Spring 2014 Spring 2014

Seminars take place on Monday afternoons 14:00 - 15:00. Everyone is welcome.

Talks

  • January 27, (C.HALL), Brita Nucinkis (Royal Holloway), Centralisers of Finite Subgroups in Generalisations of R. Thompson's Groups.
  • February 3, (ARTS 3.02), Mark Kambites (Manchester), The Geometry of Tropical Matrices.
  • February 24, (ARTS 01.02), Andrew Brooke-Taylor (Bristol), The Specker phenomenon, Evasion, and Large Cardinals.  
  • March 3, (ARTS 01.02), Volodymyr Mazorchuk (Uppsala), Representations of Lie Algebras vs Lie Superalgebras
  • March 24, (ARTS 3.02), PhD Student Talks (UEA),
  • March 31, (ARTS 01.02), Shaun Stevens (UEA), Blocks for Modular Representations of Inner Forms of p-adic GL(n)
  • April 28, (ARTS 01.02), Peter Holy (Bristol), The Outer Model Programme
  • May 16, FRIDAY (S3.05), Vincent Secherre (Versailles), Counting Congruent Cuspidal Representations
  • May 19, (SCI 3.05), Martin Dunwoody (Southampton), Structure Trees and Networks
  • June 9, 15:30 - 16: 30, (SCI 3.05), Maarten Solleveld (Radboud Universiteit Nijmegen), The Local Langlands Correspondence for Inner Forms of SL(n,F)
  • June 16, (SCI 3.05), David Helm (Imperial), The Integral Bernstein Centre for GL(n)

Abstracts

Brita Nucinkis: 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).

Mark Kambites: 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.

Andrew Brooke-Taylor: 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.

Volodymyr Mazorchuk: 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.

Professor Shaun Stevens: 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.

Peter Holy: 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.

Vincent Secherre: 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.

Martin Dunwoody: 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.

Maarten Solleveld: 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.

David Helm: 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".

Autumn 2013 Autumn 2013

Seminars take place on Monday afternoons 14:00-15:00. The first seminar will be on Monday 7 October. Everyone is welcome.

Talks

  • October 7 (ARTS 01.02), Joel Nagloo (Leeds), On Transcendence and Generic Painlevé equations.
  • October 14 (ARTS 01.02), Lewis Topley (UEA), Nilpotent Orbits and Representation Theory.
  • October 28 (ARTS 01.02), Vanessa Miemietz (UEA), Introduction to 2-representation Theory.

  • November 4 (ARTS 01.02), Nik Ruskuc (St Andrews), Permutations and Words.
  • November 11 (ARTS 01.02), Joseph Chuang (City), Algebra with Surfaces.
  • November 18 (ARTS 01.02), Agelos Georgakopoulos (Warwick), Discrete Riemann Mapping and the Poisson Boundary.
  • November 25 (ARTS 01.02), Chris Bowman (City), The Partition Algebra and the Kronecker Problem.
  • December 2 (ARTS 01.02), Paul Martin (Leeds), Fun with Partition Categories.
  • December 9 (ARTS 2.02), Brent Everitt (York), Partial Mirror Symmetry.
  • December 16 (SCI 3.05), Marianne Johnson (Manchester), The Automorphism Group of a (nice) Tropical Polytope.

Abstracts

Joel Nagloo: The Painlevé equations are nonlinear 2nd order ODE and come in six families P_I--P_VI, where P_I consists of the single equation y'' = 6y^2 + t, and P_II--P_VI come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications including for example random matrix theory and general relativity.
In this talk we are interested in the following classical transcendence problem: Suppose y'' = f(y, y', t) is one of the generic Painlevé equation from the class II to VI and let y_1, ... , y_n be distinct solutions. Are y_1, y_1' , ... , y_n, y_n' algebraically independent over C(t)?
I explain how one can use model theory, a branch of mathematical logic, to positively answer this question. It is worth mentioning that these results are the culmination of the work started by P. Painlevé (over 100 years ago), the Japanese school and many others on transcendence and the Painlevé equations.

Lewis Topley: A simple algebraic group acts upon its Lie algebra and the subset of nilpotent elements decomposes into conjugacy classes, which we call nilpotent orbits. These orbits appear to be subtly connected to the representation theory of Lie theoretic objects, such as Weyl groups. I shall begin by sketching some of these connections.

Over the past 12 years the emerging theory of finite W-algebras has offered a new perspective on these connections and allowed for progress to be made in understanding the (ordinary and modular) representation theory of semisimple Lie algebras. I shall describe some of these advances and (hopefully) mention one of my recent results, relating the representation theory of the finite W-algebras to the sheets of orbits in a classical Lie algebra.

Vanessa Miemietz: I will try to motivate why one would want to study 2-representations of 2-categories (other than to have some fun) and to explain some of the basic concepts and difficulties that one encounters in doing so.

Nik Ruskuc: ​In this talk I will explore how concepts from theoretical computer science -- automata and languages -- can be utilised in a combinatorial context such as the theory of pattern avoidance classes of permutations.

Joseph Chuang: Frobenius algebras give rise to topological invariants of surfaces. I will review this idea (two-dimensional topological field theory) and describe joint work with Andrey Lazarev on a similar construction.

Agelos Georgakopoulos: ​Answering a question of Benjamini & Schramm, we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which circle arises from a discrete version of Riemann's mapping theorem. I will start by explaining this discrete Riemann map, drawing various analogies between the discrete and continuous case.
The talk will be self contained, assuming no prior knowledge of the mentioned terminology, and will contain many pictures.

Chris Bowman: ​The Kronecker problem asks for a combinatorial understanding of the tensor products of simple modules for the symmetric group.  We shall introduce the partition algebra as a natural setting in which to study this problem and discuss new results concerning its representation theory.  This is based on joint work with M. De Visscher, O. King, and R. Orellana.

Paul Martin: ​The Brauer category sits inside the partition category - both having elementary set-theoretic constructions. The Temperley-Lieb category sits inside these categories (in at least two different ways), but it's construction has a more geometrical flavour. We will consider geometrically defined extensions of the TL category in the Brauer and partition categories. These constructions are motivated in part by applications in computational physics, but here we will consider them from a representation theory perspective.

Brent Everitt: ​In this talk (but possibly nowhere else) "mirror symmetry" means the theory of reflection groups. These are groups of linear maps generated by reflections. Partial mirror symmetry is then the theory of "reflection monoids": inverse monoids of partial linear maps generated by partial reflections. This is a subject in its infancy. I'll start at the beginning and try to give a flavour of what has been achieved so far.

Marianne Johnson: Tropical linear spaces can look a bit strange. In case you've not come across these objects before, I'll start with the basics and show you some examples. I'll then say a bit about the group of automorphisms of a finitely generated tropical linear space (ortropical polytope for short).