Forthcoming events in this series


Tue, 26 Apr 2016

14:15 - 15:30
L4

Multiserial and Special Multiserial Algebras

Sibylle Schroll
(Leicester)
Abstract

The class of multiserial algebras contains many well-studied examples of algebras such as the intensely-studied biserial and special biserial algebras. These, in turn, contain many of the tame algebras arising in the modular representation theory of finite groups such as tame blocks of finite groups and all tame blocks of Hecke algebras. However, unlike  biserial algebras which are of tame representation type, multiserial algebras are generally of wild representation type. We will show that despite this fact, we retain some control over their representation theory.

Tue, 08 Mar 2016

14:15 - 15:30
L4

Strongly dense subgroups of semisimple algebraic groups.

Emmanuel Breuillard
(Orsay and Munster)
Abstract

A subgroup Gamma of a semisimple algebraic group G is called strongly dense if every subgroup of Gamma is either cyclic or Zariski-dense. I will describe a method for building strongly dense free subgroups inside a given Zariski-dense subgroup  Gamma of G, thus providing a refinement of the Tits alternative. The method works for a large class of G's and Gamma's. I will also discuss connections with word maps and expander graphs. This is joint work with Bob Guralnick and Michael Larsen.

Tue, 01 Mar 2016

14:15 - 15:30
L4

There And Back Again: A Localization's Tale.

Sian Fryer
(Leeds)
Abstract

The prime spectrum of a quantum algebra has a finite stratification in terms
of a set of distinguished primes called H-primes, and we can study these
strata by passing to certain nice localizations of the algebra.  H-primes
are now starting to show up in some surprising new areas, including
combinatorics (totally nonnegative matrices) and physics, and we can borrow
techniques from these areas to answer questions about quantum algebras and
their localizations.    In particular, we can use Grassmann necklaces -- a
purely combinatorial construction -- to study the topological structure of
the prime spectrum of quantum matrices.

Tue, 23 Feb 2016

14:15 - 15:30
L4

Discrete triangulated categories

David Pauksztello
(Manchester)
Abstract
This is a report on joint work with Nathan Broomhead and David Ploog.
 
The notion of a discrete derived category was first introduced by Vossieck, who classified the algebras admitting such a derived category. Due to their tangible nature, discrete derived categories provide a natural laboratory in which to study concretely many aspects of homological algebra. Unfortunately, Vossieck’s definition hinges on the existence of a bounded t-structure, which some triangulated categories do not possess. Examples include triangulated categories generated by ‘negative spherical objects’, which occur in the context of higher cluster categories of type A infinity. In this talk, we compare and contrast different aspects of discrete triangulated categories with a view toward a good working definition of such a category.
 

 
Tue, 16 Feb 2016

14:15 - 15:15
L4

Formal degrees of unipotent discrete series representations of semisimple $p$-adic groups

Dan Ciubotaru
(Oxford)
Abstract

The formal degree is a fundamental invariant of a discrete series representation which generalizes the notion of dimension from finite dimensional representations. For discrete series with unipotent cuspidal support, a formula for formal degrees, conjectured by Hiraga-Ichino-Ikeda, was verified by Opdam (2015). For split exceptional groups, this formula was previously known from the work of Reeder (2000). I will present a different interpretation of the formal degrees of unipotent discrete series in terms of the nonabelian Fourier transform (introduced by Lusztig in the character theory of finite groups of Lie type) and certain invariants arising in the elliptic theory of the affine Weyl group. This interpretation relates to recent conjectures of Lusztig about `almost characters' of p-adic groups. The talk is based on joint work with Eric Opdam.

Tue, 26 Jan 2016

14:15 - 15:30
L4

Extensions of modules for graded Hecke algebras

Kei Yuen Chan
(Amsterdam)
Abstract

Graded affine Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. In particular, some problems about extensions of representations of p-adic groups can be transferred to problems in the graded Hecke algebra setting. The study of extensions gives insight to the structure of various reducible modules. In this talk, I shall discuss some methods of computing Ext-groups for graded Hecke algebras.
The talk is based on arXiv:1410.1495, arXiv:1510.05410 and forthcoming work.

Tue, 01 Dec 2015

14:15 - 15:15
L4

Uniform exponential growth for linear groups

Peter Varju
(Cambridge)
Abstract

Abstract: This is a joint work with E. Breuillard.

A conjecture of Breuillard asserts that for every positive integer d, there is a positive constant c such that the following holds. Let S be a finite subset of GL(d,C) that generates a group, which is not virtually nilpotent. Then |S^n|>exp(cn) for all n.
Considering an algebraic number a that is not a root of unity and the semigroup generated by the affine transformations x-> ax+1, x-> ax+1, the above conjecture implies that the Mahler measure of a is at least 1+c' for some c'>0 depending on c. This property is known as Lehmer's conjecture.

I will talk about the converse of this implication, namely that Lehmer's conjecture implies the uniform growth conjecture of
Breuillard.

Tue, 17 Nov 2015
14:15
L4

Representation theory related to some infinite permutation groups.

Peter Neumann
(Oxford)
Abstract

Our work (which is joint with Simon Smith) began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations.
 

