Seminars take place on Monday afternoons 14:00–15:00. Everyone is welcome.
September 14, (SCI 3.05), Fenjin Liu (Jiao Tong University and Chang’an University, Xi’an, P.R. China), The spectrum of a graph and its walk matrix
October 8, (LT4), Francesca Fedele (University of Newcastle), A (d+2)-angulated generalisation of a theorem by Brüning
October 15,(EFRY 01.02), Jason Semeraro (University of Leicester), Weight conjectures for fusion systems
October 22, (LT4), Amit Shah (University of Leeds), Cluster categories and partial cluster-tilted algebras
October 29, (LT4), Amit Hazi (University of Leeds), Indecomposable tilting modules for the blob algebraitle
November 7, 1pm, (SCI 3.05), Laure Daviaud (University of Warwick), Tropical algebra and applications [please note change to usual time and day]
November 12, (LT4), Jesper Grodal (University of Copenhagen), Modular representation theory via homotopy theory
November 19, (ARTS 2.02), Robert Kurinczuk (Imperial College, London), Local Langlands in Families
November 26, (LT4), Sam Morley (UEA), Extensions of uniform algebras
December 3, (LT4), Peter Latham (Kings College London), Towards the unicity of types for essentially tame cuspidals
Many of the conjectures of current interest in the representation theory of finite groups in characteristic p are local-to-global statements, in that they predict consequences for the representations of a finite group G given data about the representations of the p-local subgroups of G. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of Külshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples (S,F,alpha) consisting of a saturated fusion system F on a finite p-group S and a compatible family alpha.
Amit Shah (October 22):
In this talk, I will first try to indicate why some people care about cluster algebras and cluster categories. Then I will focus on a specific cluster category, namely the cluster category coming from a Dynkin graph of type A. Although the formal definition for a cluster category can look a bit intimidating, we will see that the category has a nice Auslander-Reiten quiver — pictorial description of the category — (at least for type A) and so can be easily understood. Hopefully there will be time to give the definition of a partial cluster-tilted algebra, give an idea of how I’ve been trying to study them.
Amit Hazi (October 29):
The blob algebra is an example of a diagram algebra, that is to say, an algebra whose generators and relations can be described pictorially. In this talk I will describe the representation theory of this algebra and give a description of its indecomposable tilting modules. This new result is related to KLR algebras, categorification, and (diagrammatic) Soergel bimodules.
Laure Daviaud (November 07):
Tropical algebra is the counterpart of linear algebra when the addition and multiplication operators are replaced by the minimum (or maximum) and addition operators. Tropical algebra is thus particularly suitable to deal with optimisation problems, such as public transport scheduling. In this talk, I will give an introduction to tropical algebra and present some applications, for example in computer science (with the analysis of running time of programs) and in biology (phylogenetic).
Jesper Grodal (November 12):
The modular representation theory of finite groups is a wild, and sometimes bewildering, place: Indecomposable representations may not be simple, and the indecomposable modules can be shown to be, in general, "unclassifiable", in a precise sense. In this harsh and unforgiving climate there are however islands of structure and beauty: One such is the class of so-called endo-trivial modules, modules M such that End(M) is isomorphic to a trivial module plus a projective module. These modules occur in many parts of representation theory as "almost 1-dimensional modules". I'll describe the quest to classify such modules spanning the last 40 years, starting from the work of Dade in the 70s and leading into the present, where methods from homotopy theory come to play a role.
Robert Kurinczuk (November 19):
The local Langlands correspondence provides a remarkable connection between the representation theory of Galois groups and general linear groups over p-adic fields. For a prime \ell different to p, this correspondence was interpolated in \ell-adic families in the recent pioneering work of Emerton-Helm-Moss. I will explain what all of this means, and then at the end describe current joint work with Jean-François Dat, David Helm, and Gil Moss on a conjectural generalisation to a quite general class of reductive p-adic groups.
Sam Morley (November 26):
The theory of algebraic extensions of Banach algebras is well established, particularly for commutative Banach algebras, and have been used to solve many problems. In his thesis, Cole constructed algebraic extensions of a certain uniform algebra to give a counterexample to the peak point conjecture. Cole’s method for extending uniform algebras ensures that certain properties of the original algebra are preserved by the extension. In this talk, we discuss the general theory of uniform algebra extensions and a certain class of uniform algebra extensions which generalise Cole’s construction.
Peter Latham (December 03):
Let R be a cuspidal representation of a semisimple, simply connected p-adic group G. It is possible that the restriction of R to a compact subgroup K of G contains a type: a component which can only be contained in R-isotypic representations of G. Due to work by Yu, Kim and Fintzen, under mild assumptions on p, given such an R one may always find a maximal compact K such that the restriction of R to K contains a type. Conjecturally, these are the only possible types; if so then we say that the unicity of types holds for R. I will present joint work with Monica Nevins (University of Ottawa), which reduces the unicity of types for essentially tame cuspidals to an open question regarding the representation theory of finite groups of Lie type. When the split rank of G is small, this question is often manageable; this allows us to obtain many new examples of groups for which the unicity of types holds.