## Previous Pure Mathematics Colloquia - 2009 to 2010

Previous Pure Mathematics Colloquia from: 2013/14, 2012/13, 2011/12, 2010/11, 2009/10, 2008/09, 2007/08### Thursday, 20th of May 2010, 4pm, Theatre C

Vince Vatter
*(Dartmouth)
***Growth rates of permutation classes
**

A permutation class is a set of permutations closed under the natural combinatorial notion of subpermutation. The study of permutation 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. I will discuss the problem of characterizing the growth rates which can occur.

### Thursday, 13th of May 2010, 4pm, Theatre C

Robert Brignall
*(University of Bristol)
***Antichains and the Structure of Permutation Classes
**

Permutation classes, the analogue of hereditary properties of graphs for permutations, are defined as downsets in the permutation containment partial ordering, and are most commonly described as the collection avoiding some set of permutations, cf forbidden induced subgraphs for hereditary graph properties. Much of the emphasis in their recent intensive study has been on exact and asymptotic enumeration of particular families of classes, but an ongoing study of the general structure of permutations is yielding remarkable results which typically also have significant enumerative consequences.

In this talk I will describe a number of these structural results useful in studying the question of partial well-order -- i.e. the existence or otherwise of infinite antichains in a given permutation class. The building blocks of all permutations are "simple permutations", and we will see how these on their own contribute to the partial well-order problem. Seemingly independently, we will see how "grid classes", a construction used to express large complicated classes in terms of smaller easily-described ones, also plays its part. Finally, I will present recent and ongoing work in combining these two concepts, first describing a general construction for antichains using "grid pin sequences", and second proving where certain families of grid classes are partially well-ordered.

### Thursday, 6th of May 2010, 4pm, Theatre C

Kay Magaard
*(University of Birmingham)
***On Characters of Unitriangular Groups
**

A Sylow p-subgroup $U$ of $GL_n(p^a)$ can be represented by the set of upper triangular matrices. To this day the representations of $U$ are not known. I will report on recent activity in this area.

### Thursday, 29th of April 2010, 4pm, Theatre C

Colva Roney-Dougal
*(University of St Andrews)
***Minimal and probabilistic generation of finite groups
**

Let G be a finite group. Then by d(G) we denote the smallest number of elements that generate G. I will start by surveying some results on upper bounds for d(G). For example, we could be given information about the structure of G, or told that G can be represented as a matrix group in dimension n, or as a permutation group on n points. I will present some work in progress on this topic (joint with Derek Holt), and conclude with some discussion of the number of random elements required to generate G with given probability.

### Thursday, 22th of April 2010, 4pm, Theatre C

Sergey Shpectorov
*(University of Birmingham)
***to be announced
**

### Thursday, 15th of April 2010, 4pm, Theatre C

James East
*(University of Sydney)
***Braids and transformation semigroups
**

I'll introduce a collection of monoids which contain Artin's braid groups, and map homomorphically onto various transformation semigroups. Many of these monoids arise from natural geometric considerations such as the simplification of a knot/link by cutting a string, or allowing one string to pass through another. The main themes I'll focus on will be presentations by generators and relations, and solutions to word problem. If time permits (and it should), I'll end with a speculative discussion of the newly discovered "sprout monoids", which map onto the partition algebras.

### Tuesday, 13th of April, 3pm, Theatre 1A

Toby O'Neil
*(Open University)
***Universal singular sets in the calculus of variations
**

For regular one-dimensional variational problems, Ball and Nadirashvilli introduced the notion of the universal singular set of a Lagrangian L and established its topological negligibility. This set is defined to be the set of all points in the plane through which the graph of some absolutely continuous L-minimizer passes with infinite derivative. Motivated by Tonelli's partial regularity results, the question of the size of the universal singular set in measure naturally arises. We show that universal singular sets are characterized by being essentially purely unrectifiable --- that is, they intersect most Lipschitz curves in sets of zero length and that any compact purely unrectifiable set is contained within the universal singular set of some smooth Lagrangian with given superlinear growth. This talk is based on joint work with Marianna Csornyei, Bernd Kirchheim, David Preiss and Steffen Winter

### Tuesday, 13th of April 2010, 4.15pm, Theatre D

Ritva Hurri-Syrjanen
*(University of Helsinki)
***Inequalities in domains with irregular boundaries
**

We consider classical inequalities such as Poincare-, Sobolev-, Trudinger-type inequalities, which are known to be valid in domains with sufficiently regular boundaries. We discuss the validity of these inequalities in some classes of domains which have irregular boundaries.

