Convexity and Applications

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AbstractAward: DMS 1504701, Principal Investigator: Elisabeth M. Werner These research projects are in asymptotic geometric analysis and affine convex geometry. One main emphasis of the principal investigator's research is on high-dimensional objects and phenomena. This leads to applications of her research in areas as diverse as physics, biology and medicine, computer science, optimization and economics, and material science: Indeed, a mathematical description of a scientific or engineering question often requires lots of independent numbers, leading to a geometric space of high dimension. For example, if one wants to specify the location of one gas molecule in a room then you need to report the front/back, side-to-side, and up/down locations of the molecule, using three numbers. The direction and speed of the molecule's motion takes another three numbers, and so to describe enough of the molecule's current state to allow one to predict its future motion from position and velocity one would need six separate numbers in all. If one wants to track 100 distinct molecules of the air in the room then one will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase then the difficulty of sampling and computation go up rapidly, a phenomenon scientists and mathematicians sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases which are not visible in low dimensions. We can exploit those patters thus taking advantage of the "curse of dimensionality" to make it the "blessing of dimensionality". It is one purpose of this grant to study such high dimensional phenomena.Important features of the proposed project are, on the one hand, emphasis on affine invariant and high dimensional objects and phenomena, and, on the other hand, links with other areas of mathematics and mathematical sciences (such as probability, statistics, information theory and quantum information theory). Topics to be studied include: Entropies for convex bodies and log concave functions, affine invariants and their inequalities, structural properties of convex bodies, probabilistic methods in convexity and geometric aspects related to optimization, Banach spaces, and quantum information theory. A focus of this research program is the development of affine invariants and their related inequalities. The principal investigator and her collaborators started the systematic study of (affine invariant) functionals associated with convex bodies and log concave functions and their corresponding inequalities. Among the most important such functionals for convex bodies are affine surface area and p-affine surface area. Those were recently extended to log concave functions by the PI and her collaborators. The affine isoperimetric inequalities related to the affine surface areas are more powerful than their Euclidean relatives and related to other important inequalities (e.g., the Santalo- and Inverse Santalo-inequalities). The latter is related to Mahler's conjecture, which is still open in dimension three and higher. Connections, observed recently by the PI, between affine surface areas and Renyi entropy allow interactions between information theory, and convex affine geometry.