The proposed research covers core topics in probability, convex geometry and analysis. Convex geometry is the bedrock for studying isoperimetric principles, laws that govern fundamental relationships between shapes and their size. The most famous such example is the classical isoperimetric inequality asserting that among all shapes of a given perimeter, circles enclose the largest area. Such principles underlie a wealth of extremal problems, e.g., in mathematical physics, information theory, optimization, among others. The PI and his coauthors have shown that global geometric features can be consequences of local random structure; his probabilistic tools reveal new and more quantitative information than is apparent from the standard principles. Conversely, the geometry of the underlying shapes can have implications in probability. This arises, for example, by replacing independence conditions by broader dependence structures, thereby making probabilistic results more broadly applicable. A major goal of the research is to use geometric considerations such as the presence of many symmetries as a guide for the replacement of independence. In a closely related direction, in applied sciences the "curse of dimensionality" refers to the notion that increasing a system's dimension comes with an unwieldy increase in complexity. On the other hand, a distinguishing feature of modern probability and convex geometry is that increasing the dimension brings unexpected benefits: patterns must arise simply by virtue of high-dimensionality. Mathematically, this is referred to as the "concentration of measure phenomenon" and it is fundamental in dealing effectively with large data sets, compression of signals, reducing complexity of algorithms, to name a few. The PI and co-PI will develop new tools to give considerably more accurate information on refined asymptotic scales, which are especially needed for applications. Here, as above, isoperimetric principles guide the development of the theory. The PI and co-PI will teach graduate courses on these topics, which will serve as excellent venues for engaging students in current research. Such courses may be of interest to students in computer science, statistics, or engineering whose research depends vitally on mathematics.
The project centers on concentration properties of high-dimensional probability laws, particularly for marginal laws due to their connection to small deviation inequalities and non-asymptotic random matrix theory. Using affine isoperimetric principles, the PIs will investigate criteria for well-boundedness of marginal distributions, especially under non-independence regularity assumptions such as affine invariance properties. They will also study concentration properties of norms on high-dimensional Euclidean spaces, building on the co-PI's refinements of Milman's random version of Dvoretzky's theorem for some classical normed spaces, circumvent the standard approach via Lipschitz constants by using super-concentration techniques and other refined tools. The PI and co-PI will extend the study of concentration of functionals to the multi-dimensional setting of Grassmannian manifold of linear subspaces of Euclidean space. This is a natural unified setting for the various problems above: marginals of probability distributions, the asymptotic theory of convex bodies, stochastic geometry and randomized isoperimetric inequalities. Consequently, a better understanding of the associated randomness on the Grassmannian will have diverse applications.