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

Jim Howie
*(Heriot-Watt University)
***Generalised Triangle Groups
**

A generalised triangle group is one with a presentation of the form $\langle x,y~|~x^p=y^q=W(x,y)^r=1\rangle$ with $p,q,r\ge 2$. When $W=xy$ this is an ordinary triangle group. Interest in such groups stems from the work of Culler, Gordon, Luecke and Shalen on Dehn surgery on knots in the 1980's. Rosenberger conjectured that they satisfy a version of the Tits alternative, and this conjecture has been verified for most values of the parameters $p,q,r$. I shall survey the known results to date on the Rosenberger conjecture, and sketch the proof of a new result in support of it.

### Thursday, 11th of February 2010, 4pm, Theatre C

Martin Bridson
*(University of Oxford)
***On the difficulty of presenting finitely presentable groups
**

A group G is finitely presentable if it has a finite presentation, i.e. there is a map from a finitely generated free group onto G such that the kernel is the normal closure of a finite set -- equivalently, G is the fundamental group of a compact manifold. In this talk I shall discuss the curious distinctions and interesting geometry associated to the following problems: (i) given a group with a solvable word problem, can one decide which of its finitely generated subgroups are finitely presentable?; (ii) given a group with a solvable word problem, can one find finite presentations for its finitely presentable subgroups?

### Thursday, 21st of January 2010, 4pm, Theatre C

Tuomas Sahlsten
*(University of Helsinki)
***Porous measures and their tangents
**

In this talk we consider three fundamental tools in geometric measure theory that reflect the local geometry of measures: porosity, doubling condition and tangent measures. A priori, these tools are defined very differently, but we will end up finding many connections between them. I will also present some open problems related to these connections. All the needed definitions will be explained during the presentation and only basic knowledge of measure theory will be assumed.

### Thursday, 17th of December 2009, 4pm, Theatre C

John Macquarrie
*(University of Manchester)
***Profinite modular representation theory
**

The modular representation theory of a finite group G concerns the study of finitely generated modules over the group algebra kG, where k is a field of positive characteristic p. Such modules can be organised using the concepts of relative projectivity, vertex and source. Basic results of Green and Higman along these lines constitute fundamentals upon which the subject is built.

A profinite group is the inverse limit of an inverse system of finite groups. While such groups are set-wise "big", the inverse system gives profinite groups a close relationship with finite groups - a conduit through which important finite results can flow. Having defined all the (maths) words above we will see that many of the foundational results of modular representation theory of finite groups pass (with varying degrees of reluctance) through the inverse system, to the limit.

### Thursday 19th of November, 4pm, Theatre C

Csaba Schneider
*(Centre of Algebra University of Lisbon (CAUL))
***On permutation groups that define regular or idempotent generated semigroups
**

In this talk I will present some recent results on finite permutation groups that give rise to regular or idempotent generated transformation semigroups. More specifically, we study finite permutation groups G with the property that either the transformation semigroup <G,a> is regular for all singular transformations a, or the transformation semigroup <G,a>-G is idempotent generated for all singular transformations a. It has been known for a while that the symmetric groups and the alternating groups satisfy these properties and been conjectured that these are the only such groups. In our research we confirmed a weak form of this conjecture that, apart from the alternation groups and the symmetric groups, the property above is valid only for a small number of groups. The results in this presentation were achieved in collaboration with Joao Araujo (Lisbon) and James Mitchell (St Andrews).

### Thursday 5th of November, 4pm, Theatre C

Pablo Shmerkin
*(University of Manchester)
***Local entropy averages and projections of fractal measures
**

The projection results of Marstrand-Kaufman-Mattila that relate the dimension of a set or measure in Euclidean space to that of its orthogonal projections are a jewel of geometric measure theory.

These results are measure-theoretic in nature, and no topological analogues can be expected: for example, the dimension of a projection is in general highly discontinuous as a function of the projecting map.

I will describe a class of measures for which we are able to show that the dimension of projections behaves in a semicontinuous way. This class includes many natural objects in conformal dynamics, such as deterministic and stochastic self-similar sets.

When the measure has some additional ``rotational'' structure, the semicontinuity implies that the dimension of the projection is in fact constant across all linear, and even smooth, projections (aside from some obvious exceptions).

As an application, we solve a problem of Furstenberg related to measures invariant under multiplication by two and three on the circle.

This is joint work with M. Hochman.

### Thursday 29th of October, 4pm, Theatre C

Meinolf Geck
*(University of Aberdeen)
***Generic representations of finite groups of Lie type
**

Finite groups of Lie type are analogues of real or complex Lie groups over finite fields; they are also the main source of finite simple groups. We discuss some major trends and open problems in the representation theory of these groups.

### Thursday 22nd of October, 4pm, Theatre C

Nik Ruskuc
*(University of St Andrews)
***Growth of generating sets in direct products**

### Thursday 15th of October, 4pm, Room 1A

Tara Brendle
*(University of Glasgow)
***Mapping Class Groups of Surfaces**