CAREER: Mixing and Equidistribution in Number Theory and Geometry

动力学是研究系统在控制其行为的转换规则下随时间演变的学科。它包含了各种各样的例子，如行星运动、疾病的传播和导电材料中的电流流动。事实证明，数论和几何中的许多基本问题也可以用某些代数起源的动力系统的长期行为来理解。此外，这些系统的代数性质使得可以使用各种数学学科的工具来进行研究。本项目旨在发展代数动力系统理论的新方法，以解决丢芬图近似和曲面几何领域的核心问题。教育部分旨在培训早期职业数学家，并为各级学生提供指导。这包括一个讲习班，旨在培训研究生和博士后积极的研究方向动力学，以及外联讲习班，旨在鼓励来自代表性不足的背景的学生追求数学科学的职业生涯。该研究项目有三个相互关联的目标。一个目标是从齐次动力学的角度研究自相似集附近有理点的分布。PI还将研究阿贝尔微分模空间上环不变测度的极限。第三个目标是建立负弯曲流形上测地线流的混合率。在这些问题上的进展将涉及在表示理论、几何和加性组合学的交叉领域发展动态方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamics is the study of the evolution of a system under a transformation rule governing its behavior over time. It encompasses such varied examples as planetary motion, the spread of disease and the flow of electric currents in conductive material. It turns out that many fundamental problems in number theory and geometry can also be understood in terms of the long-time behavior of certain dynamical systems of algebraic origin. Furthermore, the algebraic nature of these systems makes it possible to employ tools from a wide array of mathematical disciplines for their investigation. This project aims to develop new methods in the theory of algebraic dynamical systems with the goal of resolving central questions in the fields of Diophantine approximation and surface geometry. The educational component aims at training early-career mathematicians and providing mentorship to students at all levels. This includes a workshop geared towards training graduate students and postdocs on active research directions in dynamics, as well as outreach workshops aimed at encouraging students from underrepresented backgrounds to pursue careers in the mathematical sciences.The research program has three interrelated goals. One goal is to study the distribution of rational points near self-similar sets from the perspective of homogeneous dynamics. The PI will also investigate limits of horocycle-invariant measures on moduli spaces of Abelian differentials, A third aim is to establish rates of mixing of geodesic flows on negatively curved manifolds. Progress on these questions will involve development of dynamical methods at the intersection of representation theory, geometry, and additive combinatorics.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.