Dynamical Methods in Counting Questions and Diophantine Approximation

动力学系统模拟了各种各样的例子，如行星运动、疾病传播和导电材料中的电流流动。数学动力系统的领域包括研究系统在控制系统行为的变换规则下随时间的演化。动力系统的研究方法，当应用于代数起源的系统并承认许多对称性时，可能会引人注目地揭示数学中一些最古老和研究最充分的问题，即多项式方程的整数或整数解，以及任意真实的数的分数近似。反过来，由动力学方法驱动的数论的发展有着有趣和令人惊讶的应用，最近影响了无线通信技术。该项目旨在深化动力学和数论之间富有成效的联系，特别是通过开发新方法来计算某些类型的高度对称多项式方程的整数解的数量。该项目的一个重要组成部分是面向培养研究生在这一领域，在动力系统和数论的交叉点，通过研究和专业指导。 该项目的目标有四个方面：1）发展齐次动力学的方法，以解决有关流形和自相似集附近有理点分布的突出问题; 2）发展随机游动理论和代数群线性表示的技术，旨在研究仿射齐次簇上整点的计数问题; 3）发展了研究Teichmueller测地线流上Kontsevich-Zorich上圈动力学的谱工具，沿着应用于阿贝尔微分模空间上的单圈流的刚性问题;四、为研究负曲率的无限体积局部对称空间上测地线流的混合性质开发新方法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

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.