Spring 2024
Time & Location: All talks are on Monday in Gibson Hall 308 at 2:00 PM unless otherwise noted.
Organizer: Komendarczyk, Rafal; Co-organizer: Mahir Can, Tai Ha, Kalina Mincheva
March 4
Title: Knot Invariants, Categorification, and Representation Theory
Arik Wilbert - University of South Alabama
Abstract: I will provide a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry central to my research. I will recall basic facts about the representation theory of the Lie algebra sl2 and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. Time permitting, I will discuss how this picture might generalize to other Lie types beyond sl2.
Location: Gibson Hall 308
Time: 2:00
