Dynamical Methods in Counting Questions and Diophantine Approximation

计数问题的动力学方法和丢番图近似

• 批准号: 2055364
• 负责人: Osama Khalil
• 金额: $ 14.37万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2022-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055364&HistoricalAwards=false
• 关键词: Dynamical; Methods; Counting; Questions; Diophantine

Dynamical systems model such varied examples as planetary motion, spread of disease and the flow of electric currents in conductive material. The area of mathematical dynamical systems consists of the study of the evolution over time of a system under a transformation rule that governs the behavior of the system. Methods from the study of dynamical systems, when applied to systems of algebraic origin and admitting a lot of symmetries, shed light, perhaps strikingly, on some of the oldest and most well-studied questions in mathematics, namely solutions in whole numbers, or integers, of polynomial equations, and approximations of arbitrary real numbers by fractions. In turn, developments in number theory, driven by dynamical methods, have had interesting and surprising applications, reaching recently as far as impacting wireless communication technologies. This project aims to deepen these fruitful connections between dynamics and number theory by, in particular, developing new methods for counting the number of integer solutions to certain types of highly symmetric polynomial equations. An important component of this project is geared towards training graduate students in this area, at the intersection of dynamical systems and number theory, through research and professional mentoring. The goal of this project is fourfold: 1) develop methods in homogeneous dynamics to resolve outstanding questions regarding the distribution of rational points near manifolds and self-similar sets; 2) develop techniques in the theory of random walks and linear representations of algebraic groups aimed at studying counting problems of integral points on affine homogeneous varieties; 3) develop spectral tools for the study of the dynamics of the Kontsevich-Zorich cocycle over the Teichmueller geodesic flow, along with applications to rigidity problems for horocycle flows on moduli spaces of Abelian differentials; 4) develop new methods for the study of the mixing properties of the geodesic flow on infinite volume locally symmetric spaces of negative curvature.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.

动力系统对行星运动、疾病传播和导电材料中的电流流动等各种例子进行建模。数学动力系统领域包括在控制系统行为的变换规则下研究系统随时间的演化。动力系统研究的方法，当应用于代数起源的系统并承认大量对称性时，也许会令人惊讶地揭示数学中一些最古老和研究最深入的问题，即多项式方程的整数或整数的解，以及任意实数的分数逼近。反过来，由动力学方法驱动的数论的发展也产生了有趣且令人惊讶的应用，最近甚至对无线通信技术产生了影响。该项目旨在加深动力学和数论之间的富有成效的联系，特别是开发计算某些类型的高度对称多项式方程的整数解的数量的新方法。该项目的一个重要组成部分是通过研究和专业指导，在动力系统和数论的交叉领域培训研究生。 该项目的目标有四个：1）开发齐次动力学方法来解决有关流形和自相似集附近有理点分布的突出问题； 2）开发随机游走和代数群线性表示理论的技术，旨在研究仿射齐次簇上积分点的计数问题； 3) 开发谱工具，用于研究 Teichmueller 测地流上 Kontsevich-Zorich 余循环的动力学，以及在阿贝尔微分模空间上的 horocycle 流刚性问题的应用； 4) 开发研究无限体积负曲率局部对称空间上测地流混合特性的新方法。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。