Concentration, Convexity and Structure
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In various scientific disciplines such as mathematics, statistical mechanics, quantum information, and others, high-dimensional structures play a central role. It has been observed that these distinct areas share the common feature that basic probabilistic principles govern the underlying high-dimensional behavior. In most cases, efficient approximation and study is facilitated by (non-asymptotic) high-dimensional probability. The investigator intends to work on several questions related to the most widely applied principle in high-dimensional probability: the concentration of measure phenomenon. This principle is commonly the main reason behind the frequently-observed tendency of high-dimensional systems to congregate around typical forms. To quantify this phenomenon, one needs precise inequalities for high-dimensional objects (for instance, measures or random vectors), where independence properties can be lacking. The questions under study have a strong geometric component. Results of the study will have implications in disciplines that depend vitally on high-dimensional objects, including asymptotic geometric analysis, geometric probability, machine learning, sparse recovery, random matrices, and random polynomial theory. The main goal of the project is to find the quantities or to isolate characteristics of a function that govern its concentration (say with respect to the Gaussian measure); in particular, to determine the quantities that control small fluctuations (variance) and small ball probabilities. The project undertakes a systematic study of this problem and initiates some new methods to compute deviation inequalities (especially in the small ball regime). It is planned to test these methods on more general measures such as log-concave probability measures. The project will also investigate limit theorems for geometric quantities that complement concentration inequalities at the asymptotic level. This award reflects NSF''s statutory mission and has been deemed worthy of support through evaluation using the Foundation''s intellectual merit and broader impacts review criteria.