Forthcoming events in this series

Tue, 21 Jun 2022

14:00 - 15:00
L6

### The orbit method and normality of closures of nilpotent orbits

Dan Barbasch
(Cornell University, USA)
Abstract

The work of Kraft-Procesi classifies closures of nilpotent orbits that are normal in the cases of classical complex Lie algebras. Subsequent work of Ranee Brylinsky combines this work with the Theta correspondence as defined by Howe to attach a representation of the corresponding complex group. It provides a quantization of the closure of a nilpotent orbit. In joint work with Daniel Wong, we carry out a detailed analysis of these representations viewed as (\g,K)-modules of the complex group viewed as a real group. One consequence is a "representation theoretic" proof of the classification of Kraft-Procesi.

Tue, 14 Jun 2022

14:00 - 15:00
L6

### Invariable generation and totally deranged elements of simple groups

Scott Harper
(Bristol)
Abstract

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

Tue, 07 Jun 2022

14:00 - 15:00
L6

### How to restrict representations from a complex reductive group to a real form

Lucas Mason-Brown
((Oxford University))
Abstract

Let G(R) be the real points of a complex reductive algebraic group G. There are many difficult questions about admissible representations of real reductive groups which have (relatively) easy answers in the case of complex groups. Thus, it is natural to look for a relationship between representations of G and representations of G(R). In this talk, I will introduce a functor from admissible representations of G to admissible representations of G(R). This functor interacts nicely with many natural invariants, including infinitesimal character, associated variety, and restriction to a maximal compact subgroup, and it takes unipotent representations of G to unipotent representations of G(R).

Tue, 31 May 2022

14:00 - 15:00
L6

### Towards 3d mirror symmetry for characteristic classes

Richard Rimanyi
(UNC Chapel Hill)
Abstract

In the first half of the talk, we will explore the concept of a characteristic class of a subvariety in a smooth ambient space. We will focus on the so-called stable envelope class,  in cohomology, K theory, and elliptic cohomology (due to Okoukov-Maulik-Aganagic). Stable envelopes have rich algebraic combinatorics, they are at the heart of enumerative geometry calculations, they show up in the study of associated (quantum) differential equations, and they are the main building blocks of constructing quantum group actions on the cohomology of moduli spaces.

In the second half of the talk, we will study a generalization of Nakajima quiver varieties called Cherkis’ bow varieties. These smooth spaces are endowed with familiar structures: holomorphic symplectic form, tautological bundles, torus action. Their algebraic combinatorics features a new powerful operation, the Hanany-Witten transition. Bow varieties come in natural pairs called 3d mirror symmetric pairs. A conjecture motivated by superstring theory predicts that stable envelopes on 3d mirror pairs are equal (in a sophisticated sense that involves switching equivariant and Kahler parameters). I will report on a work in progress, with T. Botta, to prove this conjecture.

Tue, 24 May 2022

15:30 - 16:30
L6

### On centralizers in Azumaya domains

Thomas Bitoun
(University of Calgary)
Abstract

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.

Tue, 24 May 2022

14:00 - 15:00
L5

### Dirac index and associated cycles for Harish-Chandra modules

Salah Mehdi
(Université de Lorraine)
Abstract

Since their introduction in 1928 by Paul A. Dirac, Dirac operators have been playing essential roles in many areas of Physics and Mathematics. In particular, they provide powerful and efficient tools to clarify (and sometimes solve) important problems in representation theory of real Lie groups, p-adic groups or Hecke algebras, such as classification, unitarity and geometric realization. In this representation theoretic context, the Dirac index of a Harish-Chandra module is a virtual module induced by Vogan’s Dirac cohomology. Once we observe that Dirac index commutes with translation functors, we will associate a polynomial (on a Cartan subalgebra) with a coherent family of Harish-Chandra modules. Then we shall explain how this polynomial can be used to connect nilpotent orbits, associated cycles and the leading term of the Taylor expansion of the characters of Harish-Chandra modules. This is joint wok with P. Pandzic, D. Vogan and R. Zierau.

