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This expository paper is based on the lectures given at the program `Modular Representation Theory of Finite and $p$-adic Groups' at the National University of Singapore. We are concerned with recent results on representation theory and…
These are notes for a summer course given at the PIMS Summer School on Geometric and Topological Aspects of the Representation Theory of Finite Groups in Vancouver, July 27-30 2016.
The article covers developments in the representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and…
The aim of this lecture is to present the concept of C-algebra and to illustrate its applications in two contexts: the study of reflection groups and their folding on the one hand, the structure of rational conformal field theories on the…
These notes are based on a series of lectures given by the author at the Centre Bernoulli (EPFL) in July 2016. They aim at illustrating the importance of the mod-$\ell$ cohomology of Deligne--Lusztig varieties in the modular representation…
Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The…
These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the…
The past decade has witnessed two important new developments in the study of linear series on algebraic varieties. First, vector bundles have emerged as powerful tools for analyzing linear series on curves and surfaces. More recently, the…
These are the notes from an Oberwolfach Seminar which we ran from 23--29 May 2010.
In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain…
This is a semi--expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so…
These are lecture notes prepared for a minicourse given at the Cimpa Research School "Algebraic and geometric aspects of representation theory", held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction…
This paper is an overview of my recent work on abstract homomorphisms of algebraic groups. It is based on a talk given at the Conference on Group Actions and Applications in Geometry, Topology, and Analysis held in Kunming in July 2012.
These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism…
This paper surveys basic properties of finite presentation in groups, Lie algebras and rings. It includes some new results and also new, more elementary proofs, of some results that are already in the literature. In particular, we discuss…
Primarily this paper presents an expository report on alternatives to the traditional methods of classifying representations of finite dimensional algebras. Some new results illustrating such alternatives for algebras with only finitely…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but…
The aim of this paper is to study the cohomology theory of Reynolds Lie algebras equipped with derivations and to explore related applications. We begin by introducing the concept of Reynolds LieDer pairs. Subsequently, we construct the…
This paper surveys results on the connections between the cohomology for algebraic groups, finite groups and Frobenius kernels that were presented at the Workshop and Summer School on Lie and Representation Theory at East China Normal…