Past Representation Theory Seminar

22 November 2012
14:00
Dr Oleg Chalykh
Abstract
To any complex smooth variety Y with an action of a finite group G, Etingof associates a global Cherednik algebra. The usual rational Cherednik algebra corresponds to the case of Y= C^n and a finite Coxeter group G < GL_n. I will consider the special case of Etingof's definition when Y=X^n, with X a smooth algebraic curve, and G is the symmetric group S_n acting naturally on Y. I will explain how this algebra is related to the deformed preprojective algebras of Crawley-Boevey, how this leads to its representation by generators and relations, and will relate two exisitng definitions of Calogero-Moser spaces in this case.
  • Representation Theory Seminar
15 November 2012
14:00
David Jorgensen
Abstract
We will define certain Verdier quotients of the singularity category of a ring R, called defect categories. The triviality of these defect categories determine, for example, whether a commutative local ring is Gorenstein, or a complete intersection. The dimension (in the sense of Rouquier) of the defect category thus gives a measure of how close such a ring is to being Gorenstein, respectively, a complete intersection. Examples will be given. This is based on joint work with Petter Bergh and Steffen Oppermann.
  • Representation Theory Seminar
1 November 2012
14:00
David Stewart
Abstract
In 1977, Cline Parshall, Scott and van der Kallen wrote a seminal paper `Rational and generic cohomology' which exhibited a connection between the cohomology for algebraic groups and the cohomology for finite groups of Lie type, showing that in many cases one can conclude that there is an isomorphism of cohomology through restriction from the algebraic to the finite group. One unfortunate problem with their result is that there remain infinitely many modules for which their theory---for good reason---tells us nothing. The main result of this talk (recent work with Parshall and Scott) is to show that almost all the time, one can manipulate the simple modules for finite groups of Lie type in such a way as to recover an isomorphism of its cohomology with that of the algebraic group.
  • Representation Theory Seminar
25 October 2012
14:00
Johan Steen
Abstract
A triangulated category admits a strong generator if, roughly speaking, every object can be built in a globally bounded number of steps starting from a single object and taking iterated cones. The importance of strong generators was demonstrated by Bondal and van den Bergh, who proved that the existence of such objects often gives you a representability theorem for cohomological functors. The importance was further emphasised by Rouquier, who introduced the dimension of triangulated categories, and tied this numerical invariant to the representation dimension. In this talk I will discuss the generation time for strong generators (the least number of cones required to build every object in the category) and a refinement of the dimension which is due to Orlov: the set of all integers that occur as a generation time. After introducing the necessary terminology, I will focus on categories occurring in representation theory and explain how to compute this invariant for the bounded derived category of the path algebras of type A and D, as well as the corresponding cluster categories.
  • Representation Theory Seminar
18 October 2012
14:00
Petter Bergh
Abstract
By classical results of Thomason, the Grothendieck group of a triangulated category classifies the triangulated subcategories. More precisely, there is a bijective correspondence between the set of triangulated subcategories and the set of subgroups of the Grothendieck group. In this talk, we extend Thomason's results to "higher" triangulated categories, namely the recently introduced n-angulated categories. This is joint work with Marius Thaule.
  • Representation Theory Seminar
31 May 2012
14:00
to
16:00
Prof Joel Kamnitzer
Abstract
<p>Mirkovic-Vilonen polytopes are a combinatorial tool for studying<br />perfect bases for representations of semisimple Lie algebras. &nbsp;They<br />were originally introduced using MV cycles in the affine Grassmannian,<br />but they are also related to the canonical basis. &nbsp;I will explain how<br />MV polytopes can also be used to describe components of Lusztig quiver<br />varieties and how this allows us to generalize the theory of MV<br />polytopes to the affine case.</p>
  • Representation Theory Seminar
10 May 2012
15:00
Abstract
<p>The geometric Langlands correspondence relates rank n integrable connections on a complex Riemann surface $X$ to regular holonomic D-modules on&nbsp; $Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.&nbsp; If we replace $X$ by a smooth irreducible curve over a finite field of characteristic p then there is a version of the geometric Langlands correspondence involving $l$-adic perverse sheaves for $l\neq p$.&nbsp; In this lecture we consider the case $l=p$, proposing a $p$-adic version of the geometric Langlands correspondence relating convergent $F$-isocrystals on $X$ to arithmetic $D$-modules on $Bun_n(X)$.<br /><br /></p>
  • Representation Theory Seminar
10 May 2012
15:00
Alex Paulin
Abstract

The geometric Langlands correspondence relates rank n integrable connections 
on a complex Riemann surface $X$ to regular holonomic D-modules on 
$Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace 
$X$ by a smooth irreducible curve over a finite field of characteristic p 
then there is a version of the geometric Langlands correspondence involving 
$l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the 
case $l=p$, proposing a $p$-adic version of the geometric Langlands 
correspondence relating convergent $F$-isocrystals on $X$ to arithmetic 
$D$-modules on $Bun_n(X)$.

  • Representation Theory Seminar
8 March 2012
15:00
John Duncan
Abstract
In April 2010 Eguchi--Ooguri--Tachikawa observed a fascinating connection between the elliptic genus of a K3 surface and the largest Mathieu group. We will report on joint work with Miranda Cheng and Jeff Harvey that identifies this connection as one component of a system of surprising relationships between a family of finite groups, their representation theory, and automorphic forms of various kinds Mock modular forms, and particularly their shadows, play a key role in the analysis, and we find several of Ramanujan's mock theta functions appearing as McKay--Thompson series arising from the umbral groups.
  • Representation Theory Seminar

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