Sept. 16, 2024
Speaker: Sumit Singh
Title: Introduction to Lie algebras and representation theory
Abstract: I will talk about the Lie algebra sl(2,C). The structure and representation theory of sl(2,C) illuminates the general theory of semisimple Lie algebras. The talk will be based on Humphreys's book.
Sept. 23, 2024
Speaker: Carlos Tapp Monfort
Title: Group Representation Theory
Abstract: I will present the basics of group representation theory. I will define representation and cover the most elementary tools such as Maschke's Theorem. After that I will talk about the character table. My emphasis will be on how the table provides a lot of information about the group, finally I will try to sketch a proof of Burnside's (p^a)(q^b) theorem.
Sept. 30, 2024
Speaker: Alex Day
Title: Modular Group Representation Theory
Abstract: I will present some of the core results of modular representation theory. We will see that for modular representations we have similar but usually weaker analogues of results in the characteristic 0 case, including: Clifford's theorem instead of Maschke's, Brauer's theorem on the number of irreducible representations, and Brauer characters instead of usual ones. I will finish by presenting an application of modular representation theory to a group-theoretic result of Nikolov, Segal and Abért (which was the focus of my Master's dissertation).
Oct. 28, 2024
Speaker: Dr. Forrest Thurman
Title: Lambda-rings and Plethysm
Abstract: Let S(R) be the set of formal power series over R with constant coefficient 1. S(R) can be given a ring structure in which addition is the standard product on formal power series. A Lambda-ring will be defined as a ring R equipped with a section to S(R). Lambda-rings are quite ubiquitous and appear whenever there is a notion of exterior power on elements of a ring. We will see what this has to do with symmetric functions, number theory, and the plethysm operation from representations of GL(n).
Nov. 4, 2024
Speaker: Dan Tan
Title: Representations of vertex operator algebras and conformal field theory
Abstract: Vertex operator algebras are infinite dimensional algebras with infinitely many operations. While their axioms are notoriously hard to write down, the main ideas are easily motivated by the physical notion of a conformal field theory. In this talk, we will introduce the notion of a (chiral) conformal field theory and explain how it relates to the representation theory of vertex operator algebras. The example of a free bosonic string will be given to illustrate the main ideas. You don't need to know any of these words to attend!
Nov. 11, 2024
Speaker: Hong Chen
Title: Partitions and Symmetric Functions
Abstract: I will talk about the monomial, elementary, complete homogeneous and power-sum functions. As an application, I will show that the q-binomial coefficients are generating functions for partitions.
Nov. 18, 2024
Speaker: Nick Backes
Title: GL(2,R) and the Principal Series Representations
Abstract: We will construct representations of GL(2,R) by parabolic induction from a Borel subgroup. These representations form the principal series; many of these are irreducible already, and every irreducible representation is a component of some principal series representation. Principal series representations are very explicit, which makes them extremely useful for computations and applications of representation theory to number theory and other fields.
Nov. 25, 2024
Speaker: Hong Chen
Title: Representations of symmetric groups
Abstract: The conjugacy classes of S_n are indexed by partitions of n, and so are its irreducible modules (over C), called Specht modules. I will talk about these modules, and how they are related to symmetric functions, namely, Schur functions.
Fall 2022 Talks (Songhao)
Spring 2021 Talks (Jason Saied)