Daniel Wilson

Personal photo - Daniel Wilson

Daniel Wilson

Doctoral Student at the

Rex Richards Building, South Parks Road, Oxford OX1 3QU

E-mail: daniel(dot)wilson(at)dtc(dot)ox(dot)ac(dot)uk

Michaelmas Timetable

Background and Current Research

October, 19, 2015

My background is in Mathematics and I spent the last four years in Oxford studying for the masters degree MMath. I specialised in applied mathematics with a focus on stochastic modelling and numerical methods. A few of the most relevant modules I took were: stochastic models of biological processes; probability and statistics for network analysis; stochastic models in mathematical genetics; applied probability; numerical solutions to ordinary differential equations and finite element methods for partial differential equations.

In the Summer of 2014 I undertook a project with Prof Ruth Baker in the Wolfson Centre for Mathematical Biology where we investigated the potential for patterning in on lattice models with volume exclusion. We first considered the potential for Turing patterns and derived the necessary conditions for diffusion driven instabilities to occur in a volume exclusion model. Then we considered a variety of different reaction networks, and using the chemical master equation derived the particular reaction-diffusion system of equations. With both analytical and computational methods we tried to find the region of parameter space where the all the conditions for Turing patterns held. We discovered that this was much harder than anticipated and for a few well known reaction networks we showed that Turing patterns under volume exclusion were impossible. Because of this we moved on to consider other motility mechanisms such as adhesion and chemotaxis. One of the most important outcomes of the project was that I gained considerable proficiency in MatLab, especially writing stochastic simulation algorithms for cellular automata. My supervisor allowed me to attend the weekly journal club where we discussed recent publications, it was a great opportunity to discuss content but also discuss what is important to create a well written paper.

In my fourth year I produced a dissertation, again under the supervision of Prof Ruth Baker, on the multi-level Monte Carlo method. We investigated the potential for the method to be used to reconstruct the entire distribution of a particular species of a biochemical reaction network at a fixed terminal time. Then we used work by mathematicians Chang-han Rhee and Peter Glynn to extend the method to approximating stationary distributions. The dissertation concluded with a novel application in mathematical genetics to produce accurate approximate distributions to systems where analytical approaches are not feasible.