Contact
Biography
I am an Assistant Professor of Pure Mathematics, specialising in number theory. I joined the school in September 2022, following postdoctoral positions at Warwick (2019-22) and Imperial (2016-2019). Before that, I obtained my PhD from Warwick in 2016.
My research is currently supported by EPSRC Fellowship EP/T001615/1.
My personal website is more likely to be up to date than this one.
Teaching Summary
I've written a number of undergraduate and graduate courses in algebraic number theory. An up-to-date list of courses, and associated lecture notes, can be found [ here ].
Research Summary
I am broadly interested in algebraic number theory, with a specialism in Iwasawa theory (the general study of p-adic variation of automorphic forms). I have written a number of introductions to… read more
Selected Publications
Current Research
I am broadly interested in algebraic number theory, with a specialism in Iwasawa theory (the general study of p-adic variation of automorphic forms). I have written a number of introductions to Iwasawa theory/p-adic L-functions, for advanced undergraduates and above, which can be found on my personal webpage. I'm always happy to discuss this area of mathematics with those who are interested!
The following is an edited lay summary from my recent EPSRC Fellowship.
L-functions are fundamental mathematical objects that encode deep arithmetic information. Their study goes back centuries, and they are the subject of the two biggest unsolved problems in modern number theory, namely the Riemann hypothesis and the Birch and Swinnerton-Dyer (BSD) conjecture.
The BSD conjecture predicts that the number of rational solutions of a cubic equation (defining an 'elliptic curve') is controlled by a value of an analytic L-function. This prediction, providing a mysterious bridge between the fields of arithmetic geometry and complex analysis, has since been hugely generalised in the Bloch-Kato conjectures. There has been much recent success in attacking such problems by changing the way we look at this bridge. In particular, by considering different notions of 'distance' between two numbers, we are able to build a whole array of different algebraic connections between arithmetic and analysis, and these have allowed us to build parts of the bridge required for BSD and Bloch-Kato.
The distance in question is the 'p-adic' distance, where two numbers are very close if their difference is very divisible by a prime p (for example, the numbers 1 and 1,000,000,001 are very close 2-adically, since their difference is divisible by 2 nine times). For each prime p, there should be a p-adic version of the Bloch-Kato conjectures - known as 'Iwasawa main conjectures' - and each of these gives another crucial connection between arithmetic and analysis. Such connections depend absolutely on the existence of p-adic versions of L-functions.
In addition to their utility in solving important conjectures, p-adic L-functions are beautiful objects in their own right: their existence means there are discrete p-adic congruences between the values of L-functions. It still seems miraculous to me that one can find remarkable algebraic structures in the values of a transcendental, complex analytic function, and I've been captivated by this field ever since I saw the first example (Kubota and Leopoldt's p-adic Riemann zeta function).
It is expected that for every L-function there is a p-adic version, but as they can be extremely difficult to construct, we are very far from reaching this goal. A focus of my research is constructing new p-adic L-functions, particularly in higher-dimensional settings. Some of my recent research highlights include giving the first general construction of p-adic L-functions for dimension 3 (in joint work with David Loeffler) and the most general known constructions for dimension 4, 6, 8, etc. (with Daniel Barrera and Mladen Dimitrov).