An LMS online lecture course in groups, semigroups and algebras.
An LMS online lecture course in Vinberg theory.
In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (e.g., in the work of Romano-Thorne) to the local Langlands correspondence (e.g., in the work of Reeder-Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper Vinberg's representations and arithmetic invariant theory and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.
This series of talks is based on joint works with Oppermann, Grimeland, Labardini and Plamondon. Cluster categories are triangulated categories where quiver mutation appears as a natural operation. A first class of example is given by cluster categories associated with surfaces with marked points. A second class is constructed using the derived category of finite-dimensional algebras of global dimension 2. Mixing both constructions, one may consider surface cut algebras, that are algebras of global dimension 2 constructed from a surface and show how cluster combinatorics permits to deduce information on their derived category.
Let R be a commutative Noetherian ring. Denote by D-(R) the derived category of cochain complexes X of finitely generated R-modules with Hi(X)=0 for i>>0. Then D-(R) has a structure of a tensor triangulated category with tensor product ⊗RL and unit R. In this series of lectures, we study thick tensor ideals of D-(R), i.e., thick subcategories closed under the tensor action by each object in D-(R), and investigate the Balmer spectrum Spc D-(R) of D-(R), i.e., the set of prime thick tensor ideals of D-(R). Here is a plan.
• We give a complete classification of the (co)compactly generated thick tensor ideals of D-(R), establishing a generalized version of the Hopkins--Neeman smash nilpotence theorem.
• We construct a pair of maps between the Balmer spectrum Spc D-(R) and the prime spectrum Spec R, and explore their topological properties.
• We compare several classes of thick tensor ideals of D-(R), relating them to specialization-closed subsets of Spec R and Thomason subsets of Spc D-(R).
If time permits, I would like to talk about the case where R is a discrete valuation ring. My lectures are based on joint work with Hiroki Matsui.
The commutative algebraic groups over a prescribed field k form an abelian category Ck; the finite commutative algebraic groups form a full subcategory Fk, stable under taking subobjects, quotients and extensions. This mini-course will study the categories Ck and Ck/Fk (the isogeny category) from a homological viewpoint, emphasizing the analogies and differences with categories of representations. In particular, we will show that Ck/Fk has homological dimension 1, and we will describe the projective and the injective objects in Ck and Ck/Fk.
Modular representation theory of finite groups seeks to understand, and possibly classify, the algebras - called block algebras of finite groups - which arise as indecomposable direct factors of finite group algebras over a complete local principal ideal domain with residue field of prime characteristic p. The expectation is that 'few' algebras should arise in this way, and that this should in turn lead to significant structural connections between finite groups and their block algebras.
The key feature of block algebras of finite groups is the dichotomy of invariants attached to these algebras.
On the one hand, they have all the typical algebra-theoretic invariants - module categories, their derived categories and stable categories, as well as numerical invariants such as the numbers of isomorphism classes of simple modules, and cohomologivcal invariants such as their Hochschild cohomology.
On the other hand, they have p-local invariants, due to their provenance from group algebras - reminiscent of the local structure of a finite group which includes its Sylow p-subgroups and its associated fusion systems.
Essentially all prominent conjectures which drive modular representation theory revolve around the interplay between these two types of invariants. We describe this interplay with a focus on Hochschild cohomology and analogous cohomology rings which are defined p-locally. This involves a variety of angles - Hochschild cohomology is graded commutative, hence methods and notions from commutative algebra will play a role. Hochschild cohomology in positive degree is also a Lie algebra. We will investigate connections between the algebra structure of block algebras and the Lie algebra structure of its first Hochschild cohomology space.
We'll study the global structure of the stable module category StMod G or, equivalently, the category of singularities of representations of a finite group scheme G over a field of positive characteristic p. The goal of the lectures will be to classify the tensor ideal localizing subcategories in StMod G. The techniques involved in the classification include the theories of support and cosupport in modular representation theory, detection of projectivty for modules, Benson-Iyengar-Krause theory of local cohomology functors, and new methods inspired by commutative algebra which allow to relate local cohomology at closed and arbitrary points. This is based on joint work with Eric Friedlander and Dave Benson, Srikanth Iyengar and Henning Krause.
The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Plücker coordinates, which are described by certain diagrams in a disk, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure.
In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.
It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.
These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In particular, we will meet the fundamental building blocks of the theory: threefolds fibered in ADE surfaces. In the second lecture, we will learn what geometric representation theory says about these building blocks, and, in particular, meet the present day incarnation of the Weyl group, which is really a fundamental groupoid of a certain periodic hyperplane arrangement, associated to a certain geometrically defined infinite-dimensional Lie algebra. This Weyl group completely determines the curve counts, and so seems like a very fitting topic for Hermann Weyl lectures. In the third lecture, I plan to introduce some of the geometric ideas that go into the actual technical construction of the theory.
A two-hour course on expanders, thin subgroups of Lie groups, and superstrong approximation.