I finished my PhD under the supervision of Florian Theil and Ian Melbourne in February 2018. The primary interests of my work were in kinetic theory and the justification of the Boltzmann equation.
Ph.D. Thesis (2014-17)
The purpose of my Ph.D. thesis was to provide a first full proof of the justification of the linear Boltzmann equation with interactions given by a long range potential. This can also be found at the following arxiv preprint
I prove that if one has a unique particle interacting with a free flowing background where the interaction is given by a potential with sufficiently fast decay, then the low density limit of the density of the unique particle is a weak solution of the linear Boltzmann equation.
Masters Dissertation (2014)
I worked with Florian Theil for my masters dissertation. It is based on an idea by Richard D James called objective molecular dynamics. He suggests the use of this structure as a generalisation of spatial homogeneity for the Boltzmann equation. Explicitly, one assumes an ansatz of the form which corresponds to assuming there is some spatial periodicity of the solution.
Within this type of solution, the particular form of gives the explicit evolution of energy and entropy in terms of the initial distribution and .
The project culminates in showing that there shouldn't be a solution of this form when one takes for two perpendicular vectors .
Following on from this, we then transformed the problem into a mathematically tractable problem, that of approximation of a function. We then applied least squares approximation to this, with reasonable results.
The report of the project can be found here, which explains in details the methods
We also gave a presentation on the project, and the slides used in this can be found here. We also presented a poster as well. We also have a webpage with some other information on it, whose link can be found here.
MMath Dissertation (2012-13)
In my final year at Warwick I wrote a dissertation on the Interacting Bose Gas on the Lattice with Daniel Ueltschi.
Second Year Essay
In my second year at Warwick I wrote an essay on Hilbert's Third Problem. This is the problem pertaining to scissors congruence of polyhedra.
- Probability, Analysis and Dynamics, Univeristy of Bristol, April 2014
- YRM 2014, University of Warwick, June-July 2014
- Berlin Summer School:Applied Analysis for Materials, TU Berlin, August-September 2014
- Masdoc-CCA conference, University of Warwick, April 2015
- Nonlinear evolutions: Kinetic equations and defect dynamics, HCM Universitat Bonn, July 2015
- Interplay of Analysis and Probability in Applied Mathematics, MFO Workshop, July 2015
- M&MKT '16 , Porto Ercole, June 2016, where I delivered a poster
I have latexed various different lecture courses I went to. They are in various states of repair, and any comments, or corrections, would be most gratefully received.
I play squash for the University and also enjoy playing badminton and cricket, the latter of which I play for the ITS cricket team.
I can be contacted at the address of m dot egginton at warwick dot ac dot uk
or in room B3.04