Forthcoming events in this series


Tue, 28 Nov 2017
14:15
L4

Dirac induction for rational Cherednik algebras

Marcelo De Martino
(Oxford University)
Abstract

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.

Tue, 14 Nov 2017

14:15 - 15:15
L4

Representations of pseudo-reductive groups

Dr David Stewart
(School of Mathematics & Statistics Newcastle University)
Abstract

Pseudo-reductive groups are smooth connected linear algebraic groups over a field k whose k-defined unipotent radical is trivial. If k is perfect then all pseudo-reductive groups are reductive, but if k is imperfect (hence of characteristic p) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by Conrad-Gabber-Prasad has made it possible to reduce the classification of the simple representations of pseudo-reductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.

Tue, 31 Oct 2017
14:15
L4

Multiplicity-free primitive ideals and W-algebras

Alexander Premet
(University of Manchester)
Abstract

In my talk I will explain how to relate 1-dimensional representations of finite W-algebras with multiplicity free primitive ideals of universal enveloping algebras and representations of minimal dimension of the corresponding reduced enveloping algebras (Humphreys' conjecture). I will also mention some open problems in the field.

Tue, 24 Oct 2017

14:15 - 15:15
L4

Dimers with boundary, associated algebras and module categories

Karin Baur
(Graz)
Abstract

Dimer models with boundary were introduced in joint work with King and Marsh as a natural
generalisation of dimers. We use these to derive certain infinite dimensional algebras and
consider idempotent subalgebras w.r.t. the boundary.
The dimer models can be embedded in a surface with boundary. In the disk case, the
maximal CM modules over the boundary algebra are a Frobenius category which
categorifies the cluster structure of the Grassmannian.

 

Tue, 17 Oct 2017

14:15 - 15:15
L4

From classical tilting to 2-term silting

Aslak Buan
(Trondheim)
Abstract

We give a short reminder about central results of classical tilting theory, 
including the Brenner-Butler tilting theorem, and
homological properties of tilted and quasi-tilted algebras. We then discuss 
2-term silting complexes and endomorphism algebras of such objects,
and in particular show that some of these classical results have very natural 
generalizations in this setting.
(joint work with Yu Zhou)

Tue, 16 May 2017
14:15
L4

Cherednik algebras at infinity

Maxim Nazarov
(York University)
Abstract

Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at N → ∞ . These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.

Wed, 03 May 2017

14:00 - 15:00
L3

On finiteness properties of the Johnson filtrations

Mikhail Ershov
(Virginia)
Abstract

Let $A$ denote either the automorphism group of the free group of rank $n$ or the mapping class group of an orientable surface of genus $n$ with at most 1 boundary component, and let $G$ be either the subgroup of IA-automorphisms or the Torelli subgroup of $A$, respectively. I will discuss various finiteness properties of subgroups containing $G_N$, the $N$-th term of the lower central series of $G$, for sufficiently small $N$. In particular, I will explain why
(1) If $n \geq 4N-1$, then any subgroup of G containing $G_N$ (e.g. the $N$-th term of the Johnson filtration) is finitely generated
(2) If $n \geq 8N-3$, then any finite index subgroup of $A$ containing $G_N$ has finite abelianization.
The talk will be based on a joint work with Sue He and a joint work with Tom Church and Andrew Putman

Tue, 02 May 2017
14:15
L4

Representations of p-adic groups via geometric invariant theory

Beth Romano
(Cambridge University)
Abstract

Let G be a split reductive group over a finite extension k of Q_p. Reeder and Yu have given a new construction of supercuspidal representations of G(k) using geometric invariant theory. Their construction is uniform for all p but requires as input stable vectors in certain representations coming from Moy-Prasad filtrations. In joint work, Jessica Fintzen and I have classified the representations of this kind which contain stable vectors; as a corollary, the construction of Reeder-Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G(k). 

Tue, 07 Mar 2017
14:15
L4

The rationality of blocks of quasi-simple finite groups

Niamh Farrell
(City University London)
Abstract

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Morita Frobenius numbers were introduced by Kessar in 2004 in the context of Donovan’s Conjecture in block theory. I will present the latest results of a project in which we aim to calculate the Morita Frobenius numbers of the blocks of quasi-simple finite groups. I will also discuss the importance of a recent result of Bonnafe-Dat-Rouquier for our methods, and explain the relationship between Morita Frobenius numbers and Donovan’s Conjecture. 