One is the generalisation in which point stabilisers are merely assumed to satisfy min-{\sc N}, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on the socle of~$M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

Tue, 10 Nov 2015

14:15 - 15:15
L4

Some infinite permutation groups

Cheryl Praeger
(UWA)
Abstract

Our work (which is joint with Simon Smith) began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-{\sc N}, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on the socle of~$M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

Tue, 03 Nov 2015

16:00 - 17:00
C5

Equivalence relations for quadratic forms

Detlev Hoffmann
(Dortmund)
Abstract

We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.

Tue, 27 Oct 2015

14:15 - 15:30
L4

Symplectic resolutions of quiver varieties.

Gwyn Bellamy
(University of Glasgow)
Abstract

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution.  This result is based on work of Crawley-Bouvey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.
 

Tue, 13 Oct 2015

14:15 - 15:15
L4

CANCELLED!

Stefan Witzel
(Bielefeld)
Abstract

 If $R = F_q[t]$ is the polynomial ring over a finite field
then the group $SL_2(R)$ is not finitely generated. The group $SL_3(R)$ is
finitely generated but not finitely presented, while $SL_4(R)$ is
finitely presented. These examples are facets of a larger picture that
I will talk about.

Tue, 16 Jun 2015

17:00 - 18:00
C2

Growth of homology torsion in residually finite groups

Nikolay Nikolov
(Oxford)
Abstract

I will report on recent progress towards understanding the growth of the torsion of the homology of subgroups of finite index in a given residually finite group G.

The cases I will consider are when G is amenable (joint work with P, Kropholler and A. Kar) and when G is right angled (joint work with M. Abert and T. Gelander).

Tue, 09 Jun 2015

17:00 - 18:00
C2

TBA

Benjamin Klopsch
(Duesseldorf)
Tue, 19 May 2015

17:00 - 18:00
C2

Diagonalizable algebras of operators on infinite-dimensional vector spaces

Manuel Reyes
(Bowdoin)
Abstract

Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n. 

In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic.  I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V).  I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.

Tue, 12 May 2015

17:00 - 18:00
C2

Permutation groups, primitivity and derangements

Tim Burness
(Bristol)
Abstract

Let G be a transitive permutation group. If G is finite, then a classical theorem of Jordan implies the existence of fixed-point-free elements, which we call derangements. This result has some interesting and unexpected applications, and it leads to several natural problems on the abundance and order of derangements that have been the focus of recent research. In this talk, I will discuss some of these related problems, and I will report on recent joint work with Hung Tong-Viet on primitive permutation groups with extremal derangement properties.

Thu, 16 Apr 2015

14:00 - 15:00
N3.12

D-modules and arithmetic: a theory of the b-function in positive characteristic.

Thomas Bitoun
(HSE Moscow)
Abstract

We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.

Tue, 10 Feb 2015

17:00 - 18:00
C2

Spin projective representations of Weyl groups, Green polynomials, and nilpotent orbits

Dan Ciubotaru
(Oxford)
Abstract

The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras.  The talk is partly based on joint work with Xuhua He.

Tue, 27 Jan 2015

17:00 - 18:00
C2

Regular maps and simple groups

Martin Liebeck
(Imperial College London)
Abstract

A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.

Tue, 02 Dec 2014

17:00 - 18:00
C2

Branch groups: groups that look like trees

Alejandra Garrido
(Oxford)
Abstract

Groups which act on rooted trees, and branch groups in particular, have provided examples of groups with exotic properties for the last three decades. This and their links to other areas of mathematics such as dynamical systems has made them the object of intense research.
One of their more useful properties is that of having a "tree-like" subgroup structure, in several senses. 
I shall explain what this means in the talk and give some applications.

Tue, 25 Nov 2014

17:00 - 18:00
C2

On universal right angled Artin groups

Ashot Minasyan
(Southampton)
Abstract
A right angled Artin group (RAAG), also called a graph group or a partially commutative group, is a group which has a finite presentation where 
the only permitted defining relators are commutators of the generators. These groups and their subgroups play an important role in Geometric Group Theory, especially in view of the recent groundbreaking results of Haglund, Wise, Agol, and others, showing that many groups possess finite index subgroups that embed into RAAGs.
In their recent work on limit groups over right angled Artin groups, Casals-Ruiz and Kazachkov asked whether for every natural number n there exists a single "universal" RAAG, A_n, containing all n-generated subgroups of RAAGs. Motivated by this question, I will discuss several results showing that "universal" (in various contexts) RAAGs generally do not exist. I will also mention some positive results about universal groups for finitely presented n-generated subgroups of direct products of free and limit groups.
Tue, 18 Nov 2014

17:00 - 18:00
C2

Commuting probabilities of finite groups

Sean Eberhard
(Oxford)
Abstract

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Not all rationals between 0 and 1 occur as commuting probabilities. In fact Keith Joseph conjectured in 1977 that all limit points of the set of commuting probabilities are rational, and moreover that these limit points can only be approached from above. In this talk we'll discuss a structure theorem for commuting probabilities which roughly asserts that commuting probabilities are nearly Egyptian fractions of bounded complexity. Joseph's conjectures are corollaries.

Mon, 17 Nov 2014

17:00 - 18:00
C2

Nielsen realisation for right-angled Artin groups

Dawid Kielak
(Bonn)
Abstract

We will introduce both the class of right-angled Artin groups (RAAG) and
the Nielsen realisation problem. Then we will discuss some recent progress
towards solving the problem.