimensional manifolds. Contact geometry has since entered a renaissance after featuring prominently in the resolution of several long-standing problems in low-dimensional topology. It is in this context that this project broadly seeks to better our understanding of how characteristics of contact structures determine either geometric properties of the spaces they live on, or influence powerful invariants used in their study. The project will have immediate impact in several fields, such as: low-dimensional topology, symplectic and contact topology, dynamics, and mathematical physics. The PI will also devote time to helping mentor graduate students and postdoctoral scholars as they transition to being independent researchers. Recall that contact structures fall into one of two categories: tight or overtwisted. Understanding which three-manifolds support tight contact structures and the number of tight contact structures supported by a given three-manifold are the paramount goals of modern contact geometry. Accordingly, a primary goal of this project is to develop effective and computable invariants capable of determining tightness and distinguishing contact structures. Since the inception of these invariants, strong evidence has steadily built suggesting deep connections between contact structures on three-manifolds and their associated Floer-theoretic invariants. For instance, each of the various Floer homologies support an invariants which are capable of detecting tightness and distinguishing contact structures. An important goal of this project is to develop and explore refinements of these invariants which are simultaneously more effective in detecting tightness, and are more easily computable. In a parallel direction, according to work on the PI and others, much of the formal algebraic structure underpinning Floer theory appears to mirror natural geometric characteristics and constructions involving contact structures. In turn, another key goal of this project is to clarify connections and correspondences between the algebraic structure of Floer-theoretic invariants and natural contact geometric phenomena and constructions. 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.