Tue, 28 Feb 2017
14:15
L4

Sklyanin algebras are minimal surfaces

Sue Sierra
(University of Edinburgh)
Abstract

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is:  what are the minimal models within a birational class?  It is not even clear a priori what the correct definition is of a minimal model in this context.

We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.
This is joint work in progress with Dan Rogalski and Toby Stafford.

 

Tue, 21 Feb 2017

14:15 - 15:15
L4

Growth, generation, and conjectures of Gowers and Viola

Aner Shalev
(Hebrew University of Jerusalem)
Abstract

I will discuss recent results in finite simple groups. These include growth, generation (with a number theoretic flavour), and conjectures of Gowers and Viola on mixing and complexity whose proof requires representation theory as a main tool.
 

Tue, 07 Feb 2017
14:15
L4

Modular W-algebras and reduced enveloping algebras

Simon Goodwin
(University of Birmingham)
Abstract

We give an overview of joint work with Lewis Topley on modular W-algebras. In particular, we outline the classification 1-dimensional modules for modular W-algebras for gl_n, which in turn this leads to a classification of minimal dimensional modules for reduced enveloping algebras for gl_n.

Tue, 24 Jan 2017

14:15 - 15:15
L4

An Euler-Poincare formula for a depth zero Bernstein projector

Allen Moy
(Hong Kong University of Science and Technology)
Abstract


Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an
Euler-Poincare formula for the r-depth Bernstein projector. We establish an Euler-Poincare formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P.  This work is joint with Dan Barbasch and Dan Ciubotaru.
 

Tue, 17 Jan 2017

14:15 - 15:15
L4

Endo-parameters and the Local Langlands Correspondence for classical groups

Shaun Stevens
(University of East Anglia)
Abstract

The local Langlands correspondence for classical groups gives a natural finite-to-one map between certain representations of p-adic classical groups and certain self-dual representations of the absolute Weil group of a p-adic field (and more). On both sides of the correspondence, the description of the representations involves a ``wild part'' of more arithmetic nature and a ``tame part'' of more geometric nature, and the notion of endo-parameter (due to Bushnell--Henniart for general linear groups) is designed to describe the ``wild part'' of the Langlands correspondence. I will explain what this means and the connection with representations of affine Hecke algebras. This is joint work with Blondel--Henniart, with Lust, and with Kurinczuk--Skodlerack.

Tue, 15 Nov 2016

14:15 - 15:15
L4

Representations of finite groups over self-injective rings

Greg Stevenson
(Bielefeld)
Abstract

 For a group algebra over a self-injective ring
there are two stable categories: the usual one modulo projectives
and a relative one where one works modulo representations
which are free over the coefficient ring.
I'll describe the connection between these two stable categories,
which are "birational" in an appropriate sense.
I'll then make some comments on the specific case
where the coefficient ring is Z/nZ and give a more
precise description of the relative stable category.

Tue, 08 Nov 2016
14:15
L4

Decomposition rules for representations of p-adic groups

Max Gurevich
(Weizmann Institute)
Abstract


What are the irreducible constituents of a smooth representation of a p-adic group that is constructed through parabolic induction? In the case of GL_n this is the study of the multiplicative behaviour of irreducible representations in the Bernstein-Zelevinski ring. Strikingly, the same decomposition problem can be reformulated through various Lie-theoretic settings of type A, such as canonical bases in quantum groups, representations of affine Hecke algebras, quantum affine Lie algebras, or more recently, KLR algebras. While partially touching on some of these phenomena, I will present new results on the problem using mostly classical tools. In particular, we will see how introducing a width invariant to an irreducible representation can circumvent the complexity involved in computations of Kazhdan-Lusztig polynomials.

Tue, 01 Nov 2016

14:15 - 15:15
L4

Recovering automorphisms of quantum spaces

J Grabowski
(Lancaster)
Abstract

It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

Tue, 25 Oct 2016

14:15 - 15:15
L4

On the automorphic spectrum supported on the Borel subgroup

Marcelo De Martino
(Mathematical Institute, University of Oxford)
Abstract

In this talk, we consider a split connected semisimple group G defined over a global field F. Let A denote the ring of adèles of F and K a maximal compact subgroup of G(A) with the property that the local factors of K are hyperspecial at every non-archimedian place. Our interest is to study a certain subspace of the space of square-integrable functions on the adelic quotient G(F)\G(A). Namely, we want to study functions coming from induced representations from an unramified character of a Borel subgroup and which are K-invariant.

