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 2.30pm, unless stated otherwise. Tea is served after the seminar at 3.30 in Room 237 of the McCrea Building.

**Seminars for the summer term 2009.**

- 28th April:
**Alan Hayes**(York): Recent results about Farey fractions (abstract) - 5th May: Interviews. No seminar
- 12th May:
**Simon Blackburn**(RHUL): Distinct difference configurations and wireless sensor networks (abstract) - 19th May:
**Steven Galbraith**(RHUL): Fast elliptic curve cryptography using endomorphisms, and security considerations (abstract) - 26th May
**McCrea336**:**Jon Gonzalez**(Santander): Parameterisation of a Smooth Cubic Surface (abstract) - 2nd June:
**Reto Spöhel**(ETH Zurich): Small subgraphs in the Achlioptas process (abstract) - 9th June
**McCrea336**:**Mihyun Kang**(HU Berlin): How to count planar graphs? (abstract) - 15th June
**(Monday) 11-12: Mikhail Ershov**: Kazhdan quotients of Golod-Shafarevich groups - 16th June:
**Yvonne Buttkewitz**(RHUL) Multiplicative arithmetic functions at consecutive integers - 21st July
**(4pm, McCrea229)**:**Colin Reid**(Queen Mary): Subgroups of finite index and the just infinite property (abstract) - 28th July:
**Claas Roever**(National University of Ireland, Galway): The Commensurator of Thompson's Group (abstract)

**Abstracts:**

Alan Hayes: Recent results about Farey fractions

In the 1920's Franel and Landau proved that the Riemann Hypothesis is equivalent to a statement about the discrepancy of the Farey sequence. In 1971 M. N. Huxley proved an analogous theorem for Dirichlet L-functions. Over the last decade these results have motivated an investigation into the structure of relevant subsets of the Farey fractions. We will discuss some of the fruit of this investigation, as well as some potential applications and unsolved problems.

Simon Blackburn: Distinct difference configurations and wireless sensor networks

This seminar is based on joint work with Tuvi Etzion, Keith Martin and Maura Paterson. See the preprints at http://arxiv.org/abs/0811.3832 and http://arxiv.org/abs/0811.3896 for full details. A set X={x1,x2,...,xn of points in R^{2} is a *distinct difference configuration* if the vectors xi-xj for i≠j are all distinct. A typical question we might want to ask is: Suppose the points have integer co-ordinates, and that any pair of points are at Euclidean distance at most d. How large can n be (as a function of d)? Similar questions have been studied as part of the theory of Costas arrays and of B sequences, for example. Our motivation comes from a key predistribution scheme for wireless sensor networks that uses distinct difference configurations. I plan to introduce this key predistribution scheme, talk about some combinatorial problems this scheme motivates and give some good constructions and bounds for distinct difference arrays. I will show how our work settles aconjecture of Golomb and Taylor from 1984 on honeycomb arrays.

Steven Galbraith: Fast elliptic curve cryptography using endomorphisms, and security considerations

Public key cryptography based on the discrete logarithm problem in a finite group started with the work of Diffie and Hellman in 1976. In the mid-1980s it was suggested that the group of points on an elliptic curve might offer better security than the multiplicative group of a finite field. It turns out that elliptic curves have other features which make them attractive for cryptography. This talk will summarise the Gallant-Lambert-Vanstone trick of speeding up elliptic curve cryptography using an endomorphism. I will present some recent extensions of this idea. I will then explain why this trick motivates research on the multi-dimensional discrete logarithm problem, which is a topic I have been investigating together with my students Waldyr Benits Jr. and Raminder Ruprai.

Jon Gonzalez: Parameterisation of a Smooth Cubic Surface

A cubic surface S is the vanishing set of a homogeneous polynomial of degree 3 in complex 3-dimensional projective space. A proper parametrisation of S is a birational map from 2-dimensional projective space to S, given by homogeneous polynomials of the same degree and without any common divisors.

By a classical result of Clebsch, every cubic surface admits a proper parametrisation by cubic polynomials over the complex numbers. In my talk I will discuss how such a parameterisation can be obtained explicitly and under which circumstances it can be obtained over a smaller field, e.g. the real or the rational numbers.

Reto Spöhel: Small subgraphs in the Achlioptas process

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G_0 as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining r-1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph G_N. In a recent paper by Krivelevich, Loh, and Sudakov, the problem of avoiding a copy of some fixed graph F in G_N for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F. We disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N_0(F,r,n) for arbitrary graphs F and any fixed integer r.

Joint work with Torsten Mütze and Henning Thomas

Mihyun Kang: How to count planar graphs?

Random graphs on surfaces, in particular random planar graphs, have recently received much attention from the viewpoint of typical properties, uniform sampling and enumeration.The focus of this talk will be the enumeration problem. We briefly surveyrecent results and methods of how to count labelled planar graphs with given numbers of vertices and edges. We also discuss the power and limitations of each method.

Colin Reid: Subgroups of finite index and the just infinite property

An infinite residually finite (profinite) group G is* just infinite* if every non-trivial (closed) normal subgroup is of finite index, and* hereditarily just infinite* if every subgroup of finite index is just infinite. But which subgroups of finite index need to be just infinite to ensure that all of them are? I will present a partial answer to this question, after a brief overview of the basic types of just infinite groups and why they are of interest.

Claas Roever: The Commensurator of Thompson's Group

I shall describe how one can determine the abstract

commensurator of Thompson's group F and what is know about its structure. All terms occuring in the previous sentence will be explained during the talk; no preknowledge of commensurators orThompson's group are required.

**Useful links**:

Campus map (The McCrea Building is number 17)