Grant: Discovery Grants Program - Individual

The field of low-dimensional topology is the study of manifolds in dimensions up to four. It covers knot theory, the study of loops in three-dimensional spaces, which has rich applications to physics and even cellular biology. It also encompasses the study of smooth four-dimensional manifolds, which on the one hand includes our model of spacetime, but which, on the other hand, possess many counterintuitive properties and open problems. Thus, further developing the theory of these mathematical objects will help us gain a deeper understanding of our universe, as well as lead to many potential applications. One of the techniques I use to study low-dimensional topology is Floer homology, a large package of sophisticated tools invented in the last three decades that draws from mathematical physics and symplectic geometry. A topological quantum field theory, it assigns algebraic structures not only to three-manifolds or knots, but also to four-dimensional objects that interpolate between them, known as cobordisms. These algebraic structures have proved to be extremely useful in revealing patterns of the topological objects; at the same time, cobordisms themselves are of great theoretical interest, for example playing a major role in the recent disproof of the Triangulation Conjecture. Not only does Floer homology yield many topological applications, but there is also an immensely rich structure among its various flavors. It may also be compared to representation-theoretic theories such as Khovanov homology; both of these fit into the exciting domain of categorification. The philosophy of categorification is to lift existing mathematical constructions to a higher framework to reveal richer structures that are hidden before, thus enhancing their power. Over the next 5 years, I plan to focus on studying the structural properties of cobordisms and of Floer homology. First, I plan to use a wide variety of techniques to study cobordisms of knots and three-manifolds that arise naturally from topological and contact-geometric contexts. As part of this effort, I plan to enhance existing cobordism invariants in Floer homology in several directions, including a combinatorial version known as grid homology. I will continue my work on establishing relations among the various flavors of Floer homology in order to combine their strengths. Lastly, I will also work on further developing Floer theory through the lens of categorification, to clarify the link between Floer and Khovanov homologies. I expect the outcomes of this research to have significant impact to low-dimensional topology and contact geometry, and make new connections with representation theory and categorification. This proposal will also provide a fertile training ground for HQP, with opportunities both in establishing a deep and broad perspective in pure mathematics and in developing computational tools for research, skills highly desirable in both their future academic and non-academic careers.