| DATE |
SPEAKER |
TITLE (click to show abstract) |
| September 12 |
Maria Gorelik, The Weizmann Institute of Science |
On the simplicity of minimal W-algebras
Abstract: This is a report on an on-going project, joint with V. Kac, concerning simplicity of minimal W-algebras.
Minimal W-algebras are the simplest conformal vertex algebras. For non-integral central levels the simplicity
criterion was established 15 years ago in papers by Gorelik-Kac and Hoyt-Reif. The case of integral central
levels for superalgebras of non-zero defect is still open,
and I will report on recent progress in this area.
|
| September 19 |
Inna Entova-Aizenbud, Ben Gurion University |
Representation stability for \(GL_n(F_q)\)
Abstract: I will present some results on the Deligne categories for
the family of groups \(GL_n(F_q), n>0\), based on a joint project with T.
Heidersdorf. This family of symmetric monoidal categories interpolates
the tensor categories of complex representations of \(GL_n(F_q)\) and have
been previously constructed by F. Knop. I will describe some
properties of these categories, as well as the relation to the
category of algebraic representations of the infinite group
\(GL_{\infty}(F_q)\).
|
| September 26 |
Vladimir Hinich, University of Haifa |
Matsumoto theorem for skeleta
Abstract: This is a joint work with M. Gorelik and V. Serganova.
The classical Matsumoto theorem of 1964 asserts that two reduced
expressions in a Coxeter group
define the same element of the group iff they are "braided
equivalent". We present a more general context
(graphs with a geometric realization) for which a similar result
holds. This is applicable, in particular, to
the skeleta, graphs appearing in the description of root Lie
superalgebras, as in the recent work of the same authors.
A full proof will be presented. The only prerequisite is basic algebra.
|
| October 3 |
Daniel Klyuev, MIT |
Analytic Langlands correspondence for \(G=PGL(2,\mathbb{C})\)
Abstract: Analytic Langlands correspondence was suggested by Langlands several years ago and developed by Etingof,
Frenkel and Kazhdan. On one side of this conjectural correspondence there are \(\check{G}\)-opers on \(X\) satisfying
a certain condition, on the other a joint spectrum of certain operators on the space of functions on a moduli space of
\(G\)-bundles on \(X\). I will describe the main picture and present new results in this direction.
Partially based on joint projects with A. Wang and S. Raman
|
| October 10 |
Eric Jankowski, UC Berkeley |
Toric supervarieties
Abstract: Classically, toric varieties provide a dictionary between algebraic and convex geometry via their descriptions
as polyhedral fans. I will review this construction and describe how it generalizes to the super setting. In particular,
I will define algebraic supertori and give an analogous classification of toric supervarieties with one odd dimension.
The case of higher odd dimension appears to be a wild problem, but I will share some examples where they arise naturally.
|
| October 17 |
Tony Feng, UC Berkeley |
Modular representation theory and Langlands functoriality
Abstract: I will discuss some aspects of modular representation theory that arise in the study of the Local Langlands
correspondence, which concerns a duality between the representation theory of p-adic Lie groups and the representation
theory of Galois groups of p-adic fields. I will explain that Langlands philosophy can be used to prove some new,
purely representation-theoretic results on tilting modules. In the other direction, I will pose some problems in
representation theory whose answers would shed light on the local Langlands correspondence.
|
| October 24 |
Dmitry Kubrak, IAS |
Hodge-to-de Rham degeneration for BG in characteristic p
Over complex numbers one of the basic consequences of Hodge theory is the degeneration of the Hodge-to-de Rham (HdR)
spectral sequence for a smooth proper scheme X over complex numbers. As first noted by Mumford in 1960's, this is no
longer true in characteristic p, even for surfaces.
Nevertheless, in 1987 Deligne and Illusie showed that if X lifts modulo p^2 then the HdR spectral sequence does
degenerate in degrees up to p-1. Whether in this situation HdR spectral sequence degenerates in all degrees was not
known until very recently, when Petrov showed that it doesn't necessarily degenerate in degree p.
I will try to explain that nonetheless HdR spectral sequence in the case of the classifying stack BG with G reductive
does degenerate: this turns out to be a consequence of some intricate (but rather classical) representation-theoretic
results due to Cline-Parshall-Scott.
|
| October 31 |
Karthik Ganapathy, University of Michigan |
Equivariant commutative algebra in positive characteristic
Equivariant commutative algebra in positive characteristic
Abstract: In the presence of a large group action, even non-noetherian rings sometimes behave like noetherian rings.
For example, Cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by the orbit
of finitely many polynomials. In this talk, I will give a brief introduction to equivariant commutative algebra where we
systematically develop commutative algebra in interesting (and not necessarily rigid) tensor categories. I will mostly
focus on the category of polynomial representations of the infinite general linear group
(= category of strict polynomial functors).
|
| November 7 |
Sergei Korotkikh, UC Berkeley |
Construction of q-Hahn integrable models using representations of quantum affine algebras.
Abstract: A decade ago Povolotsky has introduced a new type of integrable models which are governed by so-called
q-Hahn weights. In my talk I will explain a new construction of these models, originating from representations of
quantum loop sl2 algebra. More precisely, I will show how q-Hahn weights appear as matrix coefficients of isomorphisms
between tensor products of Kirillov-Reshetikhin modules. Time permitting, I will also outline how this new cnstruction
leads to results about random particle systems, random polymer models and Macdonald functions.
|
| November 14 |
Ilia Nikrasov, University of Michigan |
Arboreal Tensor Categories
Abstract: Last year, Andrew Snowden (UofM) and Nate Harman (UGA) introduced a new combinatorial way of constructing tensor
categories. I will show how the general theory works in the case of an infinite symmetric group; we will arrive at
Deligne's categories Rep(S_t). Next we will explore arboreal tensor categories: an entirely new class of tensor
categories coming from a mixture of tree combinatorics and “finitary" harmonic analysis based on oligomorphic group
actions. I will draw similarities and emphasize differences with previously studied tensor categories in characteristic 0.
The talk is based on arXiv:2308.06660.
|
| November 21 |
No seminar |
|
| November 28 |
|
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| December 5 |
Monica Vazirani, UC Davis |
Skeins on Tori
Abstract: We study skeins on the 2-torus and 3-torus via the representation
theory of the double affine Hecke algebra of type A and its connection
to quantum D-modules. As an application we can compute the dimension of
the generic \(SL_N\)- and \(GL_N\)-skein module of the 3-torus for
arbitrary \(N\).
This is joint work with Sam Gunningham and David Jordan.
|