I’m now a Research Fellow at the University of Surrey.
I was a lecturer of mathematics at Lincoln College, Oxford. Information on current and past teaching can be found here.
Here are some helpful resources, and pages and articles I like and find fascinating. There are also excellent books, but unless there is a link to the files I can find, I won’t put them here — perhaps on another page, on another day.
fast singular limits in PDEs
- I’m now working on fast singular limits in of rotating shallow water equations and Boussinesq equations in the context of weather prediction.
stochastic analysis, probability theory, ergodic theory, pdes:
- Peter Fritz’s exposition is simply and generally good: Fritz on the Malliavin Calculus
- Fritz and Hairer on rough paths
- I thought this wasn’t on the internet anymore! — Moerters and Peres’s Brownian Motion is a really readable and interesting account of the basics of the subject
- long time behaviour of conservation laws by Debussche and Vovelle is something I have thought about a bit and may post my own notes on it later
- an odd one, but nevertheless interesting — Nelson’s non-standard development of probability theory
analytic number theory
- two classics by Furstenberg and friends on dynamics on the symbolic system and more specifically, the ergodic proof of Szemeredi’s theorem (and Gowers’, and Szemeredi’s original)
- Kowalski’s notes on probabilistic number theory
- Kowalski also wrote a long article on a general framework for the large sieve, though a much gentler, introductory set of notes would be Jameson’s
- Klainerman and Rodnianski’s geometric Littlewood-Paley Theory is a wonderful read on L-P theory on manifolds
two special problems (special to me)
- the Hadwiger conjecture states that if a graph is not k-colourable, then it has a complete-(k+1) minor; Robertson and Seymour havewritten lots about it, and here is a survey of results on the problem
- the Erdos-Turan conjecture states that if an infinite collection (a_i) of increasing positive integers satisfies the density condition that the sum of their reciprocals is infinite, then that collection contains arbitrarily long arithmetic sequences. this is of course related to Szemeredi’s theorem, with a much weaker density condition. Green and Tao (Conlon, Fox, and Zhao’s exposition) pushed beyond Szemeredi’s positive Banach density condition to arithemtic progessions on the primes.
I’m currently reading Arora and Barak’s book on computational complexity and it is fascinating!