Tue, 17 May 2022

14:00 - 15:00
L6

### Splitting fields of real irreducible representations of finite groups

Dmitrii Pasechnik
(Oxford)
Abstract

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb R$, is realisable over the field $E$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb Q(\zeta_n)$ to one over $E$.

Thu, 12 May 2022

15:30 - 16:30
L4

### Representations of p-adic groups – with a twist

Jessica Fintzen
(Bonn University)
Abstract

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex or mod-$\ell$) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress including joint work with Kaletha and Spice that introduces a twist to the story, and outline some applications.

Tue, 10 May 2022

14:00 - 15:00
L6

### Equivariance in Deep Learning

Sheheryar Zaidi and Bryn Elesedy
(Oxford)
Abstract

One core aim of (supervised) machine learning is to approximate an unknown function given a dataset containing examples of input-output pairs. Real-world examples of such functions include the mapping from an image to its label or the mapping from a molecule to its energy. For a variety of such functions, while the precise mapping is unknown, we often have knowledge of its properties. For example, the label of an image may be invariant to rotations of the input image. Generally, such properties formally correspond to the function being equivariant to certain actions on its input and output spaces. This has led to much research on building equivariant function classes (aka neural networks). In this talk, we survey this growing field of equivariance in deep learning for a mathematical audience, motivating the need for equivariance, covering concrete examples of equivariant neural networks, and offering a learning theoretic perspective on the benefits of equivariance.

Tue, 03 May 2022

14:00 - 15:00
L6

### Equivariant line bundles with connection on the Drinfeld upper half-space

Amy Zhu
(Cambridge)
Abstract

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.

Tue, 08 Mar 2022
14:00
L6

### Localization in the smooth representation theory in natural characteristic of p-adic Lie groups

Peter Schneider
(Muenster)
Abstract

In commutative algebra localizing a ring and its modules is a fundamental technique. In the general case of a Grothendieck abelian category or even a triangulated category with small direct sums this is replaced by forming the quotient category by a localizing subcategory. Therefore the classification of these localizing subcategories becomes an important problem. I will begin by recalling the case of the (derived) module category of a commutative noetherian ring due to Gabriel and Hopkins/Neeman, respectively, in order to give an idea how such a classification can look like.

The case of interest in this talk is the derived category D(G) of smooth representation in characteristic p of a p-adic Lie group G. This is motivated by the emerging p-adic Langlands program. In joint work with C. Heyer we have some modest initial results if G is compact pro-p or abelian. which I will present.

Tue, 18 Jan 2022
14:00
Virtual

### Dimensions of Iwasawa algebras and their representations

James Timmins
(Oxford)
Abstract

The Iwasawa algebra of a compact $p$-adic Lie group is fundamental to the study of the representations of the group. Understanding this representation theory is crucial in progress towards a (mod p) local Langlands correspondence. However, much remains unknown about Iwasawa algebras and their modules.

In this talk we'll aim to measure the size of the Iwasawa algebra and its representations. I'll explain the algebraic tools we use to do this - Krull dimension and canonical dimension - and survey previously known examples. Our main result is a new bound on these dimensions for the group $SL_2(O_F)$, where $F$ is a finite extension of the p-adic numbers. When $F$ is a quadratic extension, we find the Krull dimension is exactly 5, as predicted by a conjecture of Ardakov and Brown.

Tue, 30 Nov 2021
14:00
Virtual

### Braids, Unipotent Representations, and Nonabelian Hodge Theory

Minh-Tâm Trinh
(MIT)
Abstract

A complex plane curve singularity gives rise to two objects: (1) a moduli space that representation theorists call an affine Springer fiber, and (2) a topological link up to isotopy. Roughly a decade ago, Oblomkov–Rasmussen–Shende conjectured a striking identity relating the homology of the affine Springer fiber to the so-called HOMFLYPT homology of the link. In unpublished writing, Shende speculated that it would follow from advances in nonabelian Hodge theory: the study of transcendental diffeomorphisms relating “Hitchin” and “Betti” moduli spaces. We make this dream precise by expressing HOMFLYPT homology in terms of the homology of a “Betti”-type space, which, we conjecture, deformation-retracts onto the affine Springer fiber. In doing so, we recast the whole story in terms of an arbitrary semisimple group. We give evidence for the nonabelian Hodge conjecture at the numerical level, using a mysterious formula that involves rational Cherednik algebras and the degrees of unipotent principal-series representations.

