This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This project is in the field of asymptotic geometric analysis, which deals with high-dimensional phenomena in convex geometry and functional analysis, and more specifically in applications of probability theory in this field. The project has two separate areas of emphasis, the spectral behavior of large random matrices and the distribution of volume in high-dimensional convex bodies. The proposed techniques for studying these problems include both traditional probabilistic tools of random matrix theory and asymptotic geometric analysis, like the method of moments and the concentration of measure phenomenon; as well as novel techniques from theoretical probability, like Stein's method. In the area of random matrices the proposer studies the fluctuations of norms and eigenvalues of random matrices. One main goal here is to sharpen and extend concentration results due to the proposer and others. The proposer will also continue his study of the behavior of large random Toeplitz matrices, a class of random matrices which has only recently been investigated in the literature, and related random matrix ensembles. In the area of convex geometry the proposer will continue his work on Gaussian approximation theorems for volumes of sections of high-dimensional convex bodies, as well as related problems about how the volume is distributed in high-dimensional convex bodies. An important feature of the problems considered here, which is typical of the field, is their intrinsically high-dimensional nature. For example, many quantitative geometric questions are trivial in a sufficiently low number of dimensions, but deep and unexpected phenomena can arise when the dimension becomes very large. In the context of random matrices, the primary interest is in results which are nontrivial for large finite matrices, as opposed to more-traditional limit results as the size becomes infinite. This high- but finite-dimensional aspect is crucial for potential applications to fields like geometry, statistics, or computer science.
High-dimensional phenomena arise whenever one studies quantitative problems involving a large number of parameters. Besides areas in pure mathematics like convex geometry and functional analysis which are the focus of this project, such problems naturally arise in fields as diverse as statistics, computer science, mathematical biology, and physics. The so-called curse of dimensionality is familiar in many quantitative fields, indicating the potential difficulty of dealing with such problems. The goal of asymptotic geometric analysis, on the other hand, is to identify regularity or patterns that arise in systems that depend on a very large number of parameters, but which are not apparent for a small number of parameters. Thus high-dimensionality in some ways becomes a blessing rather than a curse. Probability theory is a central tool in this field, both because many problems involve randomness in an explicit way, and because many a priori deterministic problems can be clarified by introducing a probabilistic viewpoint. That is, patterns that arise in high dimensions become apparent when one looks at things in an appropriately random way. This project deals with both these types of applications of probability to high-dimensional phenomena.
