## Previous Pure Mathematics Colloquia - 2010 to 2011

Previous Pure Mathematics Colloquia from: 2013/14, 2012/13, 2011/12, 2010/11, 2009/10, 2008/09, 2007/08### Thursday, 21th of July 2011, 4pm, Theatre C

Vince Vatter
*(University of Florida at Gainesville)
***Geometric grid classes of permutations
**

A geometric grid class consists of those permutations that can be "drawn" on a specified set of line segments of slope +/- 1 in the plane. Thus geometric grid classes are permutation classes, meaning that they are closed downward under the permutation containment order. I will discuss recent work with Albert, Atkinson, Bouvel, and Ruskuc which, using a mixture of geometric and language theoretic methods, establishes that geometric grid classes are among the best behaved permutation classes. In particular, these classes can be specified by finite sets of forbidden patterns, are partially well-ordered (i.e., don't contain infinite antichains), and have rational generating functions.

### Thursday, 9th of June 2011, 4pm, Theatre C

Maria Chlouveraki
*(University of Edinburgh)
***The representation theory of Hecke algebras from the perspective of Cherednik algebras
**

In this talk, we will try to give answers to important questions on the representation theory of Hecke algebras, such as the parametrisation and decomposition of simple modules, through the study of the category O for the rational Cherednik algebra.

### Tuesday, 31th of May 2011, 4pm, Theatre C

Karma Dajani
*(University of Utrecht)
***Two special invariant measures for the random beta-transformation
**

It is well known that if beta is a non-integer greater than 1, then almost every point has uncountably many expansions in base beta. In this talk, we will introduce a transformation, the so called random beta transformation, whose iterations produce all possible expansions in base beta. We exhibit two natural ergodic invariant measures for this transformation, give their properties and prove that these measures are mutually singular.

### Tuesday, 17th of May 2011, 4pm?, Theatre C?

Henk Bruin
*(University of Surrey)
***Renormalization and Thermodynamic Formalism in Subshifts
**

A well-known fact, going back to work of Hofbauer in the 1970s, is that the Manneville-Pomeau map ($f(x) = x/(1-x)$ on the interval $[0,1/2]$ and $f(x) = 2x-1$ on $(\frac12,1]$) has a phase-transition for the geometric potential $V = -t \log |f'|$, at $t = 1$, where continuous equilibrium states are supplanted by an atomic measure at $x = 0$.

A discovery, probably due to Artur Lopes, is that $V$ is the fixed point of a natural renormalisation process on the level of potentials. This leads to the question whether there is some underlying structure explaining the connection between renormalisation and phase transitions. In order to explore this, we investigate a similar renormalisation operator for prime examples of renormalisable dynamics: the Thue-Morse substitution in symbolic dynamics and the Feigenbaum map in one-dimensional dynamics; these systems are closely related to each other. We identify a variety of renormalisation-invariant potentials, and investigate their thermodynamics properties. A far richer picture is obtained than for the Manneville-Pomeau map, and although phase transition don't seem to happen for these potentials, it does give a new point of few for thermodynamic formalism of dissipative dynamical systems.

This is joint work with Renaud Leplaideur (University of Brest).

### Tuesday, 26th of April 2011, 4pm, Theatre D

James East
*(University of Western Sydney)
***Idempotent generators in partition monoids
**

We characterise the elements of the partition monoid which are products of idempotent partitions. The finite and infinite cases require separate treatment. Our results are similar in character to those obtained by Howie in his 1966 paper on full transformation semigroups. Howie's results may also be obtained from ours. This is joint work with Des Fitzgerald.

### Wednesday, 20th of April 2011, 3pm, Theatre D

Wolfgang Kimmerle
*(University of Stuttgart)
***On the unit group of integral group rings of finite groups
**

Let G be a finite group and let ZG be its integral group ring. V(ZG) denotes the normalized unit group, i.e. the units of ZG with augmentation 1. The object of the talk is the structure of V(ZG). In particular the question which torsion subgroups of V(ZG) are determined by subgroups of G is discussed. Moreover normalizers and centralizers of such subgroups are considered. The talk reports partially on joint work with A. Bächle [1] and on progress concerning problems 15, 19-21 of [2].

[1] A.Bächle and W.Kimmerle, On torsion subgroups in integral group rings of finite groups, J. of Algebra 326 (2011) 34-46.

[2] E.Jespers, W.Kimmerle, Z.Marciniak, G.Nebe, Oberwolfach Reports, Vol. 4 No. 4, Reports No. 55/2007, Mini-Workshop: Arithmetik von Gruppenringen EMS, Zürich, 2007, pp.3149-3179.