Tue, 16 Nov 2021
14:00
L3

### Homology torsion growth in finitely presented pro-p groups

Nikolay Nikolov
(Oxford University)
Abstract

Let $G$ be a finitely presented residually finite group. We are interested in the growth of size of the torsion of $H^{ab}$ as a function of $|G:H|$ where $H$ ranges over normal subgroups of finite index in $G$. It is easy to see that this grows at most exponentially in terms of $|G:H|$. Of particular interest is the case when $G$ is an arithmetic hyperbolic 3-manifold group and $H$ ranges over its congruence subgroups. Proving exponential lower bounds on the torsion appears to be difficult and in this talk I will focus on the situation of finitely presented pro-$p$ groups.

In contrast with abstract groups I will show that in finitely presented pro-$p$ groups torsion in the abelianizations can grow arbitrarily fast. The examples are rather 'large' pro-$p$ groups, in particular they are virtually Golod-Shafarevich. When we restrict to $p$-adic analytic groups the torsion growth is at most polynomial.

Tue, 09 Nov 2021
14:00
L5

### TBA

Marek Kaluba
(Karlsruher Institute für Technologie)
Abstract

In this leisure talk I will show how a sum of squares decomposition problem can be transformed to a problem of semi-definite optimization. Then the practicality of such reformulations will be discussed, illustrated by an explicit example of Artin's solutions to Hilberts 17th problem. Finally I will show how a numerical solution could be turned into a mathematically certified one, using the order structure on the cone of sums of squares.
The talk requires no pre-requisite knowledge of neither optimization or programming and only undergraduate mathematics.

Tue, 02 Nov 2021
14:15
L5

### Solving semidecidable problems in group theory

Giles Gardam
(Münster)
Abstract

Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.

Tue, 15 Jun 2021
14:15
Virtual

### Harish-Chandra Lefschetz principle for branching laws of general linear groups

Kei Yuen Chan
(Fudan University)
Abstract

The Harish-Chandra Lefschetz principle asserts representation theory for real groups, p-adic groups and automorphic forms should be placed on an equal footing. A particular example in this aspect is that Ciubotaru and Trapa constructed Arakawa-Suzuki type functors between category of Harish-Chandra modules and category of graded Hecke algebra modules, giving an explicit connection on the representation categories between p-adic and real sides.

This talk plans to begin with comparing the representation theory between real and p-adic general linear groups, such as unitary and unipotent representations. Then I shall explain results in more details on the p-adic branching law from GL(n+1) to GL(n), including branching laws for Arthur type representations (one of the non-tempered Gan-Gross-Prasad conjectures). The analogous results and predictions on the real group side will also be discussed. Time permitting, I will explain a notion of left-right Bernstein-Zelevinsky derivatives and its applications on branching laws.

Tue, 08 Jun 2021
14:15
Virtual

### Kaplansky's conjectures

Giles Gardam
(University Muenster)
Abstract

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.

Tue, 01 Jun 2021
14:15
Virtual

### p-Kazhdan—Lusztig theory for Hecke algebras of complex reflection groups

Chris Bowman
(University of York)
Abstract

Riche—Williamson recently proved that the characters of tilting modules for GL_h are given by non-singular p-Kazhdan—Lusztig polynomials providing p>h.  This is equivalent to calculating the decomposition numbers for symmetric groups labelled by partitions with at most h columns.  We discuss how this result can be generalised to all cyclotomic quiver Hecke algebras via a new and explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson’s diagrammatic endomorphism algebras of Bott–Samelson bimodules.

This allows us to give an elementary and explicit proof of the main theorem of Riche–Williamson’s recent monograph and extend their categorical equivalence to all cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza’s categorical blob conjecture.  Furthermore, it allows us to classify and construct the homogeneous simple modules of quiver Hecke algebras via BGG resolutions.

This is joint work with A. Cox, A. Hazi, D.Michailidis, E. Norton, and J. Simental.

