Rational points on algebraic varieties

数论的基本问题之一是描述丢番图方程的有理或整数解的集合，丢番图方程是具有整数系数的多元多项式方程。 我的研究项目调查丢番图方程系统的有理解的分布，在几个方向上。Paul Vojta根据丢番图方程解集的几何性质，对丢番图方程应该有什么样的解作了广泛的讨论。 在我未来的研究中，我建议研究这些公式，改进我以前对它们的各种特殊情况的证明，并利用现有的结果来进一步了解丢番图方程的解。特别是，我对K3曲面上有理点 * 的分布感兴趣。 我已经证明了这个领域的许多结果，包括（与Logan和货车Luijk）证明了许多对角四次曲面上的有理点在真实的和Zenski拓扑中是稠密的，以及著名的Batyrev-Manin猜想的证明，该猜想是Vojta的主要猜想的条件。我与迈克尔·罗斯（Michael Roth）共同研究了两个具有有理坐标的点如何根据它们所在的物体的几何形状相互接近。甚至，我们还得到了一些结果，其中一个点不具有有理坐标，而是具有有理系数多项式的根。 事实证明，这是一项深入而有趣的工作，我将继续努力证明这一领域更有趣的结果。 *****

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. *****