Presentations
Here are the slides and video recording of my most recent talk at the Graduate Algebra and Representation Theory Seminar (GARTS) at Rutgers University
Abstract: This talk is a continuation of my talk last week in which we defined ribbon categories. We will proceed to establish graphical notation for ribbon categories, then define the geometric objects of interest: colored ribbon graphs! The definition of RT-invariants will then be natural to define, and I’ll make a final appeal to why quantum groups are worth studying with some “examples” of RT-invariants and closing remarks. Also, I will briefly recap what a ribbon category is at the start, so aside from this there really is no expectation of prior knowledge!
Abstract: The goal of my sequence of talks is to showcase how useful quantum groups can be through one of my favorite applications: (quantum) knot invariants! These are an instance of something more general called Reshetikhin-Turaev invariants, or RT invariants for short. In order to define and discuss these objects, we will need some background on a couple of topics; this is why I will give a couple of talks! The first installment of my talks will predominantly be the necessary algebraic background to construct RT invariants. This will feature plenty of category theory, but all motivated and exemplified from familiar categories e.g. kVect, Rmod, Rep(G), Rep(g), and Rep(U_q(g)). Specifically, we will work toward the definition of ribbon categories and a notion of trace therein. I will then use the remaining time to lay the foundation for next week’s geometric part of these presentations by introducing a nice pictorial tool to work with ribbon categories.
Abstract: I will focus on the categorical significance of U_q(sl2)-modules, which I hope to discuss more explicitly, and its applications, in future talks. The present talk will be motivated by the representation theory of Lie algebras and of groups. Along the way I will discuss the Hopf algebras working behind the scenes in each of these studies, and present a couple of interesting results about Hopf algebras. From here, I will then emphasize the important categorical differences between U_q(sl2)-modules versus modules of a Lie algebra or a group.
In the Spring of 2020 I was in a Maple programming course taught by Prof. Zeilberger, and one day none other than Neil Sloane gave a talk on some interesting sequences he and a colleague of his had been considering. He tasked the class with working on analyzing the sequences, and he was quite interested with the plots I produced; you can find them here on the OEIS!