Tue, 18 May 2021
14:15
Virtual

### Categorification of the elliptic Hall algebra

Alistair Savage
(Ottawa)
Abstract

The elliptic Hall algebra has appeared in many different contexts in representation theory and geometry under different names.  We will explain how this algebra is categorified by the quantum Heisenberg category.  This diagrammatic category is modelled on affine Hecke algebras and can be viewed as a deformation of the framed HOMFLYPT skein category underpinning the HOMFLYPT link invariant.  Using the categorification of the elliptic Hall algebra, one can construct large families of representations for this algebra.

Tue, 11 May 2021
14:15
Virtual

### C*-blocks and crossed products for real and p-adic reductive groups

Anne-Marie Aubert
(Sorbonne Université - Université de Paris)
Abstract

Let G be a real or a p-adic connected reductive group. We will recall the description of the connected components of the tempered dual of G in terms of certain subalgebras of its reduced C*-algebra.

Each connected component comes with a torus equipped with a finite group action. We will see that, under a certain geometric assumption on the structure of stabilizers for that action (that is always satisfied for real groups), the component has a simple geometric structure which encodes the reducibility of the associate parabolically induced representations.

We will provide a characterization of the connected components for which the geometric assumption is satisfied, in the case when G is a symplectic group.

This is a joint work with Alexandre Afgoustidis.

Tue, 27 Apr 2021

14:15 - 15:15
Virtual

### An upper bound for the nonsolvable length of a finite group in terms of its shortest law

Orazio Puglisi
(Università degli Studi di Firenze)
Abstract

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in G, is called the nonsolvable length $\lambda(G)$ of $G$. In recent years several authors have investigated this invariant and its relation to other relevant parameters. E.g. it has been conjectured by Khukhro and Shumyatsky (as a particular case of a more general conjecture about non-$p$-solvable length) and Larsen that, if $\nu(G)$ is the length of the shortest law holding in the finite group G, the nonsolvable length of G can be bounded above by some function of $\nu(G)$. In a joint work with Francesco Fumagalli and Felix Leinen we have confirmed this conjecture proving that the inequality $\lambda(G) < \nu(G)$ holds in every finite group $G$. This result is obtained as a consequence of a result about permutation representations of finite groups of fixed nonsolvable length. In this talk I will outline the main ideas behind the proof of our result.

Tue, 09 Mar 2021
14:15
Virtual

### Coadmissible modules, bornologies, and derived categories II

Andreas Bode
(ENS Lyon)
Abstract

Coadmissible modules over Frechet-Stein algebras arise naturally in p-adic representation theory, e.g. in the study of locally analytic representations of p-adic Lie groups or the function spaces of rigid analytic Stein spaces. We show that in many cases, the category of coadmissible modules admits an exact and fully faithful embedding into the category of complete bornological modules, also preserving tensor products. This allows us to introduce derived methods to the study of coadmissible modules without forsaking the analytic flavour of the theory. As an application, we introduce six functors for Ardakov-Wadsley's D-cap-modules and discuss some instances where coadmissibility (in a derived sense) is preserved.

Tue, 02 Mar 2021
14:15
Virtual

### Graded Clifford-Drinfeld algebras

Kieran Calvert
(Manchester University)
Abstract

We combine the notions of graded Clifford algebras and Drinfeld algebras. This gives us a framework to study algebras with a PBW property and underlying vector space $\mathbb{C}[G] \# Cl(V) \otimes S(U)$ for $G$-modules $U$ and $V$. The class of graded Clifford-Drinfeld algebras contains the Hecke-Clifford algebras defined by Nazarov, Khongsap-Wang. We give a new example of a GCD algebra which plays a role in an Arakawa-Suzuki duality involving the Clifford algebra.

Tue, 23 Feb 2021

14:15 - 15:15
Virtual

### From braids to transverse slices in reductive groups

Dr Wicher Malten
(University of Oxford)
Abstract

We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties.

Our perspective furnishes a common generalisation, essentially solving the problem. We also give a geometric criterion for Weyl group elements to yield strictly transverse slices.