## Previous Pure Mathematics Colloquia - 2008 to 2009

Previous Pure Mathematics Colloquia from: 2013/14, 2012/13, 2011/12, 2010/11, 2009/10, 2008/09, 2007/08

### Thursday 23rd of April, 4pm, Room 1A

Sara Munday
(University of St Andrews)
Hausdorff dimensions of Good sets and strict Jarnik sets for Fuchsian groups with parabolic elements.

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarnik limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. The talk will describe generalisations of these Jarnik sets. In particular, we will show that a natural generalisation of these sets, which we call strict Jarnik limit sets, gives rise to generalised weak multifractal spectra. We will also give number-theoretical interpretations of these results in terms of continued fractions.

### Thursday 12th of March, 4pm, Room 1A

Frank Lübeck
(RWTH Aachen)
Representations of Small Degree of Finite Groups of Lie Type

An important tool for studying a (finite) group G is to consider its representations, i.e., homomorphisms G -> GL(n,F) from G into groups of invertible nxn-matrices over various fields F. The n is called the degree of the representation. In this talk we consider as groups G finite groups of Lie type - these will be introduced informally. They are closely related to many of the finite simple groups which were classified in the 1980's. An example are the general linear groups GL(k,q) of invertible kxk-matrices over a finite field with q elements. We will consider the following question: For a given such group G and a given (algebraically closed) field F, what is the smallest degree of a non-trivial representation of G over F? The known answers to this question rely on quite deep mathematics, and the theoretical background is very different depending on the characteristic of the field F: F is the complex numbers, or of prime characteristic l dividing the order of G, in the latter case the defining characteristic (l divides q in the example above) and non-defining characteristic must be considered separately.

### Thursday 5th of March, 4pm, Room 1A

Richard Sharp
(University of Manchester)
Distortion and entropy for automorphisms of free groups

Recently, several numerical characteristics have been introduced to quantify the distortion induced by an automorphism of a free group: generic stretch (Kaimanovich, Kapovich and Schupp), curl (Myasnikov and Shpilrain), conjugacy distortion spectrum (Kapovich). We shall describe how these may be unified by interpreting them in terms of an entropy function of a kind familiar in thermodynamic ergodic theory and multifractal analysis.

### Thursday 8th of January, 4pm, Room 1A

Eric Jespers
(Brussels)
Finitely presented algebras and groups defined by permutation relations

In recent joint work with F. Cedo and J. Okni\'nski, the class of finitely presented algebras $R$ over a field $K$ with a set of generators $a_{1},\ldots , a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{\sigma (a)} a_{\sigma (2)} \cdots a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group $\Sym_{n}$ of degree $n$, is introduced.

We mainly studied three cases: (1) $H$ a cyclic group, (2) $H=\mbox{Sym}_{n}$, the symmetric group of degree $n$, and $H=\mbox{Alt}_{n}$, the alternating group of degree $n$.

We investigate (1) the underlying monoid, defined by the same (monoid) presentation, (2) The associated group, defined by the same (group) presentation and (3) the algebra $R$. The results obtained are presented in this talk.

### Thursday 27th of November at 4pm in Room 1A

Johannes Orlob
(RWTH Aachen/University of Aberdeen)
Tensor products of simple modules of Symmetric Groups

Recently developed computational methods of modular representation theory provide an insight into the structure of modules for group algebras. I will present some results concerning the (inner) tensor products of simple modules of the Symmetric Groups with respect to the decomposition into indecomposable modules.

### Thursday 13th of November at 4pm in Room 1A

Peter Cameron
(QMUL)
Synchronization, homomorphisms, and cores

### Monday 10th of November at 4pm in Room 1A

Peter Cameron
(QMUL)
Synchronization and permutation groups

### Thursday 6th of November at 3pm in Room 1A

Jim Wright
(University of Edinburgh)
Isoperimetric(-type) inequalities in Harmonic Analysis

### Thursday 30th of October

Graham Ellis
(NUI Galway)
Polytopes, perturbations and cohomology calculations

The talk will describe some attempts at using computers to determine cohomological properties of a range of finite and infinite groups. It will cover techniques such as Wythoff polytopes, a perturbation lemma, Groebner bases, a lemma of Whitehead and the Gauss-Bonnet formula.

### Thursday 16th of October

Jan Kristensen
(University of Oxford)
On the problem of regularity in the calculus of variations

### Thursday 9th of October

Victor Bovdi
(University of Debrecen, Hungary)
Lie nilpotency indices of modular group algebras

Let K be a field of positive characteristic p and KG be the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G'|+1, where |G'| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. In our talk we determine G for which upper (or lower) Lie nilpotency index are maximal or almost maximal, or the next highest possible value.

### Thursday 2nd of October

Michael Albert
(University of Otago, New Zealand)
Growth rates of pattern classes

A pattern class is a set of permutations closed under the natural combinatorial notion of subpermutation. The study of pattern classes, and in particular their enumeration has been an active area of research; spurred initially by the observation of strange coincidences in their enumerative sequences. The resolution, early this century, by Marcus and Tardos of the Stanley-Wilf conjecture has focused attention on the exponential growth rates of these classes. In addition to the question of which growth rates can occur, it seems natural to investigate circumstances under which the enumeration of a pattern class can be accomplished "automatically", typically owing to some natural underlying correspondence with a regular or context-free language. We will introduce, survey, and look ahead to the future of this area.

### Thursday 25nd of September

John Bamberg on behalf of Jan De Beule, Philippe Cara and Michel Lavrauw
(Ghent University and Vrije Universiteit Brussel)
A GAP package for finite geometry

Finite geometry is an area of mathematics that is especially amenable to computational investigations, and it is the purpose of this talk to give an overview of what finite geometry is and to present some details on the GAP package which is in development. No prior knowledge of finite geometry or the computer algebra system GAP is required.