Our goal is to describe how the decomposition of such space can be related with the Plancherel decomposition of a graded affine Hecke algebra (GAHA).

The main ingredients are standard analytic properties of the Dedekind zeta-function as well as known properties of the so-called residue distributions, introduced by Heckman-Opdam in their study of the Plancherel decomposition of a GAHA and a result by M. Reeder on the support of the weight spaces of
the anti-spherical  discrete series representations of affine Hecke algebras. These last ingredients are of a purely local nature.


This talk is based on joint work with V. Heiermann and E. Opdam.

Tue, 18 Oct 2016

14:15 - 15:15
L4

Tensor diagrams and Chebyshev polynomials

Lisa Lamberti
(Oxford)
Abstract

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional. 
In this talk, I report on recent developments in the 3-dimensional setting of this theory.

Tue, 11 Oct 2016
14:15
L4

Categorical matrix factorizations

Petter Bergh
(NTNU Trondheim)
Abstract

We define categorical matrix factorizations in a suspended additive category, 
with respect to a central element. Such a factorization is a sequence of maps 
which is two-periodic up to suspension, and whose composition equals the 
corresponding coordinate map of the central element. When the category in 
question is that of free modules over a commutative ring, together with the 
identity suspension, then these factorizations are just the classical matrix 
factorizations. We show that the homotopy category of categorical matrix 
factorizations is triangulated, and discuss some possible future directions. 
This is joint work with Dave Jorgensen.

Wed, 01 Jun 2016

15:00 - 16:00
L6

Homology torsion growth in right angled groups

Miklos Abert
(Renyi Institute Budapest)
Abstract

Torsion in homology are invariants that have received increasing attention over the last twenty years, by the work of Lück, Bergeron, Venkatesh and others. While there are various vanishing results, no one has found a finitely presented group where the torsion in the first homology is exponential over a normal chain with trivial intersection. On the other hand, conjecturally, every 3-manifold group should be an example.

A group is right angled if it can be generated by a list of infinite order elements, such that every element commutes with its neighbors. Many lattices in higher rank Lie groups (like SL(n,Z), n>2) are right angled. We prove that for a right angled group, the torsion in the first homology has subexponential growth for any Farber sequence of subgroups, in particular, any chain of normal subgroups with trivial intersection. We also exhibit right angled cocompact lattices in SL(n,R) (n>2), for which the Congruence Subgroup Property is not known. This is joint work with Nik Nikolov and Tsachik Gelander.

Tue, 24 May 2016

14:15 - 15:15
L4

Thurston and Alexander norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups

Dawid Kielak
(Bielefeld)
Abstract

We will introduce the Thurston norm in the setting of 3-manifold groups, and show how the techniques coming from L2-homology allow us to extend its definition to the setting of free-by-cyclic groups.
We will also look at the relationship between this Thurston norm and the Alexander norm, and the BNS invariants, in particular focusing on the case of ascending HNN extensions of the 2-generated free group.

Tue, 17 May 2016

14:15 - 15:15
L4

Bounds of Minkowski type for finite complex linear groups - the answer to a question of Serre

Michael Collins
(Oxford)
Abstract


In 1878, Jordan showed that there is a function f on the set of natural numbers such that, if $G$ is a finite subgroup of $GL(n,C)$, then $G$ has an abelian normal subgroup of index at most $f(n)$. Early bounds were given by Frobenius and Schur, and close to optimal bounds were given by Weisfeiler in unpublished work in 1984 using the classification of finite simple groups; about ten years ago I obtained the optimal bounds. Crucially, these are "absolute" bounds; they do not address the wider question of divisibility of orders.

In 1887, Minkowski established a bound for the order of a Sylow p-subgroup of a finite subgroup of GL(n,Z). Recently, Serre asked me whether I could obtain Minkowski-like results for complex linear groups, and posed a very specific question. The answer turns out to be no, but his suggestion is actually quite close to the truth, and I shall address this question in my seminar. The answer addresses the divisibility issue in general, and it turns out that a central technical theorem on the structure of linear groups from my earlier work which there was framed as a replacement theorem can be reinterpreted as an embedding theorem and so can be used to preserve divisibility.