Number Theory, Geometry, and Dynamics

数论、几何和动力学

• 批准号: 2302641
• 负责人: Alex Kontorovich
• 金额: $ 35万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302641&HistoricalAwards=false
• 关键词: Number; Theory; Geometry; Dynamics

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.

这个项目旨在探索分析，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研究的算术性质的圆和球包装，一个肥沃的研究领域，技术可以开发，然后适用于更广泛的背景。与该项目的研究组成部分相结合的是一些教育和推广工作。PI参与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视频接触更广泛的受众，这些视频涉及高水平但可访问的数学主题。在这个项目的第二年和第三年，PI将为来自新玩法地区的初中和高中学生带来罗格斯大学的数学军团暑期项目。PI研究计划的一个主要组成部分是研究瘦群体中的局部-全局原理及其在遥远的数学分支中看似无关的问题的无数应用。其结果包括更好地理解封闭测地线长度的渐近性质的无限体积双曲曲面，台球轨迹在平面上，和一套半径的圆和球的包装所产生的群体行动。PI还将致力于（1）分类哪些凸多面体，当几何化为有限的共体积3倍时，是算术的，（2）攻击齐次动力学和丢番图近似中的一些问题，以及（3）该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准。