Career: Geometry of Measures In High Dimensions
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The PI''s long term research goal is to investigate the properties of high dimensional probability measures and understand the geometry that lies underneath. In pursuit of this goal the PI plans to quantify the concentration phenomena that appears, by further investigating the structure and geometry of the family of centroid bodies that is generated by the measure. This leads to several research directions that explore the connections of the theory of high-dimensional measures with Convex geometry (and local theory of Banach spaces), classical Analysis (PDE''s and isoperimetric inequalities) and Probability theory (including generic chaining). The general principle in high dimensional systems is that they have the tendency to congregate around typical forms. For general measures this is a relatively new discovery. The proposal is dedicated to the investigation of these phenomena. This theory has already found significant applications in probability, combinatorics, mathematical physics and informatics, with the recurrent theme being that of extending known results to a much broader setting without any assumptions on "randomness". Any progress in the direction of this proposal will be of great importance to every field interested in high dimensional phenomena.