Rational points on algebraic varieties

数论的基本问题之一是描述 丢番图方程的有理解或整数解集，其中 是具有整数的多个变量的多项式方程 系数。 我的研究计划调查了 丢番图方程组的有理解，有几个 方向。 保罗·沃伊塔 (Paul Vojta) 对什么类型的问题做出了一些广泛的猜想 丢番图方程应该有的解，基于几何 这些方程的解集的性质。 在我的未来 研究，我建议研究这些猜想，以提高我的 以前对它们的各种特殊情况的证明，并使用现有的 结果以进一步了解丢番图的解决方案 方程。 我特别对有理点的分布感兴趣 在 K3 表面上。 我已经在这方面证明了很多成果， 包括（与 Logan 和 van Luijk 一起）证明有理点 许多对角四次曲面在实数和 Zariski 中都是稠密的 拓扑学，以及著名的巴特列夫-马宁猜想的证明 以 Vojta 的主要猜想为条件。 我与迈克尔·罗斯 (Michael Roth) 合作，根据点所在物体的几何形状，研究了具有有理坐标的两个点彼此之间的距离有多近。 更重要的是，我们获得了一些结果，其中一个点没有有理坐标，而是具有有理系数多项式的根坐标。 事实证明，这是一项深入而有趣的工作，我将继续努力证明该领域更有趣的结果。

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.