### Thursday, 14th of April 2011, 4pm, Theatre C

Mike Todd
*(University of St Andrews)
***Transience in dynamical systems
**

The definition of an attractor for a dynamical system can be defined in a topological or a `metric' way. In `good cases' these notions coincide. I will discuss two elementary classes of interval maps within which there are maps which have some transient phenomena which means that these notions do not coincide. These examples can be understood via a countable state Markov chain, where a change in parameter leads to a phase transition from recurrence to transience. (Joint work with Henk Bruin.)

### Tuesday, 22th of March 2011, 4pm, Theatre D

Richard Sharp
*(University of Manchester)
***Pair correlations and length spectra on negatively curved surfaces
**

A central problem in the area of mathematical physics known as "quantum chaos" is to understand the spacings between eigenvalues of the Laplacian on compact negatively curved surfaces. An analogous and related problem is to understand the spacings between lengths of closed geodesics on the surface. The purpose of the talk will be to discuss the two cases and present new results in the geodesic case. A key idea is to order closed geodesics by the word length associated to generators of the fundamental group, rather than geometric length. (Joint work with Mark Pollicott.)

### Thursday, 17th of March 2011, 4pm, Theatre C

Mark Dukes
*(University of Strathclyde)
***Upper triangular matrices, (2+2)-free posets, pattern avoiding permutations, matchings and ascent sequences
**

In this talk I will present a collection of correspondences between the objects in the title. These correspodences arose by considering a generalisation of generalised permutation patterns. Labelings, and certain restricted labelings, give rise to a hierarchy of correspondences.

### Tuesday, 1st of February 2011, 2pm, Theatre C

Jose Burillo
*(University of Barcelona)
***The automorphism group of Thompson's group F
**

In this talk we will go over the definition of Thompson's group F with the goal of describing its group of automorphisms. The group Aut F will be completely described algebraically and geometrically, and we will compute a presentation for it. Finally, some interesting subgroups will be constructed, giving some nice actions of F onto itself. Metric properties and distortion of these subgroups will also be studied.

This is joint work with Sean Cleary.

### Thursday, 16th of December 2010, 4pm, Theatre C

Derek Holt
*(University of Warwick)
***Algorithms and generator numbers for matrix groups
**

This talk will be in two parts. In the first part, we give a brief survey of of the "Composition Tree" method for analysing potentially large matrix groups over finite fields. In this approach, we need to find generating sets for the kernels of homomorphisms from matrix groups, which we do by choosing a collection of random elements of the kernel. This motivates the second part of the talk, which is joint work with Colva Roney-Dougal, in which we investigate how many random elements of a finite matrix group we need in order to generate it with high probability.

### Thursday, 25th of November 2010, 4pm, Theatre C

Graham A. Niblo
*(University of Southampton)
***Topological Superrigidity
**

The geometric superrigidity theorem states broadly, that for $\Gamma$ in a wide class of uniform lattices, a non-constant $\Gamma$-equivariant harmonic map, with target in a suitable non-positively curved manifold is, up to renormalisation, a totally geodesic embedding. In this paper we propose a topological analogue: for a wide class of even dimensional manifolds every codimension-$1$, $\pi_1$-injective map into an aspherical, orientable manifold is, up to homotopy, a finite cover of an embedding.

### Thursday, 18th of Nov 2010, 4pm, Theatre C

Peter Moerters
*(University of Bath)
***Random networks with concave preferential attachment
**

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function of its current degree. We use approximation by branching random walks to find necessary amd sufficient criteria for the existence and robustness of a giant component in these networks. The talk is based on joint work with Steffen Dereich (Marburg).

### Thursday, 28th of October 2010, 4pm, Theatre C

Andrei Krokhin
*(University of Durham)
***The complexity of the list homomorphism problem for graphs
**

An edge-preserving mapping from a graph G to another graph H is called a homomorphism from G to H, or an H-colouring of G. If, in addition, every vertex x of G has a list L_x of allowed target vertices in H then an H-colouring of G that respects the lists is called a list H-colouring of G. We consider the list H-colouring problem for a fixed graph H, denoted LHOM(H): given a graph G and a list L_x for every vertex x of G, is there a list H-colouring of G? We provide a series of results of the following form: Theorem. For any graph H, the following conditions are equivalent: 1) LHOM(H) is complete for complexity class A; 2) LHOM(H) can be defined in logic B; 3) H has combinatorial property C; 4) H has algebraic property D. In particular, we show that, for every graph H, the problem LHOM(H) is either NP-complete, NL-complete, L-complete or is definable in first order logic. Only very basic knowledge of complexity theory is assumed.

### Thursday, 30th of September 2010, 4pm, Theatre C

Collin Bleak
*(University of St Andrews)
***Cantor Set Distractions
**

We tell tales of the lands of Cantor, and regale listeners with why we love them so. Along the way, we naively pause at the fields of Language, Algebra, and Dynamics, and, forgetting our place, harvest small flowers.