CAREER: High-Dimensional Geometry and Its Applications

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The project is in the area of High-dimensional Geometric Analysis, which comprises a family of relatively new and very active areas of mathematics, arising at the interface of Harmonic and Functional Analysis, Convex Geometry and Probability. The primary focus of these areas is the study of high-dimensionality; the consideration often focuses around geometric objects such as convex bodies and hypersurfaces, convex and concave functions, as well as random vectors with certain geometric characteristics. Our experience in low dimensions seems to suggest that when the dimension becomes very large, the geometric properties of objects become more and more complicated and difficult to study. However, many nice, and sometimes surprising, properties arise in high dimensions. Such properties are informally called ``high-dimensional phenomenon''. The study of this phenomenon has been crucial for many applications in computer science, in particular in questions regarding the speed of certain algorithms, as well as in data science. The educational component of this project focuses on supporting junior researchers, with the particular emphasis placed on encouraging female mathematicians. The principal investigator will organize two workshops for junior researchers, featuring research discussions during allocated time, and short lecture courses by leading experts in the field. These workshops are designed to help junior mathematicians to develop new interests and create new collaborations. In addition, a seminar for women in mathematics in Northern Georgia is run by the principal investigator jointly with Yulia Babenko from Kennesaw State University.The principal investigator has been working on several aspects of the geometry in high dimensions. An important direction of this project concerns the study of the inequalities of Brunn-Minkowski type. More specifically, the intriguing question is how those inequalities improve under certain symmetry and convexity assumptions. The techniques involved in studying such questions involve ideas from Harmonic Analysis and Convex Geometry. In addition, the principal investigator shall continue to study small-ball inequalities and their applications to Information theory. One of the important objects studied by the principal investigator in the past is the noise sensitivity of distributions with respect to convex sets, and the principal investigator shall continue to study this quantity and its relations to the central problems in the field. Finally, a different aspect of the project concerns combinatorial properties of convex sets, such as the illumination number. The principal investigator has studied this number in the past, and is working on improving current known estimates on this quantity.