Diophantine Approximation to Closed Subschemes and Integral Points on Varieties

该项目研究的主题是算术和数论的核心，主要集中在丢番图近似的主题上。丢番图逼近的核心是研究接近给定真实的数的有理数。这个话题有着古老的历史，至少可以追溯到π的第一个有理近似，在现代已经导致了数论和整个数学中的几个深入应用。PI将研究几个不等式的推广和改进，特别关注施密特的子空间定理及其与封闭子方案几何的关系。PI将继续应用于该主题的中心猜想，Vojta猜想，以及与最近涉及最大公约数的不等式的联系。在一个相关的方向，PI将探索有效地确定多项式方程组的整数解集的经典和基本问题，重点是技术开发和理解较少的高维问题。这些项目对数论以外的不同数学领域有着额外的密切联系和影响，包括几何和复分析。 该项目将支持范围广泛的指导活动和研究机会，包括本科生、研究生和博士后研究人员的培训。特别是，PI计划继续创建和监督高中和本科生的研究项目，从PI的研究计划得出。 丢番图近似研究的最新路线涉及高度不平等的研究与封闭的子计划，而不是经典的设置与除数的高度。PI计划发展这一理论的丢番图逼近封闭的子计划，并探讨应用该理论的整点品种。在一个方向上，PI将研究封闭子格式的施密特子空间定理的改进和改进，包括对m-次一般位置设置的扩展和Nochka-Ru-Wong定理的推广。在另一个方向上，PI将研究和开发最近涉及最大公约数的不等式及其与Vojta猜想的联系，并开发函数场模拟和应用。最后一组项目集中在发现新的丢番图近似不等式在品种上的整点的应用，包括开发研究整点的有效方法，特别是在高维品种上。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The project studies topics central to arithmetic and number theory, with a primary focus on the subject of Diophantine approximation. At its core, Diophantine approximation consists of the study of rational numbers which closely approximate a given real number. This topic has an ancient history, going back to at least the first rational approximations for pi, and in modern times has led to several deep applications in number theory and throughout mathematics. The PI will study generalizations and improvements of several inequalities in the subject, with a particular focus on Schmidt’s Subspace Theorem and its relation to the geometry of closed subschemes. The PI will pursue applications to a central conjecture in the subject, Vojta’s conjecture, as well as connections to recent inequalities involving greatest common divisors. In a related direction, the PI will explore the classical and fundamental problem of effectively determining the set of integer solutions to a system of polynomial equations, with an emphasis on higher-dimensional problems where the techniques are less developed and understood. The projects have additional close connections and consequences for diverse areas of mathematics beyond number theory, including geometry and complex analysis. The project will support a wide range of mentoring activities and research opportunities, involving the training of undergraduate students, graduate students, and postdoctoral researchers. In particular, the PI plans to continue creating and supervising high school and undergraduate research projects, drawn from the PI’s research program. A recent line of research in Diophantine approximation studies inequalities involving heights associated to closed subschemes, as opposed to the classical setting of heights associated to divisors. The PI plans to develop this theory of Diophantine approximation to closed subschemes, and to explore applications of the theory to integral points on varieties. In one direction, the PI will study refinements and improvements of the Schmidt Subspace Theorem for closed subschemes, including extensions to the setting of m-subgeneral position and generalizations of the Nochka-Ru-Wong theorem. In another direction, the PI will study and develop recent inequalities involving greatest common divisors and their connections with Vojta’s conjecture, and develop function field analogues and applications. A last set of projects are centered on discovering applications of the new Diophantine approximation inequalities to integral points on varieties, including developing effective methods for studying integral points, particularly on higher-dimensional varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.