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*是系数为*的多变量多项式方程。我的研究项目调查了丢番图方程组的*有理解在几个*方向上的分布。*Paul Vojta根据这些方程的解集的几何*性质，对丢番图方程应该有哪些*解做出了一些广泛的猜测。在我未来的*研究中，我建议研究这些猜想，改进我以前对这些猜想的各种特殊情况的证明，并利用已有的*结果进一步深入研究丢番图方程的解。我已经证明了这一领域的许多结果，包括(与Logan和van Luijk)证明了许多对角四次曲面上的有理点*在实拓扑和Zariski拓扑中是稠密的，以及著名的Batyrev-Manin猜想的证明，即*是以Vojta的主要猜想为条件的。*我与Michael Roth合作，研究了两个有理坐标的点如何接近另一个点，就点所在对象的几何而言。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. *****