GARTS Fall 2022

Dec. 12, 2022

Speaker: Nilava Metya

Title: Quiver representations

Abstract: Quivers are (finite) directed graphs. Representations are, well... representations (I'll assume people know the definition of modules over an algebra). One usually likes to find all irreducible representations (i.e., simple modules)... but they're not so interesting when things are not 'semisimple'. So one can instead look at indecomposables - things which cannot be broken into sum of submodules. Usually there is a finite list of simples (in case of semisimple finite dimensional algebras). We (time permitting) prove a characterization of quivers which admit only finitely many indecomposables in terms of (undirected graphs). This is widely known as Gabriel's theorem. We'll see some examples and try to understand at least the basics. Maybe some algebraic varieties will appear, but (non-technical) geometric intuition is enough for this portion.

Dec. 5, 2022

Speaker: Nick Backes

Title: Representation theory of p-adic groups

Abstract: We will explore the representations of the general linear group of p-adics. First, I will introduce p-adic numbers. We will discuss their structure, including the characters. Then we will study algebraic and admissible representations on the general linear groups of p-adics by induction and by the Jacquet functor. We will discuss principal series representations and cuspidal representations. No prior experience with p-adics will be assumed.

Nov. 21, 2022 (ON ZOOM)

Speaker: Sylvester Zhang

Title: Greene–Kleitman Correspondence, Jordan Forms, and Flag Varieties

Abstract: The Greene–Kleitman correspondence associates each finite poset a Young diagram (a partition), and is related to many areas of algebra and combinatorics. In this talk, I will survey several well-known interpretations and applications of GK correspondence. In particular, I will discuss its connection to the Robinson–Schensted correspondence (due to Greene) and Jordan canonical forms of upper triangular matrices (due to Gansner and Saks), and give a geometric interpretation using the relative positions of pairs of flag varieties (due to Steinberg). Finally, if time permits, I will discuss a current work in progress (joint with P. Pylyavskyy) towards a generalization to affine type A.

Nov. 14, 2022

Speaker: Lee Tae Young

Title: The Steinberg Representation of Finite Groups of Lie Type

Abstract: The Steinberg representation is a representation of finite groups of Lie type (such as SL_n(q)) with many remarkable properties and applications. In this talk, I will give construction(s) and basic properties of Steinberg representation, and briefly mention some interesting facts and applications.

Nov. 7, 2022

Speaker: Zeyu Shen

Title: An introduction to algebraic K-theory

Abstract: In 1957, Grothendieck reformulated the Riemann-Roch theorem in algebraic geometry by using Grothendieck groups of coherent sheaves and locally free sheaves on a variety. This marked the beginning of the subject of algebraic K-theory. In this brief introduction to algebraic K-theory, I am going to state the definitions and examples of K_0 for a ring, abelian categories, exact categories and Waldhausen categories. I will discuss some sample computations, and state some basic definitions and theorems for higher K-theory of rings and schemes. Some open problems in algebraic K-theory will be mentioned.

Oct. 31, 2022

Speaker: Gautam Krishnan

Title: Nilpotent orbits in semisimple Lie algebras

Abstract: In the study of representations of simple Lie groups, nilpotent orbits serve as useful combinatorial data that can be associated with the representations. In this talk, we will go over the classical complex simple Lie algebras and see explicit combinatorial descriptions of nilpotent orbits. Then we will introduce the wave front set and the process of attaching the nilpotent orbits to representations. All the required terminology will be introduced in the talk and no prerequisite will be assumed apart from linear algebra.

Oct. 17, 2022

Speaker: Forrest Thurman

Title: The Temperley-Lieb Algebra and Planar Diagrams

Abstract: I will discuss how the algebra of planar diagrams can be described by the Temperley-Lieb algebra, as well as mention some connections of this algebra with combinatorics and knot theory.

Oct. 10, 2022

Speaker: Luochen Zhao (Johns Hopkins University)

Title: Fourier transform according to Cartier

Abstract: Let ell and p be distinct primes. In this talk I'll explain how to naturally define the p-adically valued Fourier transform on Q_ell, as first studied by W. H. Schikhof in his thesis. This employs the language of group schemes and Cartier duality in the style of Katz. I'll start by mentioning some motivations from number theory, and conclude by reporting my recent work on a p-adic local functional equation at ell using this Fourier transform.

Oct. 3, 2022

Speaker: Dan Tan

Title: From even lattices to modular tensor categories

Abstract: Huang proved that a vertex operator algebra, satisfying certain conditions, will naturally produce a modular tensor category from its category of modules. In this talk, we will first go over the definition of a modular tensor category. Then we will see some very explicit examples of modular tensor categories produced by vertex operator algebras induced from even lattices. (The definition of a vertex operator algebra will not be needed to understand these examples.)

Sept. 19, 2022

Speaker: Hong Chen

Title: A Brief Introduction to D-modules

Abstract: I'll first review some history of D-modules, then introduce Weyl algebras, and a conjecture of Dixmier's that lead to the famous Jacobian conjecture. Then I'll talk about modules over Weyl algebras and their relation with differential equations. No prerequisite will be assumed apart from basic calculus and module theory.

Spring 2021 Talks (Jason Saied)

Spring 2019 Talks (Alejandro Ginory)