Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

One of the fundamental problems of number theory is to describe the

set of rational or integer solutions to Diophantine equations, which

are polynomial equations in several variables with integer

coefficients. My research program investigates the distribution of

rational solutions to systems of Diophantine equations, in several

directions.

Paul Vojta has made some wide-ranging conjectures on what kinds of

solutions Diophantine equations should have, based on the geometric

properties of the solution sets of these equations. In my future

research, I propose to study these conjectures, to improve on my

previous proofs of various special cases of them, and to use existing

results to gain further insight into the solutions of Diophantine

equations.

In particular, I am interested in the distribution of rational points

on K3 surfaces. I have already proven many results in this area,

including (with Logan and van Luijk) a proof that the rational points

on many diagonal quartic surfaces are dense in the real and Zariski

topology, and a proof of the celebrated Batyrev-Manin Conjecture that

is conditional on Vojta's Main Conjecture.

I have, in joint work with Michael Roth, investigated how close two points with rational coordinates can get to one another, in terms of the geometry of the object the points lie on. Even more, we have obtained some results in which one of the points doesn't have rational coordinates, but instead has coordinates that are the roots of polynomials with rational coefficients. This has proven to be deep and interesting work, and I am continuing to work on proving more interesting results in this area.