High-dimensional Phenomena in Asymptotic Geometric Analysis and Finite-dimensional Random Matrices

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The proposed research program is in the area of Pure Mathematics (Modern Analysis) called Asymptotic Geometric Analysis (AGA, in short). This is a fast growing interdisciplinary area at the "crossroads" of analysis, convex and discrete geometry, several areas of probability. While AGA takes roots in classical functional analysis, it is based on a new asymptotic point of view: properties of interest appear in quantitative forms that depend on the underlying finite dimension or a finite number of other relevant parameters. Combining them with structural or geometric hypotheses specific for the problems on hand shows the tendency of high dimensional systems to gather around a "typical" behaviour, and expresses this in a quantitative manner. High-dimensional systems are very frequent in mathematics and applied sciences hence understanding high-dimensional phenomena is becoming increasingly important.