Rational points on algebraic varieties

数论的一个基本问题是描述丢芬图方程的****有理或整数解集，丢芬图方程****是具有整数****系数的多变量多项式方程。我的研究计划调查丢番图方程系统在若干****方向上的****有理解的分布。*******保罗·沃伊塔根据丢梵图方程解集的几何****性质，对哪些****解应该有一些广泛的猜想。在我未来的****研究中，我建议研究这些猜想，改进我****之前对它们的各种特殊情况的证明，并利用现有的****结果进一步深入了解丢芬图恩****方程的解。*******特别是，我对K3曲面上有理点****的分布很感兴趣。我已经证明了这个领域的许多结果，****包括（与Logan和van Luijk一起）证明了许多对角四次曲面上的有理点****在实拓扑和Zariski****拓扑中是密集的，以及著名的Batyrev-Manin猜想的证明，该猜想****以Vojta的主猜想为条件。*******我和迈克尔·罗斯合作，研究了两个有理性坐标的点能有多接近，根据点所在物体的几何形状。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，而是有多项式的根，有有理系数。这已被证明是一项深入而有趣的工作，我将继续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. *****