Away from Independence: Probability in Geometry, Topology, Number Theory and Mathematical Physics

Case Western Reserve University, Cleveland OH

Investigators

Abstract

The common thread running through the various sub-projects described here is to make use of geometric and functional-analytic perspectives in conjunction with tools of quantitative probabilistic analysis in order to investigate probabilistic phenomena in areas of mathematics in which independence is not a natural condition. The first sub-project is a continuation of the PI's ongoing investigation of the phenomenon of "typically Gaussian marginals'' of a large class of high-dimensional probability distribution. In the most recent part of this work, the PI found a sharp cut-off for the projection dimensions for which this phenomenon occurs for a very general class of distributions. One direction of proposed future work is to enrich this theory by investigating the effect of additional geometric conditions on the measure on the largest dimension onto which most projections look Gaussian. A second sub-project deals with the spectral measures of certain types of random matrices, and has connections to both geometry, mathematical physics and number theory. In very recent joint work with M. Meckes, the PI proved strong concentration results for the empirical spectral distributions of arbitrary powers of random unitary matrices. It is likely that this work will enable a detailed analysis of the asymptotic behavior of the maximal eigenvalue gap of random matrices from the compact classical groups. A third proposed project is a collaboration with P. Albin to investigate conditions on a compact manifold under which the eigenfunctions of its Laplacian have approximately Gaussian value distributions in the high-eigenvalue limit. Conjectures that such an approximation sometimes holds have mostly come out of quantum chaos, and some numerical evidence for these conjectures exists. Combining earlier work of the PI with techniques of pseudo-differential calculus appears promising. Other projects in early stages include a collaboration with C. Hughes to find new patterns in the zeroes of L-functions, inspired by a deep result of Rains on random unitary matrices; a joint project with K. Kirkpatrick to prove central limit theorems in many-body quantum dynamics using a new non-commutative version of Stein's method; and a joint project with V. de Silva to extend the PI's earlier work on limit theorems for Betti numbers of random simplicial complexes to the context of persistence homology. A main goal running through these projects is to move away from the problems that classical probability theory has traditionally focused on, so as to interface with a wide variety of other areas of mathematics. It seems that more and more fields, from Riemannian geometry to number theory to algorithm design and beyond, are finding that a probabilistic viewpoint can open doors to solutions to previously impenetrable problems, and suggest fruitful new lines of research that had not previously been considered. Central to such exploration is the role of independence. Classical probability theory focuses on situations in which independence plays a key role -- when one can describe multiple experiments and assume that their outcomes cannot affect each other, very precise analyses can be carried out in tremendous detail. However, probability arises in many fascinating and important ways in other areas of mathematics and science in which independence does not naturally occur, but where there is some other special structure. Consider, for example, a sensor network that needs a certain degree of connectivity in order to function properly. If sensors are dropped from an airplane over a region of wilderness, it is important to know how many are probably needed to gain enough connectivity for the network to function. A classical probabilistic approach might be to assume that each pair of sensors communicates with some prescribed probability, independently, but a more realistic approach is to incorporate spacial data and say that sensors communicate if they are close enough together. Understanding the likelihood that such a random network will function becomes a harder problem because of the lack of independence, but an ultimately more useful one. The goal of this project is to investigate situations like this, in which independence assumptions are gone but geometric or other considerations can be exploited instead.