Floer Invariants, Cobordisms, and Contact Geometry

点击翻译按钮获取中文摘要

Low-dimensional topology is the study of geometric shapes and spaces in dimensions up to four, which has, perhaps unintuitively, proved to be more difficult than high-dimensional topology. Within this subject lies the theory of knots, loops of tangled string that are tied together at their ends, which has various connections to physics (via quantum theory and string theory), chemistry (via molecular knots), and biology (via DNA structure, with applications to drug design). To better understand knots and other geometric objects, topologists have invented tools called invariants. Some modern invariants are inspired by theoretical physics, such as gauge theory. On the one hand, these invariants have been successfully applied to solve many important and long-standing questions in topology; on the other hand, their behaviors are far from entirely understood. The goal of this research project is to further the development of modern invariants in terms of both theory and computation, harnessing their power to explore the link between low-dimensional topology and contact geometry, a related area of mathematics that has its roots in Newtonian mechanics and that has emerged as an exciting area of research in recent years. As part of this project, the investigator will provide research training to undergraduate and graduate students, make modern invariants accessible to a wide audience, and continue efforts in mathematical outreach. Floer theory, which encompasses instanton, monopole, and Heegaard Floer homologies, is a large package of invariants for three-manifolds and knots, as well as their cobordisms, that originate from gauge theory and symplectic geometry. In recent years, Heegaard Floer homology has been shown to be algorithmically computable, using combinatorial diagrams or bordered invariants. This project aims to harness the power of Floer invariants that comes from combining theory and computation, in several related directions. First, Floer theory provides information on the existence or non-existence of cobordisms between three-manifolds and between knots, with topological or geometric constraints. It is also known to be closely related to contact geometry, giving rise to invariants that certify tightness of a contact three-manifold, and distinguish smoothly isotopic knots that are not Legendrian isotopic. One goal of the project is to further extend these applications to cobordisms and contact geometry. To do so, the investigator aims to establish naturality results that will refine isomorphism class invariants to concrete homology group elements. The combinatorial diagrams involved will also shed light on the significant yet mysterious link between Floer theory and representation-theoretic invariants, which has been established in the form of spectral sequences. Similarly, the project also aims to advance bordered Floer invariants, which will activate more topological applications and significantly augment the use of the contact invariants above. This project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR).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.