Browsing by Author "Paouris, Grigorios"

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Concentration, Convexity and Structure
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/335; National Science Foundation
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.
Topology and Measure in Dynamics and Operator Algebras
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/336; National Science Foundation
Three of the basic ingredients in the structural foundations of modern analysis and its connections with theoretical physics are the concepts of measure, topology, and group. The first of these deals with the notions of volume and size, the second with proximity and convergence, and the third with symmetry and the idea of displacement in space or time. When combined together they form the subject of dynamical systems, which in its classical origins models the time evolution of physical systems but is nowadays applicable to a wide range of phenomena involving transformations from one state to another. The project will pursue novel relationships between measure and topology that have recently begun to emerge within this dynamical framework and are tightly linked to the related field of operator algebras. The goal is to use this perspective to develop new ways of understanding how the particular symmetries of a given dynamical system may condition different types of asymptotic behavior, from the deterministic to the chaotic. While the topological and measure-theoretic perspectives have long between fruitfully intertwined in the theory of operator algebras, in the last several years this symbiosis has been reinvigorated through not only the elaboration of surprisingly far-reaching analogies but also the discovery of new kinds of applications of von Neumann algebra techniques to C*-algebras with structurally profound consequences. The central aim of the project is to reimagine these analogies and techniques in dynamical terms. This will on the one hand forge a novel line of investigation in topological dynamics that intertwines finite approximation properties with asymptotic phenomena like mean dimension, and on the other establish broader connections between group actions and C*-algebras as part of the effort to understand general types of crossed products and their K-theoretic classifiability. 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.