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

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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.