This page lists this term's Pure Mathematics Seminars. All are welcome. All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 4.00pm, unless stated otherwise. Tea is served before the seminar at 3.30 in Room 237 of the McCrea Building.

**Seminars for the spring term 2009.**

- 13th January:
**Martin Bright**(Bristol) The local-global appraoch to Diophantine equations (abstract) - 20th January:
**Peter Allen**(Warwick) Turán to Pósa-Seymour: minimum degree conditions for large subgraphs (abstract) - 27th January:
**Christopher Voll**(Southampton) - 3rd February:
**Jens Marklof**(Bristol) The Boltzmann-Grad limit of the periodic Lorentz gas and the distribution of visible lattice points (abstract) - 10th February:
**Simon Blackburn's**inaugural lecture. No seminar**.** - 12th February:
**(Thursday, 2pm McCrea229):****Ben Martin**(Canterbury) Polynomial representation growth and alternating quotiens (abstract) - 17th February:
**Maura Paterson**(RHUL) Combinatorial batch codes (abstract) - 24th February: No seminar.
- The talk by
**Alexander Kleshchev**(University of Oregon) on 'Graded representation theory of symmetric groups' (abstract) had to be canceled. - 3rd March:
**Arie Koster**(Warwick) Recent advances in integer programming - 10th March:
**Yvo Desmedt**(UCL) Applying Recreational Mathematics to Secure Mutliparty Computation - 17th March:
**Norman Biggs**(LSE) Curious curves and graph polynomials. - 24th March:
**Dudley Stark**(QMUL) Poisson approximation of the number of cliques in random intersection graphs **Thursday**26th March,**2pm, McCrea229**:**Mark Berman**(Ben-Gurion University of the Negev) Counting conjugacy classes using p-adic integrals (abstract)

**Abstracts: **

Martin Bright: The local-global appraoch to Diophantine equations

The study of whole-number solutions of polynomial equations has attracted mathematicians for thousands of years, yet we are still far from understanding anything but the simplest equations. One approach is to look at "local" information: that is, information over the real numbers and modulo prime powers; and to try to deduce information about the "global" solutions: that is, solutions in integers or rational numbers. For example, Hasse proved that a quadratic form in any number of variables has a rational zero if, and only if, it has zeros in the real numbers and modulo each prime power. More general equations do not satisfy this "Hasse principle"; I will describe ways to understand this failure.

Peter Allen: Turán to Pósa-Seymour: minimum degree conditions for large subgraphs

The rth power of a graph G is the graph G^{r} on the same vertex set, with uv an edge of G^{r} if the distance from u to v in G is at most r. Extremal graph theory begins with Turán's theorem, which guarantees that an n-vertex graph whose minimum degree is (r-1/r)n must contain a complete subgraph K_{r+1}. Generalisations (such as the Erdös-Stone theorem) guarantee the existence of all small subgraphs with chromatic number r+1 at around the same minimum degree threshold; this includes the rth power of a short path. At the other extreme Dirac's theorem guarantees that an n-vertex graph with minimum degree n/2 contains a path covering all n vertices; and the Pósa-Seymour conjecture (proved by Komlós, Sárközy and Szemerédi for large graphs) generalises this, guaranteeing the rth power of a path covering all n vertices when the minimum degree is at least (r/r+1)n. This leaves a gap: what minimum degree condition guarantees the existence of the rth power of a path on a n vertices for some 0<a<1? We answer this question for r=1 and 2, give conjectures for larger r, and mention a further generalisation to `path-like' graphs. This is joint work with Julia Böttcher and Jan Hladký.

Jens Marklof: The Boltzmann-Grad limit of the periodic Lorentz gas and the distribution of visible lattice points

The periodic Lorentz gas describes a particle moving in a periodic array of spherical scatterers, and is one of the fundamental mathematical models for chaotic diffusion in a periodic set-up. In this lecture (aimed at a general mathematical audience) I describe the recent solution of a problem posed by Y. Sinai in the early 1980s, on the nature of the diffusion when the scatterers are very small. The problem is closely related to some basic questions in number theory, in particular the distribution of lattice points visible from a given position, see e.g. Polya's 1918 paper on the visibility in a forest. The key technology in our approach is measure rigidity, a branch of ergodic theory that has proved valuable in recent solutions of other problems in number theory and mathematical physics, such as the value distribution of quadratic forms at integers, quantum unique ergodicity and questions of diophantine approximation. (This lecture is based on joint work with A. Strombergsson, Uppsala.)

Ben Martin, Polynomial representation growth and alternating quotients

Let Gamma be a group. Given a positive integer n, we define r_n(Gamma) to be the number of isomorphism classes of complex representations of Gamma of dimension at most n (note that r_n(Gamma) can be infinite). We say that Gamma has *polynomial representation growth* if there exist positive alpha and beta such that r_n(Gamma) is at most alpha*(n^beta) for every n. In this talk I will discuss a question of Brent Everitt: does there exist a finitely generated group Gamma such that (1) Gamma has polynomial representation growth; and (2) Gamma has the alternating group A_m as a quotient for infinitely many m?

Maura Paterson, Combinatorial batch codes

A batch code specifies a method to distribute a database of n items among m devices (servers) in such a way that any k items can be retrieved by reading at most t items from each of the servers. It is of interest to devise batch codes that minimize the total storage, denoted by N, over all m servers. In this talk we consider the special case t=1, under the assumption that every server stores a subset of the items. This is purely a combinatorial problem, so we call this kind of batch code a "combinatorial batch code''. We consider how batch codes that are optimal with respect to the storage requirement N can be constructed for various combinations of parameters. We also discuss uniform codes, where every item is stored in precisely c of the m servers (such a code is said to have rate 1/c), presenting interesting new results in the cases c = 2, k-2 and k-1. In addition, we describe improved existence results for arbitrary fixed c using the probabilistic method.

Alexander Kleshchev, Graded representation theory of symmetric groups

Mark Berman: Counting conjugacy classes using p-adic integrals

Abstract: Consider the problem of counting the number of finite index substructures of a given algebraic object; for example, subgroups of a finitely generated discrete group, or isomorphic sublattices in a Z_{p} lattice endowed with a ring structure. Grunewald, Segal and Smith were the first to show how to study these and related problems by associating to the algebraic object a p-adic integral. Fundamentally different types of p-adic integrals are employed in different contexts. These various integrals may be analyzed using techniques coming from logic, algebraic geometry or the theory of algebraic groups. Voll has recently proved that various zeta functions associated to groups and rings satisfy functional equations in very general settings. One of his key ideas was to introduce certain p-adic integrals generalising Igusa's local zeta function. In the first part of the talk I will give an account of the history and main results. In the second part, I will consider the concrete problem of counting the numbers of conjugacy classes in congruence quotients of the general linear group over the p-adic integers. This problem was considered for compact p-adic analytic groups in general by du Sautoy, who proved that the sequence satisfies a recurrence relation. In joint work with Onn and Paajanen, I have shown how to express this sequence in terms of a different p-adic integral more closely resembling the type of integral considered by Voll.

I will focus on the specific case of GL_{2}, for which a simple formula was found by Avni et al, and show how the integral can be effectively computed. I will outline two approaches, one based on resolution of singularities, the other based on a special decomposition of the domain which considerably simplifies the calculation.

The situation for larger n is largely unchartered territory. I will suggest some open questions and a different counting problem amenable to the same approach.

**Useful links**:

Campus map (The McCrea Building is number 17)