Rational points on algebraic varieties

数论的一个基本问题是描述 丢番图方程的有理或整数解的集合，它 是具有整数的多变量多项式方程 系数。我的研究项目调查了 丢番图方程组的有理解 方向。 保罗·沃伊塔对哪些类型的 丢番图方程的解应该有，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，以改进我的 前人对它们的各种特殊情况的证明，并使用现有的 结果进一步深入了解丢番图的解决方案 方程式。 特别是，我对有理点的分布很感兴趣 在K3曲面上。我已经在这方面证明了很多成果， 包括(与Logan和van Luijk一起)证明有理性点 许多对角线上的四次曲面在实数和Zariski中是稠密的 拓扑学，并证明著名的Batyrev-Manin猜想 是以沃伊塔的主要猜测为条件的。 我与迈克尔·罗斯合作，研究了两个有理坐标的点之间的距离，就点所在物体的几何形状而言。更重要的是，我们得到了一些结果，其中一个点没有有理坐标，而是有一个坐标是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，我正在继续努力，在这一领域证明更多有趣的结果。

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.