Floer homology and low-dimensional topology

Boston College, Chestnut Hill MA

Investigators

Abstract

The principal investigator (PI) will pursue a collection of concrete problems in low-dimensional topology, using a combination of traditional topological techniques, Heegaard Floer homology, combinatorics, and whatever else might come in handy. As a sample list, the PI hopes to address the questions of which alternating knots have unknotting number one; which alternating knots bound smoothly slice disks in the four-ball (joint work with Brendan Owens); and which 3-manifolds admit a strong Heegaard diagram (joint work with Adam Levine and John Luecke). In each of these projects, Heegaard Floer homology provides important information and sets up a challenging combinatorial problem, which in turn requires modern methods to solve. Low-dimensional topology is concerned with the properties of curves, surfaces, and 3- and 4-dimensional spaces -- objects that we can visualize, though sometimes with a bit of effort. On one hand, this field has a rich tradition in the sciences: it owes a lot of its development to physics, both classical and modern; it impacts biology, where the knotting of DNA plays a significant role; and it interacts with fields all across mathematics. On the other hand, it has a distinctly visual nature that is reflected in the arts: for example, through Celtic knots, motifs in ancient architecture, and Escher's prints. The goal of the PI's proposal is to study some attractive and simply-stated problems in low-dimensional topology. In fact, many of these problems focus on the mathematical properties of Celtic (a.k.a. alternating) knots. While some of these problems have been around for many decades, they have only recently become accessible through the advent of sophisticated techniques from nearby fields (Floer homology and combinatorics). The interplay between these different fields fascinates the PI.