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High Dimensional Phenomena
High dimensional phenomena occur in several areas of pure and applied mathematics such as in classical and free probability, graph theory, convex geometry, functional analysis, and recently data science. For instance, in random matrix theory, one of the aims is to understand the behavior of the spectrum as the dimension grows. It turns out that there usually is a universal phenomenon happening: the limiting spectral distribution does not depend on the distribution of the matrix entries.
Free probability theory plays a big role in uncovering those universal limits and unraveling some of their properties. In graph theory, understanding the structure of the graph as the number of vertices grows is of crucial importance. This can be studied by looking at several interesting quantities, such as the edge expansion constant, clustering coefficients, or also the spectrum of the adjacency matrix.
In the study of Markov chains, one aims to quantify the speed of convergence of the chain to its equilibrium measure, establishing tight bounds on the mixing time of the chain. In convex geometry, one of the celebrated high dimensional phenomena is that convex bodies contain a large Euclidien sub-structure. At the heart of all the above is a deep concept known as the concentration of measure phenomenon: functions of many independent random variables which only exhibit local variations are almost constant with high probability. Establishing such concentration inequalities in various contexts, intimately related to functional inequalities, reveals several interesting high dimensional features in those different topics.
Abu Dhabi Stochastics Seminar Series
This seminar series, organized by Pierre Youssef, brings together researchers from NYU Abu Dhabi and Sorbonne University Abu Dhabi. It covers a wide range of topics in pure and applied probability as well as mathematical physics and related areas, such as statistics and machine learning.