Low Dimensional Topology and holomorphic disks

Princeton University, Princeton NJ

Investigators

Abstract

The award supports the principal investigator's research in topology, a mathematical study of properties of spaces that are preserved under continuous deformations, such as stretching and bending. The central theme in this research is to develop a better understanding of a highly effective mathematical tool known as the knot Floer homology that is used in the study of three- and four-dimensional spaces and knotted loops. The PI and his collaborators have done foundational work in this area. The proposed project builds on their recent success in approaching knot Floer homology and related invariants of three-dimensional spaces by means of algebraic techniques, thereby making them much more tractable. In addition to applications within the abstract field of topology, the methods are closely related to other areas of mathematics that are motivated by physics, such as gauge theory and symplectic geometry. The completion of this project would lead to new bridges between different research areas and methods. The project also includes investigating the relationship between various invariants for knots, and computational problems in knot theory. Heegaard-Floer homology provides various tools to study problems in low-dimensional topology. The construction, first developed by Peter Ozsvath and the principal investigator, uses methods both from topology and symplectic geometry through the study of Heegaard diagrams, and holomorphic disks in symmetric products of Heegaard surfaces. There are special families of three-manifolds, including fibered three-manifolds and Dehn surgeries along knots, where advances in knot Floer homology and Heegaard-Floer homology are expected to lead to new topological applications. Related research directions involve concordance invariants, smoothly slice knots and smooth four-manifolds. The project also aims to further develop algebraic constructions to study problems in knot theory, and investigate relationships between different knot homology invariants.