CAREER: Link homology -- in type A and beyond

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). A link is a mathematical object that formalizes the notion of one or more knotted pieces of string. The mathematical theory of links (knot theory) is the study of such objects up to continuous deformations, that is we imagine that our link is made from a flexible material, and we consider two links to be the same if we can deform one into the other. Despite the seemingly specialized nature of knot theory, it has been shown to have applications to the theory of 3- and 4-dimensional spaces, to theoretical physics, and it has been applied to the study of DNA recombination and protein folding. The problem of distinguishing two links is challenging for two reasons: first, it can be difficult to exhibit the deformations that identify two links that look different but are indeed the same; second, given two links that we suspect to be distinct, it is difficult to directly and rigorously show that in fact one cannot be deformed to the other. One technique for solving the latter problem is via a link invariant: an assignment of a simpler mathematical object to a link that is unchanged under deformation. If two links have different invariants, we know that they are indeed distinct. The main research goal of this project is to further our understanding of powerful modern link invariants called link homology theories. The accompanying educational activities share a common theme of increasing participation in the mathematical sciences at a variety of levels, including course-development, student research, and outreach activities at the K-12 level. The broader impacts of this project aim to support the persistence of groups typically underrepresented in mathematics via research mentoring efforts, and to foster connections with the public through the UNC Science Expo. Link homologies are modern invariants of links that generalize (and moreover, categorify) quantum invariants such as the Jones polynomial. In addition to providing deep topological information in dimensions three and four, they enjoy a rich algebraic structure arising from connections to modern representation theory. Consequently, they are an important nexus for research in representation theory, topology, and related considerations in theoretical physics. Thus far, algebraic developments in link homology have focused on the Khovanov-Rozansky theory, which is associated with the Lie algebra sl(n). The PI will develop link homology beyond this (non-super) type A case. These developments are crucial for providing long sought-after bridges between quantum and classical topological structures and for studying link homology associated to simple complex Lie algebras distinct from sl(n) about which essentially nothing is known. In one line of work, the PI will construct the gl(m|n) link homologies (super type A) that have been predicted to exist by considerations in theoretical physics, and which conjecturally provide a connection between the Khovanov-Rozansky theory and knot Floer homology. In a second line of work, the PI will construct explicit and computable link homologies associated to simple complex Lie algebras outside type A. Along the way, he will also resolve the long-standing problem of finding generators-and-relations descriptions of categories of quantum group representations for non-type A simple complex Lie algebras. 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.