University of Cambridge

This talk is about the ordinary representation theory of finite groups of Lie type. I will begin by carefully reviewing algebraic groups and finite groups of Lie type and the construction and properties of (ordinary) Gelfand--Graev characters. I will then introduce generalized Gelfand--Graev characters, which are constructed using the Lie algebra of the ambient algebraic group. Towards the end I hope to give an idea of how generalized Gelfand--Graev characters can and have been used to attack Lusztig's conjecture and the role this plays in the determination of the character tables of finite groups of Lie type.

I shall report on a programme of research which is joint with Terence Tao. Our

goal is to count the number of solutions to a system of linear equations, in

which all variables are prime, in as much generality as possible. One success of

the programme so far has been an asymptotic for the number of four-term

arithmetic progressions p_1 < p_2 < p_3 < p_4 <= N of primes, defined by the

pair of linear equations p_1 + p_3 = 2p_2, p_2 + p_4 = 2p_3. The talk will be

accessible to a general audience.

Diffusion limited aggregation (DLA) is a random growth model which was

originally introduced in 1981 by Witten and Sander. This model is prevalent in

nature and has many applications in the physical sciences as well as industrial

processes. Unfortunately it is notoriously difficult to understand, and only one

rigorous result has been proved in the last 25 years. We consider a simplified

version of DLA known as the Eden model which can be used to describe the growth

of cancer cells, and show that under certain scaling conditions this model gives

rise to a limit object known as the Brownian web.

We consider multi-dimensional Gaussian processes and give a novel, simple and

sharp condition on its covariance (finiteness of its two dimensional rho-variation,

for some rho <2) for the existence of "natural" Levy areas and higher iterated

integrals, and subsequently the existence of Gaussian rough paths. We prove a

variety of (weak and strong) approximation results, large deviations, and

support description.

Rough path theory then gives a theory of differential equations driven by

Gaussian signals with a variety of novel continuity properties, large deviation

estimates and support descriptions generalizing classical results of

Freidlin-Wentzell and Stroock-Varadhan respectively.

(Joint work with Nicolas Victoir.)