Past Representation Theory Seminar

23 June 2015
14:00
Abstract

We will discuss our approach to global analytic geometry, based on overconvergent power series and functors of functions. We will explain how slight modifications of it allow us to develop a derived version of global analytic geometry. We will finish by discussing applications to the cohomological study of arithmetic varieties.

  • Representation Theory Seminar
23 June 2015
10:00
Jennifer Balakrishnan
Abstract

We describe how p-adic height pairings allow us to find integral points on hyperelliptic curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss how to carry out this ``quadratic Chabauty'' method over quadratic number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).

  • Representation Theory Seminar
27 November 2014
14:00
to
16:00
Pavel Safronov
Abstract

Both sides of the geometric Langlands correspondence have natural Hecke
symmetries. I will explain an identification between the Hecke
symmetries on both sides via the geometric Satake equivalence. On the
abelian level it relates the topology of a variety associated to a group
and the representation category of its Langlands dual group.
 

  • Representation Theory Seminar
30 October 2014
14:00
to
16:00
Alexey Sevastyanov
Abstract

I shall show that for every conjugacy class O in a connected semisimple algebraic group G over an algebraically closed field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.

  • Representation Theory Seminar
23 October 2014
14:00
Dario Beraldo
Abstract

Let $G$ be a connected reductive group and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $Bun_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called $coeff_{G}$, from the DG category of D-modules on $Bun_G$ to a certain DG category $Wh(G, ext)$, called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor  $$\mathbb{L}_G : \operatorname{IndCoh}_{N}(LocSys_{\check{G}} ) \to D(Bun_G)$$ with the Whittaker model. For $G = GL_n$ and $G = PGL_n$, we prove that $coeff_G$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.

  • Representation Theory Seminar
12 June 2014
14:00
to
16:00
Emily Cliff & Robert Laugwitz
Abstract
<p>Factorization is a property of global objects that can be built up from local data. In the first half, we introduce the concept of factorization spaces, focusing on two examples relevant for the Geometric Langlands programme: the affine Grassmannian and jet spaces.</p> <p>In the second half, factorization algebras will be defined including a discussion of how factorization spaces and commutative algebras give rise to examples. Finally, chiral homology is defined as a way to give global invariants of such objects.</p>
  • Representation Theory Seminar
10 June 2014
15:45
Abstract
The Weil representation is a central object in mathematics responsible for many important results. Given a symplectic vector space V over a finite field (of odd characteristic) one can construct a "quantum" Hilbert space H(L) attached to a Lagrangian subspace L in V. In addition, one can construct a Fourier Transform F(M,L): H(L)→H(M), for every pair of Lagrangians (L,M), such that F(N,M)F(M,L)=F(N,L), for every triples (L,M,N) of Lagrangians. This can be used to obtain a natural “quantum" space H(V) acted by the symplectic group Sp(V), obtaining the Weil representation. In the lecture I will give elementary introduction to the above constructions, and discuss the categorification of these Fourier transforms, what is the related sign problem, and what is its solution. The outcome is a natural category acted by the algebraic group G=Sp, obtaining the categorical Weil representation. The sign problem was worked together with Ofer Gabber (IHES).
  • Representation Theory Seminar

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