Dynamical Methods in Counting Questions and Diophantine Approximation

动力系统模拟各种不同的例子，如行星运动、疾病传播和导电材料中的电流流动。数学动力系统的研究领域包括在支配系统行为的变换规则下研究系统随时间的演变。来自动力系统研究的方法，当应用于代数起源的系统并允许许多对称性时，可能引人注目地揭示了数学中一些最古老和最受研究的问题，即多项式方程的整数解，以及任意实数的分数逼近。反过来，在动力学方法的推动下，数论的发展产生了有趣和令人惊讶的应用，最近甚至影响到了无线通信技术。这个项目旨在深化动力学和数论之间的这些富有成效的联系，特别是通过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。这个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点上培养这一领域的研究生。这个项目的目标有四个方面：1)发展齐次动力学的方法来解决流形和自相似集附近有理点的分布问题；2)发展随机游动理论和代数群的线性表示的技术，目的是研究仿射齐次簇上的积分点的计数问题；3)发展谱工具来研究Teichmueller测地流上的Kontsevich-Zorich余循环的动力学，以及应用于阿贝尔微分模空间上的圆环流的刚性问题；4)发展研究无限体积局部对称负曲率空间上测地线流混合性质的新方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为是值得支持的。

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.