Equidistribution in Arithmetic: Dynamics, Geometry and Spectra

算术中的均匀分布：动力学、几何和谱

• 批准号: 2302592
• 负责人: Ilya Khayutin
• 金额: $ 19万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302592&HistoricalAwards=false
• 关键词: Equidistribution; Arithmetic; Dynamics; Geometry; Spectra

A central problem in Number Theory is finding integer solutions to polynomial equations. Some systems of equations have plenty of integer solutions. This research concerns the distribution of the integer solutions inside the continuum of real solutions. A wide-range of conjectures imply that in many cases the distribution of integer solutions mimics a randomly generated set in the ambient space. Systems of equation that have a large enough group of symmetries are called homogeneous and they play a central role in this project. These symmetries can be used to relate different integer solutions to each other and facilitate the introduction of methods from the theory of dynamical systems to this research area. The dynamical methods have been immensely successful in solving long standing problems in number theory. The major objectives of this project are to make progress on long-standing problems about equidistribution in number theory, to expose undergraduate students to number theory and prepare them to graduate studies, and to prepare graduate students to conduct research at the intersection of number theory and dynamics.The focus of this project is the chaotic behavior of periodic subgroup orbits on homogeneous spaces, and the randomness exhibited in the asymptotic behavior of automorphic forms. Early breakthroughs include Linnik's results about the equidistribution of integral points on the sphere and Margulis's solution of the Oppenheim conjecture regarding the values attained by an irrational quadratic form at integer points. To make progress on these topics, this project will fuse and expand results from homogeneous dynamics and analytic number theory. Specifically, the proposed research relies on measure rigidity techniques from homogeneous dynamics, the study of relative trace formulae and the theta correspondence from the theory of automorphic forms, and sieve methods from multiplicative number theory.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.

数论中的一个中心问题是寻找多项式方程的整数解。一些方程组有大量的整数解。本文研究了整数解在实数解连续体中的分布问题。各种各样的猜测意味着，在许多情况下，整数解的分布模仿了环境空间中随机生成的集合。具有足够大的对称组的方程组被称为齐次方程组，它们在这个项目中起着核心作用。这些对称性可以用来将不同的整数解相互联系起来，并有助于将动力系统理论的方法引入到这一研究领域。动力学方法在解决数论中长期存在的问题上取得了巨大的成功。这个项目的主要目标是在数论中长期存在的均匀分布问题上取得进展，让本科生接触数论并为他们的研究生学习做准备，并为研究生在数论和动力学的交叉点上进行研究做准备。本项目的重点是齐次空间上周期子群轨道的混沌行为，以及自同构形式的渐近行为所表现出的随机性。早期的突破包括Linnik关于球面上积分点均匀分布的结果和Marguis关于Oppenheim猜想在整点上的无理二次型所获得的值的解。为了在这些主题上取得进展，这个项目将融合和扩展齐次动力学和解析数论的结果。具体地说，建议的研究依赖于齐次动力学中的测量刚性技术，自同构形式理论中的相对迹公式和theta对应的研究，以及乘数理论中的筛选方法。该奖项反映了NSF的法定使命，并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，被认为值得支持。