Rational points on algebraic varieties

数论的基本问题之一是描述 不定方程的一组有理或整数解， 是多个变量的多项式方程， 系数 我的研究项目调查了 丢番图方程组的有理解，在几个 方向 保罗·沃伊塔对什么样的 解丢番图方程应该有，基于几何 这些方程的解集的性质。 在我的未来 研究，我建议研究这些知识，以提高我的 以前的证明，他们的各种特殊情况下，并使用现有的 结果，以进一步了解丢番汀的解决方案 方程 我特别感兴趣的是有理点的分布 在K3表面上。 我已经证明了这方面的许多结果， 包括（与Logan和货车Luijk）证明， 在许多对角四次曲面上，它们在真实的和Zagliki中是稠密的 拓扑，以及著名的Batyrev-Manin猜想的证明， 是以沃伊塔的主要猜想为条件的 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.