Pure Maths Seminar Archive - Abstracts 2014
"Centralisers of finite subgroups in generalisations of Thompson's groups"
Brita Nucinkis (Royal Holloway)
January 27th 2014
We define a family of groups that generalises Thompson's groups T and G, and also those of Higman, Stein and Brin. For groups in this family we describe centralisers of finite subgroups and show, that for a given finite subgroup Q, there are finitely many conjugacy classes of finite subgroups isomorphic to Q. This also has some implications on the type of the classifying space for proper actions for these groups. (joint work with C. Martinez-Perez and F. Matucci).
"The geometry of tropical matrices"
Mark Kambites (Manchester)
February 3rd 2014
The tropical semiring is (roughly speaking) the algebraic structure formed by the real numbers under the operations of addition and maximum. It arises naturally (and indeed has often been independently rediscovered) in diverse areas of mathematics, including formal language theory, process control, combinatorial optimization and scheduling, phylogenetics and algebraic geometry. I shall discuss recent progress in understanding the multiplicative structure of matrices over the tropical semiring. This turns out to be intimately connected with the geometry of tropical convexity; indeed, almost every algebraic property seems to manifest itself in some beautiful geometric phenomenon involving tropical polytopes. I shall give an overview of how these connections arise, assuming no background knowledge but trying to minimise overlap with Marianne Johnson's recent talk.
"The Specker phenomenon, evasion, and large cardinals"
Andrew Brooke-Taylor (Bristol)
February 24th 2014
The evasion number e is an uncountable cardinal which, if the continuum hypothesis fails, may be strictly less than the cardinality of the reals. It was introduced by Blass in connection with Specker's theorem, which states that any homomorphism to the group Z from the countable product of copies of Z must be trivial on all but finitely many basis vectors. After surveying this background, I will move to the analogue of e for large cardinals, discussing my recent work with Joerg Brendle on which relations from the basic case generalise.
"Representations of Lie algebras vs Lie superalgebras"
Volodymyr Mazorchuk (Uppsala)
March 3rd 2014
In this talk I plan to compare representation theory for simple complex finite dimensional Lie algebras with that for Lie superalgebras emphasizing both similarities and differences. The main examples will include category O, simple modules, primitive spectrum and, hopefully, functorial actions on module categories.
"Blocks for modular representations of inner forms of p-adic GL(n)"
Shaun Stevens (UEA)
March 31st 2014
In the complex representation theory of p-adic groups, Bernstein gives a decomposition of the category of smooth representations into blocks -- that is, into subcategories which are indecomposable summands. When one passes to l-modular representations (i.e. over an algebraically closed field of characteristic l different from p), it is not clear that there is such a decomposition, partly because of the difference between the notions of cuspidal (not a quotient of a parabolically induced representation) and supercuspidal (not a subquotient of a parabolically induced representation). I will describe some joint work with Secherre, in which we have obtained such a decomposition for inner forms of GL(n), generalizing previous work of Vigneras on GL(n) itself. In particular, I'll try to explain what some, hopefully all, of these words mean.
"The outer model programme"
Peter Holy (Bristol)
April 28th 2014
The Outer Model Programme investigates L-like forcing extensions of the universe, where we say that a model of Set Theory is L-like if it satisfies properties of Goedel's Constructible Universe of sets L. I will introduce the Outer Model Programme, talk about its history, motivation, recent results and applications. I will be presenting joint work with Sy Friedman.
"Counting congruent cuspidal representations"
Vincent Sécherre (Versailles)
May 16th 2014
In the representation theory of p-adic reductive groups, an important case of Langlands functoriality is given by the local Jacquet-Langlands correspondence, which relates the representation theory of the group GL(n) over a p-adic field to that of its inner forms. In the classical setting, representations have complex coefficients, but it is natural, for number theoretic reasons, to search for a Jacquet-Langlands correspondence between representations having coefficients in fields of nonzero characteristic (and different from p). Dat has obtained partial results in this direction, in the case where the inner form is the multiplicative group of a division algebra. In this talk we will investigate the case of an arbitrary inner form. In order to construct a modular Jacquet-Langlands correspondence, Dat looks at l-adic cuspidal representations and isolates, thanks to a counting argument due to Vignéras, those whose reduction mod l is irreducible and supercuspidal. We will explain this crucial counting argument, and we will show that a finer computation also gives some information in the case where the reduction mod l is not irreducible supercuspidal.
"Structure trees and networks"
Martin Dunwoody (Southampton)
May 19th 2014
For any network there is a uniquely determined network based on a structure tree that provides a convenient way of determining a minimal cut separating a pair s, t where each of s, t is either a vertex or an end in the original network. There is Max-Flow Min-Cut Theorem for any network. In the case of a Cayley Graph for a finitely generated group the theory provides another proof of Stallings' Theorem on the structure of groups with more than one end.
"The local Langlands correspondence for inner forms of SL(n,F)"
Maarten Solleveld (Nijmegen)
June 9th 2014
Let F be a local non-archimedean field. A fundamental result in representation theory is the proof of the local Langlands correspondence for GL(n,F). It provides a canonical bijection between - a set of Langlands parameters for GL(n,F); - the space of irreducible smooth complex representations of GL(n,F). From this one can derive the local Langlands correspondence for some other groups, in particular for SL(n,F) and for the inner forms of GL(n,F). We will discuss the local Langlands correspondence for a more difficult class of groups, namely the inner forms of SL(n,F). Every such group looks like SL(m,D), where D is a division algebra with centre F. In this talk no prior knowledge of p-adic representation theory is assumed.
"The integral Bernstein centre for GL(n)"
David Helm (Imperial)
June 16th 2014
The Bernstein center is a ring, first studied by Bernstein and Deligne, that serves a purpose in the representation theory of p-adic groups that is analogous to the role played by the center of the group ring in the theory of finite groups. We consider the Bernstein center in the context of the modular representation theory of GL_n(F), where F is a p-adic field, in characteristics prime to p. We will prove basic structural results about the Bernstein center in this context, and describe work in progress that relates the Bernstein center to the deformation theory of Galois representations via a "local Langlands correspondence in families".