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

Indiana University, Bloomington IN

Investigators

Abstract

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.