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

该奖项支持首席研究员深化概率、群论、分析和遍历理论等数学领域之间的各种联系的研究。所有这些领域都为许多科学和技术提供了基础。公众熟悉日常生活中的概率，但往往没有意识到它在当今经济中的重要性，或者在常见智能手机应用程序中的今天计算机算法中的重要性。群论研究对称性，是现代物理学的主要理论基础。分析始于微积分，发明用于研究运动物体，现在用于整个科学和工程领域。遍历理论是这些数学分支中最不为人所知的；它始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。例如，在19世纪，Cayley引入了图来表示称为群的代数对象。人们总是希望对无限的对象有有限的近似，对无限群也是如此。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，至少对那些被称为“Sofic”的群体是如此。目前尚不清楚这种方法在多大程度上奏效。PI和Aldous一起发现，概率设置导致了这个问题的更广泛的框架，并建议了一种新的方法。如果一个人真的能够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的一系列重要猜想。国际和平研究所将继续就这一根本问题开展工作。该项目最重要的更广泛的影响之一是通过培训印第安纳大学的本科生和研究生来加强STEM教育，他们将从与该项目最高水平的尖端数学主题的密切合作中受益。相反，PI将与一群对他的研究议程有重要贡献的有才华的学生合作，以了解有限群近似，特别是在分析Cayley图上的概率对象的行为方面。在对Cayley图上的概率对象的分析中，PI将与他的研究生一起研究的一些主题包括一类随机的点过程，称为行列式。其中一些是通过正交投影从希尔伯特空间中产生的。由此产生的过程似乎提供了随机生成集。特别有趣的应用是复杂的分析。无限离散地面集的类比是早先由PI建立的。有人建议在总体上建立相同的联系，这将建立PI和Peres的猜想。还建议更好地了解当两个行列式概率度量的生成矩阵接近时，它们的接近程度。在有限情况下，这样的结果很可能也推广到无限情况，至少在上面第一个问题中研究的无限“SOFIC”情况下是这样。PI将与他的研究生合作的其他问题包括一维格子和圈以外的图上的连续时间无碰撞随机游动、首次通过渗流和Galton-Watson树上的随机游动。

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.