Principal Investigator: Bo'az B. Klartag
This project explores several challenging directions in the study of high dimensional convex bodies. The unifying theme of the project is that high dimensionality, when viewed correctly, may entail simplicity and order, rather than enormous diversity which is impossible to control, the so-called ``curse of dimensionality''. In particular, the PI will address the fundamental questions regarding distribution of mass in high dimensions, the typical shape of lower-dimensional sections of a given convex set, and the interplay between convex-geometric inequalities and functional inequalities, mostly in the category of logarithmically concave functions. While classical convexity focuses on fine properties of bodies or shapes in a fixed dimension, in the asymptotic theory one views the dimension as being finite, yet tending to infinity. We aim at bridging between these two disciplines, both by applying classical tools to the study of asymptotic questions, and by drawing corollaries of a classical flavor from asymptotic results. These corollaries are often of an almost-isometric nature, not only isomorphic.
The study of high dimensional objects is common to various scientific disciplines, including statistical mechanics, the theory of machine learning and asymptotic combinatorics, to name a few. Connections between these areas and asymptotic convex geometry have already begun to emerge. A better understanding of the high dimensional phenomena from the geometric viewpoint has the potential to contribute to the development of the aforementioned fields and to others
