Probabilistic Models Tied to Group Theory, Analysis, and Ergodic Theory

根据该奖项开展的研究将加深概率、群论、分析和遍历理论等数学领域之间的各种联系。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。分析与微积分相似，微积分是为了研究运动物体而发明的，现在在整个科学和工程中都在使用。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。作为要研究的这些联系的一个例子，我们回忆起在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。国际刑警组织将继续就这一问题开展工作。研究生将通过与这个项目相关的研究进行培训。其他方向涉及一类随机的点过程，称为行列式。这些过程最初在物理学中被考虑，然后在概率论中发现了很大的用处，最近在计算机科学中引起了兴趣(为了从大数据集中找到具有代表性的、多样化的样本)。这个项目的目标之一是更好地了解当两个行列式概率度量的生成矩阵接近时，它们有多接近。在有限的情况下，这样的结果很可能也推广到无限的情况，至少在无限的“SOFIC”情况下。另一方面，无论是离散时间还是连续时间，随机游动都是概率论的一个基本研究对象。PI发现了连续时间随机游动的许多非常令人费解的行为，特别是在Cayley图上。人们建议确定这种令人费解的行为是否受限于它能在多大程度上与直觉相矛盾。作为最后一个例子，“因素”的概念是遍历理论的基础。当行动小组服从时，因子图很容易理解，但当小组更大时就不是这样了。然而，树上的因素在组合学和计算机科学中也很重要。建议更好地了解哪些地图是IID(独立且同分布)过程的因素，哪些不是。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The research carried out under this award will 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 is familiar 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 of these connections to be studied, we recall that 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 his work on this question. Graduate students will be trained through research related to this project.Other directions concern a class of random processes of points, known as determinantal. These processes were first considered in physics, then found much use in probability theory, and recently have become of interest in computer science (in order to find representative, diverse samples from large data sets). One of the goals of this project is 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. In another direction, random walks are a basic object of study in probability, whether in discrete time or continuous time. The PI has discovered much very puzzling behavior of continuous-time random walks, especially on Cayley graphs. It is proposed to determine whether such puzzling behavior is limited in how much it can contradict intuition. As a final example, the notion of "factor" is basic to ergodic theory. Factors maps are very well understood when the acting group is amenable, but not when the group is larger. Yet factors on trees are also important in combinatorics and computer science. It is proposed to understand better which maps are factors of IID (independent and identically distributed) processes and which are not.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.