Short Courses in Algebra

Charlotte Chan: Deligne-Lusztig theory: examples and applications

20th May, 2024

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural 'partition' of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

Ivan Losev: Quantum category 𝒪

1st March, 2024

The representation theory of quantum groups including at roots of unity is an important part of Lie representation theory. In this talk, we will study one of categories of representations: the quantum category 𝒪, which is a suitable analogue of the classical Bernstein-Gelfand category 𝒪. We will relate it to a model representation category, the affine Hecke category, more precisely to the heart of the new t-structure on that category (all these terms will be defined in the lectures).

Quoos Luciane: Semigroups and algebraic geometry codes

6th November, 2023

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Asilata Bapat: Triangulations, rigid motions, and applications to representation theory

27th June, 2022

The course will begin with a brief survey of the theory of triangulations of a convex n-gon, which appear in several different places in mathematics. This course will focus on one such, possibly unexpected, appearance, namely in the theory of rigid motions of points in the plane. With this perspective, we will move to non-convex arrangements of points, and discuss the appropriate replacement of a triangulation. Finally, we will say a word about how these constructions are related to representation theory, via configuration spaces of points in the plane and a certain category of quiver representations. I will also mention some open questions in this direction.

Geordie Williamson: Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics

27th June, 2022

This will be a course on the representation theory of algebraic groups, and relations to the representation theory of symmetric groups. Reductive algebraic over finite fields and their algebraic closures are fascinating objects: one the one hand they look like Lie groups, but on the other hand they look like finite groups. Thus they mix two very different areas of mathematics. I will outline some of the basic theory, and then move on to questions of current interest.

Ting Xue's lectures in the previous week will provide essential background. I will aim to point out connections to the modular representation theory of finite groups. Although not essential, some background in algebraic geometry (e.g. the first three chapters of Hartshorne's Algebraic Geometry) will help with understanding latter parts of the course.

Uri Onn: Representation zeta functions

20th June, 2022

Algebraic groups are fundamental objects in representation theory and number theory. The course will discuss the structure theory of linear algebraic groups over algebraically closed fields. Topics include tori, parabolic subgroups and Borel subgroups, Lie algebras, root data, Weyl group, and classification of simple algebraic groups. If time permits, we will briefly discuss relation between compact Lie groups and algebraic groups.

Ting Xue: Introduction to linear algebraic groups

20th June, 2022

Algebraic groups are fundamental objects in representation theory and number theory. The course will discuss the structure theory of linear algebraic groups over algebraically closed fields. Topics include tori, parabolic subgroups and Borel subgroups, Lie algebras, root data, Weyl group, and classification of simple algebraic groups. If time permits, we will briefly discuss relation between compact Lie groups and algebraic groups.

Rose Morris-Wright: Artin groups: algebraic and geometric techniques

28th June, 2021

Artin groups are a broad class of groups whose presentations all follow a particular pattern. They are generalizations of braid groups and are closely related to Coxeter groups. Artin groups provide examples of groups with many interesting properties but there is very little that is known about ALL Artin groups.

In the first lecture, we’ll define Artin groups, talk about different types of Artin groups, and give a summary of known results and open questions. In the second lecture, we’ll focus on algebraic techniques for studying Artin groups, including the Garside structure, parabolic subgroups, and, if time permits, the Artin monoid. In the third lecture, we’ll discuss geometric techniques for studying Artin groups, including the Deligne complex and newer complexes such as the Clique-cube complex and the systolic Artin complex.

Sahana Balasubramanya: Acylindrically and relatively hyperbolic groups

28th June, 2021

The class of acylindrically hyperbolic groups has been of immense interest in recent times. It is an extremely large class of groups, containing many interesting examples. Yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context. The goal of this mini course is to provide an introduction to this class of groups, and focus on some important techniques.

In the first lecture, we will define acylindrical actions and talk about the motivation to study them. We will then define acylindrically hyperbolic groups and discuss some examples and properties of this class of groups. The second lecture will focus on the notion of hyperbolically embedded subgroups and discuss relative hyperbolicity in this context. In the last lecture, we will discuss the concept of group theoretic Dehn filling and some of its applications. Time permitting, I will also talk a bit about my own research.