Forthcoming events in this series


Tue, 19 Oct 2010

17:00 - 18:00
L2

Homological finiteness Bredon properties for groups

Desi Kochloukova
(UNICAMP)
Abstract

We discuss homological finiteness Bredon types FPm with respect to the class of finite subgroups and seperately with respect to the class of virtually cyclic subgroups. We will concentrate to the case of solubles groups and if the time allows to the case of generalized R. Thompson groups of type F. The results announced are joint work with Brita Nucinkis

(Southampton) and Conchita Martinez Perez (Zaragoza) and will appear in papers in Bulletin of LMS and Israel Journal of Mathematics.

Tue, 12 Oct 2010

17:00 - 18:00
L2

Duality for representations and quantum isogenies

Kevin McGerty
(Oxford)
Abstract

Recently Frenkel and Hernandez introduced a kind of "Langlands duality" for characters of semisimple Lie algebras. We will discuss a representation-theoretic interpretation of their duality using quantum analogues of exceptional isogenies. Time permitting we will also discuss a branching rule and relations to Littelmann paths.

Tue, 15 Jun 2010

17:00 - 18:00
L2

Bilinear Forms and Differential Forms under Field Extensions

Detlev Hoffmann
(Nottingham)
Abstract

An important problem in algebra is the study of algebraic objects

defined over fields and how they behave under field extensions,

for example the Brauer group of a field, Galois cohomology groups

over fields, Milnor K-theory of a field, or the Witt ring of bilinear

forms over

a field. Of particular interest is the determination

of the kernel of the restriction map when passing to a field extension.

We will give an overview over some known results concerning the

kernel of the restriction map from the Witt ring of a field to the

Witt ring of an extension field. Over fields of characteristic

not two, general results are rather sparse. In characteristic two,

we have a much more complete picture. In this talk, I will

explain the full solution to this problem for extensions that are

given by function fields of hypersurfaces over fields of

characteristic two. An important tool is the study of the

behaviour of differential forms over fields of positive

characteristic under field extensions. The result for

Witt rings in characteristic two then follows by applying earlier

results by Kato, Aravire-Baeza, and Laghribi. This is joint

work with Andrew Dolphin.

Tue, 01 Jun 2010

17:00 - 18:00
L2

The cluster category of Dynkin type $A_\infty$

Peter Jorgensen
(Newcastle)
Abstract

\ \ The cluster category of Dynkin type $A_\infty$ is a ubiquitous object with interesting properties, some of which will be explained in this talk.

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\ \ Let us denote the category by $\mathcal{D}$. Then $\mathcal{D}$ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $D^c(C^{∗}(S^2;k))$ of the singular cochain algebra $C^*(S^2;k)$ of the 2-sphere $S^{2}$. There is also a “universal” definition: $\mathcal{D}$ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.

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\ \ Just like cluster categories of finite quivers, $\mathcal{D}$ has many cluster tilting subcategories, with the crucial difference that in $\mathcal{D}$, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects.

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\ \ The talk will show how the cluster tilting subcategories have a rich combinatorial

structure: They can be parametrised by “triangulations of the $\infty$-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.

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\ \ This will be used to show how to obtain a subcategory of $\mathcal{D}$ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $\mathcal{D}$ generalises the situation from Dynkin type $A_n$ , and how triangulations of the $\infty$-gon are new and interesting combinatorial objects.

Tue, 27 Apr 2010

17:00 - 18:00
L2

Hopf-Galois extensions and an exact sequence for H-Picard groups

Andrei Marcus
(Cluj)
Abstract

The topic of this talk is the representation theory of Hopf-Galois extensions. We consider the following questions.

Let H be a Hopf algebra, and A, B right H-comodule algebras. Assume that A and B are faithfully flat H-Galois extensions.

1. If A and B are Morita equivalent, does it follow that the subalgebras A^coH and B^coH of H-coinvariant elements are also Morita equivalent?

2. Conversely, if A^coH and B^coH are Morita equivalent, when does it follow that A and B are Morita equivalent?

As an application, we investigate H-Morita autoequivalences of the H-Galois extension A, introduce the concept of H-Picard group, and we establish an exact sequence linking the H-Picard group of A and

the Picard group of A^coH.(joint work with Stefaan Caenepeel)

Tue, 16 Feb 2010

17:00 - 18:00
L2

Monstrous moonshine and black holes

John Duncan
(Cambridge)
Abstract

\ \ In 1939 Rademacher derived a conditionally convergent series expression for the modular j-invariant, and used this expression---the first Rademacher sum---to verify its modular invariance. We may attach Rademacher sums to other discrete groups of isometries of the hyperbolic plane, and we may ask how the automorphy of the resulting functions reflects the geometry of the group in question.

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\ \ In the case of a group that defines a genus zero quotient of the hyperbolic plane the relationship is particularly striking. On the other hand, of the common features of the groups that arise in monstrous moonshine, the genus zero property is perhaps the most elusive. We will illustrate how Rademacher sums elucidate this phenomena by using them to formulate a characterization of the discrete groups of monstrous moonshine.

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\ \ A physical interpretation of the Rademacher sums comes into view when we consider black holes in the context of three dimensional quantum gravity. This observation, together with the application of Rademacher sums to moonshine, amounts to a new connection between moonshine, number theory and physics, and furnishes applications in all three fields.

Tue, 24 Nov 2009

17:00 - 18:00
L2

Base sizes for algebraic groups

Tim Burness
(Southampton)
Abstract

Let G be a permutation group on a set S. A base for G is a subset B of S such that the pointwise stabilizer of B in G is trivial. We write b(G) for the minimal size of a base for G.

Bases for finite permutation groups have been studied since the early days of group theory in the nineteenth century. More recently, strong bounds on b(G) have been obtained in the case where G is a finite simple group, culminating in the recent proof, using probabilistic methods, of a conjecture of Cameron.

In this talk, I will report on some recent joint work with Bob Guralnick and Jan Saxl on base sizes for algebraic groups. Let G be a simple algebraic group over an algebraically closed field and let S = G/H be a transitive G-variety, where H is a maximal closed subgroup of G. Our goal is to determine b(G) exactly, and to obtain similar results for some additional base-related measures which arise naturally in the algebraic group context. I will explain the key ideas and present some of the results we have obtained thus far. I will also describe some connections with the corresponding finite groups of Lie type.

Tue, 17 Nov 2009

17:00 - 18:00
L2

Finite generation of invariants over an arbitrary base

Vincent Franjou
(Nantes)
Abstract

A classic problem in invariant theory, often referred to as Hilbert's 14th problem, asks, when a group acts on a finitely generated commutative algebra by algebra automorphisms, whether the ring of invariants is still finitely generated. I shall present joint work with W. van der Kallen treating the problem for a Chevalley group over an arbitrary base. Progress on the corresponding problem of finite generation for rational cohomology will be discussed.

Tue, 27 Oct 2009

17:00 - 18:00
L2

Birational Geometry via Auslander Algebras

Michael Wemyss
(Oxford)
Abstract

I'll explain how the `Auslander philosophy' from finite dimensional algebras gives new methods to tackle problems in higher-dimensional birational geometry. The geometry tells us what we want to be true in the algebra and conversely the algebra gives us methods of establishing derived equivalences (and other phenomenon) in geometry. Algebraically two of the main consequences are a version of AR duality that covers non-isolated singularities and also a theory of mutation which applies to quivers that have both loops and two-cycles.