The Team

My primary interest is rational points on algebraic varieties, particularly their existence and distribution. I have worked extensively on failures of the Hasse principle and Manin's conjecture on rational points of bounded height. I'm very excited by the recent discovery that many problems in arithmetic statistics can be viewed as studying rational points of bounded height on algebraic stacks, most notably Malle's conjecture on number fields of bounded discriminant. I will bring my expertise in analytic number theory and algebraic geometry to build on this new perspective in the programme grant.

My interests include the Cohen—Lenstra heuristics and other aspects of arithmetic statistics, the arithmetic of elliptic curves and higher dimensional abelian varieties, classical and integral representation theory of finite groups, and topics around the question "can you hear the shape of a drum". I particularly enjoy discovering and exploring the many connections between these different areas.

I am interested broadly in topics in arithmetic geometry and arithmetic statistics. More specifically, my interests include rational points on varieties, the Hasse principle and the statistical behaviour of Selmer groups in families. Much of my recent work involves applying new results on Selmer groups to the study of the Hasse principle for Kummer surfaces and del Pezzo surfaces, via the descent-fibration method pioneered by Swinnerton-Dyer. I am also interested in the parity conjecture for abelian varieties (a mod 2 version of the Birch and Swinnerton-Dyer conjecture) and have previously worked on the explicit construction of models of curves over local fields. 

My primary interests are rational points on algebraic varieties, local-global principles (such as the Hasse principle and weak approximation) and obstructions to such principles, especially Brauer–Manin obstructions. I have also worked on local-global principles for zero-cycles, which generalise rational points. My main contributions in arithmetic statistics concern counting failures of the Hasse norm principle in families of global fields. 

I am currently a senior research fellow at the University of Bristol, moving to a prize fellowship at the University of Bath this autumn. My research spans arithmetic statistics and arithmetic geometry, including number field counting, the statistical behaviour of Selmer groups of elliptic curves, class groups, and similar objects, as well as the existence of rational points and zero cycles in families of varieties.  My work often involves combinations of algebraic, analytic, and computational techniques, and often involves Selmer groups of Galois modules.