Research

I am interested in analytic number theory, especially the Hardy-Littlewood Circle Method, and its application to questions in Diophantine Geometry. The method naturally provides answers to local-global principles on varieties, such as the well known Hasse principle. In my thesis, I have been particularly interested in weak approximation which is a rather powerful (perhaps ironically named) local-global density property that the rational points on a variety often satisfy. A nice feature of my work is that we can work over a general number field rather than just Q as analytic number theorists tend to do.

I currently have two papers on weak approximation. The first proves it for smooth models of cubic hypersurfaces of large enough dimension. A novel feature of the proof is that it uses the circle method and the fibration method in tandem, perhaps the first time this has been done explicitly.

1. Weak approximation on cubic hypersurfaces of large dimension. Algebra and Number Theory (accepted).

The second is a note to extend a theorem of Heath-Brown and Skorobogatov to a more general setting using a number field version of the circle method (with some modifications).

2. A note on a theorem of Heath-Brown and Skorobogatov. Q.J.Math. (2012)

More recently I have been working on a related area, that of counting the number of rational points on variety of bounded height (with a suitable definition of the 'height' of a rational point. This is an extremely old question, though seems to be difficult, even for some of the simplest varieties that we know.