Geometry and Randomness: Counting, Partitions, Stochastics, Shape

几何和随机性：计数、分区、随机、形状

基本信息

批准号：
2005512
负责人：
Moon Duchin
金额：
$ 19.63万
依托单位：
Tufts University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005512&HistoricalAwards=false
关键词：
Geometry Randomness Counting Partitions Stochastics

项目摘要

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.

该项目支持将数学桥接到数据科学的研究计划。该项目中心开发的关于随机结构的发展，这些构造可以大规模照亮几何形状，并且可以在网络研究中有用。示例应用程序包括流行病学，社交网络和地理空间网络。这些应用的许多想法可以从无限群体的研究中挖掘出来。但是，几何群体理论中开发的大多数技术都是渐近的，需要传递到无限的极限。在实际应用中，必须拥有既不是无限的也不渐近的中等规模技术。本研究项目将重点放在四个领域，（1）计数和统计几何形状：研究地球测量的精确几何形状（例如，诸如理性生长，统计多重多和宏观的高音和宏观curvature）的精确几何形状。（2）Nilpotent几何形状：诸如3D海森伯格组之类的尼尔植物群在几何分析，谎言理论，甚至是基于亚riemannian几何形状的控制理论的一部分中具有核心意义。在1980年代，他们还通过格罗莫夫（Gromov）出色的多项式生长定理开辟了关于几何群体理论的新远景，该定理仍在探索以供见解。这里探索的询问的新方向将几何分析与组合群体理论一起。（3）Teichmuller几何形状和台球：从随机三角形的几何形状到符号动力学中的刚性定理，该提案描述了一个由平坦和多余的几何形状相互作用构建的主动研究程序。（4）图形分区上的马尔可夫链：我们如何从图形的平衡，连接的K分区中有效采样？那优先从具有短边界的分区进行采样呢？这是许多应用领域的基本兴趣的问题，并且非常适合在大型数据集上进行探索。 PI和他/她的合作者已经实施了重组（“ IMOM”）马尔可夫链和研究计划，以理解其动态和几何形状。这为Moduli Space的Ergodic理论，等级和组合模型提供了丰富的想法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来支持的。