Interactions Among Probability, Group Theory, Analysis, and Ergodic Theory

概率、群论、分析和遍历理论之间的相互作用

• 批准号: 1612363
• 负责人: Russell Lyons
• 金额: $ 15万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612363&HistoricalAwards=false
• 关键词: Interactions; Among; Probability; Group; Theory

This award supports the principal investigator's research to deepen various connections among the mathematical areas of probability, group theory, analysis, and ergodic theory. All these areas undergird much of science and technology. The public is familiar with probability from everyday life, but often is not aware of how crucial it is in today's economy, for example, or in today's computer algorithms in common smartphone apps. Group theory studies symmetries and lies behind much of modern physics. Analysis started from calculus, invented to study moving bodies and now used throughout science and engineering. Ergodic theory is the least known of these branches of mathematics; it began in physics with the study of systems of many particles, such as gases. It now provides a unifying framework to study many disparate questions, including some in computer science. As one example, in the 19th century, Cayley introduced graphs to represent the algebraic objects known as groups. It is always desirable to have finite approximations to infinite objects, and the same holds for infinite groups. Gromov and Weiss suggested a way to use finite networks for this purpose, at least for those groups known as ``sofic". It is not known how widely this approach works. The PI discovered with Aldous that a probabilistic setting leads to a wider framework for this question and suggests a new approach to it. If one can actually succeed in making such approximations for all groups, then this would resolve a host of important conjectures in a variety of fields of mathematics. The PI will continue work on this fundamental question. One of the project's most important broader impacts is on the strengthening of STEM education, by training of undergraduate and graduate students at Indiana University, who will profit form working closely with the PI on the project's cutting-edge mathematical topics of the highest caliber. Conversely, the PI will work with a talented cohort of students who will contribute significantly to his research agenda to understand finite group approximations, particularly in analyzing the behavior of probabilistic objects on Cayley graphs.Among the analysis of probabilistic objects on Cayley graphs, some of the topics which the PI will investigate with his graduate students include a class of random processes of points, known as determinantal. Certain of these arise from Hilbert spaces via orthogonal projections. The resulting processes seem to provide random spanning sets. Especially interesting applications are in complex analysis. The analogue for an infinite discrete ground set was established earlier by the PI. It is proposed to make the same connection in general, which would establish a conjecture of the PI and Peres. It is also proposed to understand better how close two determinantal probability measures are when their generating matrices are close. Such a result in the finite case is very likely to extend to the infinite case as well, at least in the infinite "sofic" situation studied in the first problem above. Other problems on which the PI will collaborate with his graduate students include continuous-time non-colliding random walks on graphs beyond the one-dimensional lattice and cycle, first passage percolation, and random walks on Galton-Watson trees.

该奖项支持首席研究员的研究，以加深概率，群论，分析和遍历理论的数学领域之间的各种联系。 所有这些领域都是科学技术的基础。公众熟悉日常生活中的概率，但往往不知道它在当今经济中的重要性，例如，或者在今天常见的智能手机应用程序中的计算机算法中。群论研究对称性，是现代物理学的基础。分析从微积分开始，发明来研究运动的物体，现在用于整个科学和工程。遍历理论是这些数学分支中最不为人所知的;它起源于物理学中对许多粒子系统的研究，例如气体。它现在提供了一个统一的框架来研究许多不同的问题，包括计算机科学中的一些问题。作为一个例子，在19世纪，凯莱介绍了图来表示代数对象被称为团体。 对于无限对象总是希望有有限的近似，对于无限群也是如此。 格罗莫夫和韦斯提出了一种使用有限网络来实现这一目的的方法，至少对于那些被称为“sofic”的群体来说是这样。目前尚不清楚这种方法的适用范围有多广。 PI和Aldous一起发现，概率的设置为这个问题提供了一个更广泛的框架，并提出了一种新的解决方法，如果人们真的能成功地对所有群体进行这样的近似，那么这将解决各种数学领域中的许多重要问题。 PI将继续就这一基本问题开展工作。该项目最重要的广泛影响之一是通过培训印第安纳州大学的本科生和研究生来加强STEM教育，他们将从与PI密切合作中获益，以研究该项目最高水平的尖端数学主题。相反，PI将与一群有才华的学生一起工作，这些学生将对他的研究议程做出重大贡献，以理解有限群近似，特别是在分析Cayley图上概率对象的行为方面。在Cayley图上概率对象的分析中，PI将与他的研究生一起研究的一些主题包括一类点的随机过程，称为行列式。 其中某些是从希尔伯特空间通过正交投影产生的。 由此产生的过程似乎提供了随机生成集。特别有趣的应用是在复杂的分析。对于无限离散基集的模拟是由PI早先建立的。有人建议作出同样的连接一般，这将建立一个猜想的PI和佩雷斯。 它还建议更好地理解如何接近两个行列式概率措施时，他们的生成矩阵是接近的。有限情形下的这种结果很可能也扩展到无限情形，至少在上面第一个问题中研究的无限“sofic”情形中是这样。PI将与他的研究生合作的其他问题包括在一维晶格和循环之外的图上的连续时间非碰撞随机行走，第一次通过渗流和Galton-Watson树上的随机行走。