Feb. 6, 2023
Speaker: Songhao Zhu
Title: An overview of Lie Superalgebras
Abstract: We will take a look at the Lie objects in the category of vector superspace, i.e., what our organizer emeritus wasted his Ph.D. on. I will briefly review Lie algebras and discuss linear superalgebra as the background. Still, the zoo of simple Lie superalgebras is going to be a wild safari. If time allows, I will say a few things on supersymmetry and my work.
SlidesFeb. 24, 2023
Speaker: Nicholas Backes
Title: Kirillov Models of p-adic Representations
Abstract: The Kirillov model is a way of taking irreducible representations of GL(n, F) and realizing them as representations on a vector space of complex functions on the base field F. Among other uses, the Kirillov model is one way to define Whittaker functions. Properties of the Kirillov model can determine if a representation is cuspidal. For this talk, the base field F will be the p-adic numbers. We will discuss some properties and uses of the Kirillov model, and depending on available time we will discuss some aspects of its construction.
Mar. 3, 2023
Speaker: Jason Saied (KBR, Inc. at NASA Ames Research Center)
Title: Examples of algebra and representation theory in quantum computing
Abstract: A quantum computer is a computer that takes advantage of quantum mechanical properties, generally with the goal of someday performing certain computations faster than classical computers can. I will give an overview of some basic notions in quantum computing, then discuss several places in which algebra and representation theory appear in that field. We will study the Pauli and Clifford groups, their importance, and their relation to several groups of interest to representation theorists. We will then discuss character randomized benchmarking, a technique that uses group character theory to determine the fidelity of real-world implementations of ideal quantum gates.
Mar. 10, 2023
Speaker: Fanxin Wu
Title: Representation of self-distributive systems on digraphs
Abstract: Common examples of self-distributive (LD) operations include group conjugation and knot quandle. Another example arises from nice endomorphisms of infinite extensional digraphs. It turns out this provides a faithful representation of the free LD-system on one generator to the LD-system of endomorphisms. This has several implications such as decidability of word problem of free LD-systems, orderability of Braid groups and growth of periods in certain quotients of the free LD-system. After introducing the necessary background, we will discuss this unexpected relation between LD-systems and digraphs.
SlidesMar. 24, 2023
Speaker: Dan Tan
Title: 2d TQFTs
Abstract: An n-dimensional topological quantum field theory (TQFT) can be thought of as a representation of compact orientable n-dimensional spacetime up to topological equivalence. We will go through the relatively simple n = 2 case and prove the common folk theorem that 2d TQFTs are (almost categorically) equivalent to working with Frobenius algebras. After setting up this equivalence, we can easily provide some examples of 2d TQFTs.
Mar. 31, 2023
Speaker: Forrest Thurman
Title: Lattices, a motivating example for modular forms
Abstract: I will talk about some of the nice properties of lattices and how one can use analytic tools such as poisson summation and modular forms to study them. The theta series of certain lattices can also encode arithmetic information.
Apr. 7, 2023
Speaker: Terence Coelho
Title: Classifying root systems via walks on the Dynkin diagram
Abstract: Finite and affine root systems are geometric objects that arise eerily often in representation theory. One usually considers only the reduced root systems (those that forbid non-trivial multiples of the same root) but there are contexts in which one needs the classification of even the non-reduced root systems. In this talk, I will consider the set of indecomposable directed multigraphs with loops allowed (only requiring out-neighborhoods in-neighborhoods to be equal for all vertices), classify those that have spectral radius at most 2, then demonstrate how this corresponds exactly to a classification of the reduced and non-reduced finite and affine root systems.
Apr. 21, 2023 (zoom)
Speaker: Zeyu Shen
Title: An Introduction to Homological Dimensions
Abstract: Projective, injective and flat dimensions of a module over a ring will be introduced, together with global dimension and Tor-dimension of a ring. Some examples will be discussed. Regular local rings, characterized among commutative noetherian local rings as having finite global dimension, will be introduced. Regular rings will also be discussed.
Apr. 28, 2023 (zoom)
Speaker: Filip Dul
Title: Toward Formal Derived Geometry
Abstract: In this talk, we will introduce the Chevalley-Eilenberg (CE) cochain complex. This is a differential graded vector space defined to manage certain facts in the representation theory of Lie algebras; however, we shall see how certain CE cochains also naturally arise as function rings of certain formal derived spaces. In this way, we will see how starting with representation theory actually “gets our foot in the door” when it comes to understanding more modern notions of what a space is.
Fall 2022 Talks (Songhao)
Spring 2021 Talks (Jason Saied)