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.
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
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.)