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

Indiana University, Bloomington IN

Investigators

Abstract

It is proposed to deepen various connections among the mathematical areas of probability, group theory, graph theory, and ergodic theory. Mainly this will be achieved by probabilistic thinking about questions that arise in other areas. In group theory, the PI will work on the question of whether every group is sofic. The PI discovered with Aldous that a probabilistic setting leads to a wider framework for this question and suggests a new approach to it. In graph theory, probabilistic thinking leads to new questions and results involving inequalities for finite graphs. Namely, one often finds when counting combinatorial objects that a subgraph contains fewer of them than the whole graph. But if the subgraph is based on fewer vertices, then one ought to normalize the counts to reflect this. The PI has some partial results in this direction and proposes to find more. In ergodic theory, questions involve graphings, colorings, and factors. In fact, it may be that these investigations will lead to progress in the theory of percolation on groups. Combining several of these areas are very natural processes that are related (by previous work of the PI) to algebraic invariants known as ell-2-Betti numbers. Therefore, they suggest ways of resolving an important open question about these Betti numbers. In the 19th century, Cayley introduced graphs (networks) 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. 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 proposes to continue work on this question. Inequalities are important in most areas of mathematics. The field of graph theory and combinatorics contains many inequalities, often of the form that certain graphs contain the most (or the fewest) possible objects of a certain type among all graphs in a given class. The PI will develop novel inequalities of this type, which are inspired by a probabilistic viewpoint. Topology is the study of the shape of things. One tool is to count the number of holes of various dimensions. But if the whole space of interest is infinite, then the number of holes is often either zero or infinity. It turns out that there is a more informative way to count holes, and previous work of the PI has shown how it is related to random combinatorial objects. Therefore, the PI will attempt to use his new random objects to resolve an important open question about this hole counting.