Rational points on algebraic varieties

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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. *****