Number Theory, Geometry, and Dynamics

本项目旨在探讨分析、动力学、数论和群论之间的相互作用。一个典型的例子发生在圆和球填料的算术性质的研究中，这是一个丰富的研究领域，可以开发技术，然后应用于更广泛的背景。与这个项目的研究组成部分相结合的是一些教育和推广工作。PI在世界各地组织研讨会、讲习班和会议，与国家数学博物馆合作，并通过YouTube视频向更广泛的受众介绍高水平但通俗易懂的数学主题。在这个项目的第二年和第三年，PI将为新不伦瑞克地区的中学生和高中生带来罗格斯大学的MathCorps暑期项目。PI研究计划的一个主要组成部分是研究局部-全局原理，并将其大量应用于数学遥远分支中看似无关的问题。结果包括更好地理解无限体积双曲表面上的封闭测地线长度的渐近性质，平面上的台球轨迹，以及由群作用产生的填料中的圆和球的半径集。PI还将工作(1)分类哪些凸多面体，当几何化到有限协体积3折叠时，是算数的，(2)攻击齐次动力学和Diophantine近似中的一些问题，以及(3)为精益交互定理证明器的mathlib库做出贡献。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。

This project aims to explore interactions between analysis, dynamics, number theory, and group theory. A quintessential example of such occurs in the study of arithmetic properties of circle and sphere packings, a fertile area of research where techniques can be developed that then apply to much broader contexts. Integrated with the research components of this project are a number of educational and outreach endeavors. The PI is engaged in organizing seminars, workshops, and conferences around the world, collaborating with the National Museum of Mathematics, and reaching broader audiences with YouTube videos on high-level but accessible mathematics topics. In years 2 and 3 of this project, the PI will bring to Rutgers the MathCorps summer program for middle and high school students from the New Brunswick area.A major component of the PI's research program is the study of local-global principles in thin groups and their myriad applications to seemingly unrelated problems in distant branches of mathematics. Consequences include a better understanding of the asymptotic properties of closed geodesic lengths on infinite-volume hyperbolic surfaces, billiard trajectories on flat surfaces, and the sets of radii of circles and spheres in packings generated by group actions. The PI will also work to (1) classify which convex polyhedra, when geometrized to finite covolume 3-folds, are arithmetic, (2) attack a number of problems in homogeneous dynamics and Diophantine approximation, and (3) contribute to the Lean Interactive Theorem Prover's mathlib library.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.