Related papers: Lectures on modular Deligne--Lusztig theory
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…
We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a…
These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of…
ICM lecture on minimal models and moduli of varieties.
We give a generalisation of Deligne-Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching…
This is the companion article to the Bourbaki talk of the same name given in March 2009. The main theme of the talk and the article is to explain the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie…
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial…
For connected reductive groups together with a Frobenius root $F$, we show that the cohomology of the structure sheaf and respectively the canonical sheaf for compactified Deligne--Lusztig varieties associated to an element in the free…
If $f$ is a mod-$3$ eigenform of weight 2 and level $\Gamma_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but…
These notes provide a short, focused introduction to modelling stochastic gene expression, including a derivation of the master equation, the recovery of deterministic dynamics, birth-and-death processes, and Langevin theory. The notes were…
We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as…
We survey old and new results about the cohomology of the moduli space $A_g$ of principally polarized abelian varieties of genus $g$ and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and…
We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a…
These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and…
This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular…
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to…
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…
These are extended notes based on lectures given by Vincent Franjou, Paul Sobaje, Peter Symonds and Antoine Touz\'e at the Master Class on New Developments in Finite Generation of Cohomology that took place at Bielefeld University in…
Lecture notes. Introduction to the cohomology of algebras, Lie algebras, Lie bialgebras and quantum groups. Contains a new derivation of the classification of classical r-matrices in terms of deformation cohomology, and a calculation of the…
The modular representation theory of finite groups has its origins in the work of Richard Brauer. In this survey article we first discuss the work being done on some outstanding conjectures in the theory. We then describe work done in the…