Professor of Pure Mathematics

Professor of Pure Mathematics

**Analytic number theory, quantitative arithmetic geometry, arithmetic combinatorics:**

My research is centred on the Hardy-Littlewood (circle) method, a method based on the use of Fourier series that delivers asymptotic formulae for counting functions associated with arithmetic problems. In the 21st Century, this method has become immersed in a turbulent mix of ideas on the interface of Diophantine equations and inequalities, arithmetic geometry, harmonic analysis and ergodic theory, and arithmetic combinatorics. Perhaps the most appropriate brief summary is therefore “arithmetic harmonic analysis”. Much of my work hitherto has focused on Waring’s problem (representing positive integers as sums of powers of positive integers), and on the proof of local-to-global principles for systems of diagonal Diophantine equations and beyond. I am also involved in exploring the consequences for the circle method of Gowers’ higher uniformity norms, the use of arithmetic descent, and function field variants. The ideas underlying each of these new frontiers seem to offer viable approaches to tackling Diophantine problems known to violate the Hasse principle. Most recently of all, my work on the “efficient congruencing” method delivers close to best possible conclusions concerning a host of problems associated with translation-dilation invariant systems of equations, and has vastly improved our understanding of exponential sums.

In arithmetic combinatorics, my research centres on the use of Fourier analysis in problems of a number-theoretic flavour. Given a subset of the rational integers, or of a polynomial ring, or some other interesting algebraic set, can one infer the existence of arithmetically interesting structures merely from information concerning the number of elements in the set of a given height? The archetypal result of this flavour is due to Klaus Roth in 1953, and asserts that whenever a set of integers not exceeding N possesses no 3-term arithmetic progressions, then the set contains no more than N/log log N integers, and hence has relative density zero. Rather than linear equations, as in this example, in my work and that of my students we typically consider equations of higher degree. One arena of interest right now is the relationship between the sensational work of Gowers, as pursued by Green and Tao, over the integers, or vector spaces over finite fields, and analogous function field problems. Another is the pursuit of higher-degree analogues of Freiman’s theorem suggestive of alternatives to Gowers norms.

I am happy to discuss possible research projects by email — anything connected with the topics above (arithmetic harmonic analysis). See my web page for some surveys and other details of work with which I am currently involved: http://www.maths.bris.ac.uk/˜matdw/