Geometry of Nonhomogeneous Random Matrices, Vectors, and Processes

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High-dimensional random structures arise throughout mathematics and its applications. Many classical problems of probability theory are devoted to understanding the patterns that are generated by random noise in high-dimensional systems. For example, suppose one has a large matrix each of whose entries is the outcome of an independent fair coin flip, i.e., the matrix is filled with noise. What do the eigenvalues of such a matrix look like? Such questions have beautiful answers that are now well understood. There are however many situations, both in pure and in applied mathematics, in which there is nontrivial structure underneath the noise. For example, in the random matrix problem, we could allow each entry to have an arbitrary variance that reflects the intensity of the corresponding variable. The challenge in such problems, which are much less well understood than their classical counterparts, is to capture the interplay between the structure and the noise. The aim of this project is to develop mathematical techniques that make it possible to address such questions in considerable generality. Beside their fundamental interest, such techniques are expected to be applicable to areas such as statistics and computer science where high-dimensional random structures play an important role. Several education and outreach activities further form a key part of this project.The overarching goal of this project is to develop new mechanisms and techniques in high-dimensional probability to address problems that could be broadly characterized as having nonhomogeneous structure. The fundamental question in such problems is: how is the underlying geometric structure reflected in the probabilistic behavior of the model? Such questions are of significant importance in various pure and applied mathematical problems. This project aims to systematically investigate such problems that arise in three interrelated themes: (i) the norms of nonhomogeneous random matrices (for example, with arbitrary variance or sparsity pattern); (ii) the geometry of nonhomogeneous random processes (such as those that arise, for example, in random matrix theory or in functional analysis); (iii) geometric inequalities and dimension-free phenomena (for example, in connection with measure concentration or convexity). The project will leverage recent advances on these topics to push forward the state-of-the-art in this area of probability and its applications.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.