Oxford

The Siegel-Walfisz theorem gives an asymptotic estimate for the number of primes in an arithmetic progression, provided the modulus of the progression is small in comparison with the length of the progression. Counting primes is harder when the modulus is not so small compared to the length, but estimates such as Linnik's constant and the Brun-Titchmarsh theorem give us some information. We aim to look in particular at upper bounds for the number of primes in such a progression, and improving the Brun-Titchmarsh bound.

When modeling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation diagram.

Stochastic Simulation Algorithms (SSAs) exist which draw exact trajectories from the Chemical Master Equation (CME). However, these methods can be very computationally expensive, particularly where there is a separation of time scales of the evolution of some of the chemical species. Some of the species may react many times on a time scale for which others are highly unlikely to react at all. Simulating all of these reactions of the fast species is a waste of computational effort, and many different methods exist for reducing the system to one which only contains the slow variables.

In this talk we will introduce the conditional Gillespie algorithm, a method for sampling directly from the conditional distribution on the fast variables, given a static value for the slow variables. Using this, we will go on to describe the constrained Gillespie approach, which uses simulations of the CG algorithm to estimate the drift and diffusion terms of the effective dynamics of the slow variables.

If there is time at the end, I will briefly describe my work on another project, which involves full sampling of the posterior distributions in various problems in data assimilation using Monte Carlo Markov Chain (MCMC) methods.

Let $G $ be a compact Lie group, and consider the variety $\text {Hom} (\bb Z^k,G)$

of representations of the rank $k$ abelian free group $\bb Z^k$ into $G$. We prove

that the fundamental group of $\text {Hom} (\bb Z^k,G) $ is naturally isomorphic to direct

product of $k$ copies of the fundamental group of $G$. This is joint work with

Jose Manuel Gomez and Juan Souto.