Geometry and Randomness: Counting, Partitions, Stochastics, Shape

该项目支持一个将数学与数据科学联系起来的研究项目。在这个项目中开发的技术集中在随机结构上，这些结构可以大规模地阐明几何形状，并且可以在网络研究中发挥作用。示例应用包括流行病学、社会网络和地理空间网络。这些应用的许多想法可以从无限群的研究中挖掘出来；然而，在几何群论中发展的大多数技术是渐近的，并且需要通过无限极限。在实际应用中，必须有既不是无穷小也不是渐近的中等规模技术。这个研究项目将集中在4个方面：(1)计数和统计几何：研究测地线精确几何的方法开辟了理性增长、统计双曲和宏观里奇曲率等应用。(2)幂零几何：幂零群，如三维海森堡群，是几何分析、李理论甚至控制理论中以亚黎曼几何为中心的部分的中心兴趣。在20世纪80年代，他们还通过格罗莫夫非凡的多项式增长定理，开辟了几何群论的新视野，该定理仍在探索中。这里探索了新的研究方向，将几何分析与组合群论结合在一起。(3) Teichmuller几何和台球：从随机三角形几何到符号动力学中的刚性定理，提案描述了一个基于平面几何和双曲几何相互作用的积极研究计划。(4)图分区上的马尔可夫链：我们如何有效地从图的平衡的、连通的k分区中采样？那么从边界较短的分区中优先抽样呢？这是一个跨许多应用领域的基本问题，它很适合在大型数据集上进行探索。PI和他/她的合作者已经实现了一个重组（“ReCom”）马尔可夫链和研究计划，以了解其动力学和几何。这为模空间的遍历理论、等径和组合模型的思想提供了丰富的应用。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 supports a research program that bridges mathematics to data science. The techniques developed in this project center on randomized constructions that illuminate geometry at large scale, and which can be useful in the study of networks. Example applications include epidemiology, social networks, and geospatial networks. Many ideas for these applications can be mined from the study of infinite groups; however, most of the technology developed in geometric group theory is asymptotic, and requires passing to an infinite limit. In practical applications, it is essential to have medium-scale techniques that are neither infinitesimal nor asymptotic.This research project will focus on 4 areas, (1) Counting and statistical geometry: methods for studying the precise geometry of geodesics open up applications like rational growth, statistical hyperbolicity, and macro Ricci curvature. (2) Nilpotent geometry: nilpotent groups such as the 3D Heisenberg group are of central interest in geometric analysis, Lie theory, and even the part of control theory that centers on sub-Riemannian geometry. In the 1980s, they also opened up a new vista on geometric group theory, through Gromov's remarkable polynomial growth theorem, which is still being explored for insights. New directions of inquiry explored here bring the geometric analysis together with the combinatorial group theory. (3) Teichmuller geometry and billiards: From the geometry of random triangles to rigidity theorems in symbolic dynamics, the proposal describes an active research program built from an interplay of flat and hyperbolic geometry. (4) Markov chains on graph partitions: How can we efficiently sample from the balanced, connected k-partitions of a graph? And how about preferentially sampling from partitions with a short boundary? This is a question of fundamental interest across many application domains, and it lends itself well to exploration on large datasets. The PI and his/her collaborators have implemented a recombination ("ReCom") Markov chain and research program for understanding its dynamics and geometry. This provides a rich application for ideas from ergodic theory, isoperimetry, and combinatorial models for moduli space.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.