Colloquium - Nicolle Gonzalez

Date: December 7, 2020

Time: 1:00PM - 2:00PM

Location: Zoom

Speaker: Nicolle Gonzalez, University of California, Los Angeles

Description:
Title: Categorification: a revelation of hidden shadows
Abstract: In recent years there has been a lot of activity revolving around a series of celebrated conjectures that relate Khovanov-Rozansky homology, a homological invariant for knots and links, to Hilbert schemes. In an effort to prove these conjectures, unexpected and deep ties to the category of Soergel bimodules and the combinatorics of Catalan numbers and Macdonald polynomials have been discovered. Central to this story is the idea of diagrammatic categorification, an innovative process by which classical algebraic and topological objects are replaced by a richer categorical structure that is defined in terms of planar diagrams. I will explain what categorification is and how seemingly disparate areas like low dimensional topology, algebraic geometry, representation theory, and algebraic combinatorics come together to present a unified picture. I will then describe how joint work with Matt Hogancamp on a skein theoretic formulation and subsequent categorification of the Dyck path algebra and its polynomial representation can shed light on some of these mysterious conjectured connections.