Diophantine Approximation to Closed Subschemes and Integral Points on Varieties

该项目研究算术和数论的核心主题，主要关注丢番图近似主题。丢番图逼近的核心是研究非常接近给定实数的有理数。这个话题有一个古老的历史，至少可以追溯到圆周率的第一个有理近似，在现代导致了数论和整个数学中的几个深度应用。PI将研究这门学科中几个不等式的推广和改进，特别关注施密特子空间定理及其与闭子方案几何的关系。PI将继续应用于该主题中的一个中心猜想--Vojta猜想，以及与涉及最大公约数的最近的不平等的联系。在一个相关的方向上，PI将探索有效确定多项式方程组的整数解集合的经典和基本问题，重点放在技术发展和理解较少的高维问题上。除了数论之外，这些项目还对数学的不同领域产生了额外的密切联系和影响，包括几何和复杂分析。该项目将支持广泛的指导活动和研究机会，涉及本科生、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，这些项目来自PI的研究计划。丢番图近似最近的一系列研究涉及与闭合子格式相关的高度，而不是与除数相关的高度的经典设置。PI计划将这一丢番图逼近理论发展到闭子格式，并探索该理论在簇上的积分点上的应用。在一个方向上，PI将研究闭子方案的Schmidt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将研究和发展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变元上的积分点上的应用，包括开发研究积分点的有效方法，特别是在高维变分上。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.