Sep. 20, 2021
Speaker: Nicholas Backes
Title: Introduction to Lie Algebras and Representation Theory
Abstract: Lie Algebras are one of the essential objects of study in algebra and representation theory at Rutgers. At a minimum, they are likely to appear time and time again in future talks in this seminar. I will be introducing some background information on Lie Algebras that future talks will build off of. I will start with definitions and examples, discuss some of the key tools for studying Lie Algebras such as the adjoint representation and the killing form, and state some of the important theorems. We will build up to the representation theory of sl(2,F).
Sep. 27, 2021
Speaker: Terence Coelho
Title: Root Systems
Abstract: What makes simple Lie algebras so easy to classify is their natural bijection with root systems - a finite sets of vectors in Euclidean space with certain symmetry and integrality properties. We'll dive into this correspondence and show what it means for the structure of simple Lie algebras and their modules.
Oct. 4, 2021
Speaker: Weihong Xu
Title: Flag Manifolds
Abstract: I will briefly introduce Lie groups and then focus on flag manifolds, which are compact manifolds with a transitive action by a (nice) Lie group. Many topological and geometric properties of flag manifolds are governed by representation theory and the combinatorics of root systems. I will give some specific examples to illustrate such an interplay. If time permits, I will also say a few words about my research.
Oct. 11, 2021
Speaker: Brian Pinsky
Title: Fractal Groups and Coding Theory
Abstract: Have you ever wanted to learn coding theory, but worried it's too useful in real world applications? Even the smallest lemma gets winds up in some engineer's cure for cancer or whatever. Well, I spent this summer working on an antidote; a use of coding theory that is so obscure and technical the engineers should never dare care about it.
It starts with this question: Let G be a finitely generated group such that every element has finite order. Is G necessarily finite?
It turns out the answer is no. We'll show this with a famous example, the Grigorchuck group G. The proof will use the fractal structure of G; it turns out the groups G and G\times G are essentially the same (they have isomorphic subgroups of finite index). Every interesting property of G comes from the observation that (F_2)^3 has an interesting subspace, or code. By looking at length n codes over F_q, we can produce lots of other fractal groups. Some of them have interesting properties, and, crucially, none of them will ever be useful.
Oct. 18, 2021
Speaker: Yael Davidov
Title: Hunting for Division Algebras in Representations of Finite Groups
Abstract: Over fields of characteristic zero, finite representations of finite groups are completely reducible. Another word for completely reducible is semi-simple (i.e. made up of simple algebras). This is where I start to care, because my area of research is all about simple algebras...
We will start with some introductory material about representations of finite groups and then turn our attention to representations over some fields that aren't the complex numbers. I will then hopefully make explicit the connection to division algebras and talk about some questions Tamar and I have been thinking about with respect to this connection.
Nov. 1, 2021
Speaker: Daniel Tan
Title: Monoidal-Category-Theoretic Algebras and Their Representations
Abstract: We will start by going over the definition of a monoidal category – accompanied by some very familiar examples. Then we will be able to see how the usual notions of unital associative algebras and their representations can be generalised to the monoidal-category-theoretic setting. Pretty pictures will be provided to guide our intuitions.
Nov. 8, 2021
Speaker: Terence Coelho
Title: Lattice Realizations and Diagram Automorphisms
Abstract: The simple Lie algebras with unequal root lengths can be realized as fixed points of certain "diagram automorphisms" on the simple Lie algebras with equal root lengths. This correspondence is pivotal in understanding twisted affine Lie algebras, but can also make the root systems and Weyl groups easier to understand.
Lattice realizations of Simple Lie algebras give one concrete basis relations that are very easy to do computations with and give a beautiful way of realizing the diagram automorphisms. They also immediately give integral structure constants for the simple Lie algebras.
In this talk, I will give a sales pitch for these lattice realizations and, if there is time, share some of my relevant research.
Nov. 15, 2021
Speaker: Jason Saied
Title: Macdonald Polynomials and Double Affine Hecke Algebras
Abstract: Representation theory is hard. Often, we can learn a great deal about representations by instead studying certain (much more concrete) families of polynomials associated with them. As many of you have heard me say a million times, the motivating example is the family of Schur polynomials, which are given by a simple formula and can be used to classify the finite-dimensional (algebraic) representations of GL_n (and also S_n). My research is related to the more general family of Macdonald polynomials, whose specializations recover Schur polynomials and many other interesting families of special functions. In this talk, I will construct the family of Macdonald polynomials and show you how to work with them concretely. This involves a representation of an algebra called the double affine Hecke algebra. I will then discuss a nice combinatorial formula for Macdonald polynomials that follows easily from this perspective. In the last few minutes, I will talk about my own research, in which I give a corresponding combinatorial formula for a generalization of Macdonald polynomials due to Sahi, Stokman, and Venkateswaran.
Dec. 6, 2021
Speaker: Forrest Thurman
Title: Automorphic Forms and Borel-Eisenstein Series
Abstract: This talk will be about how to define automorphic forms and Borel-Eisenstein series and some basic tools used to study them. These include the use of differential operators, properties of the Iwasawa decomposition and the representation theory of compact groups.
Dec. 13, 2021
Speaker: Doyon Kim
Title: From number theory to automorphic forms and then to automorphic distributions
Abstract: The theory of automorphic forms is essential in studying number theory. In this talk, I will start with the connection between number theory and automorphic forms, and then briefly talk about how the study of automorphic forms can be reformulated using the language of representations. Then I will talk about how automorphic distributions arise naturally from automorphic representations and some applications of automorphic distributions to problems in number theory.
Spring 2021 Talks (Jason Saied)