Convexity and Applications

Case Western Reserve University, Cleveland OH

Investigators

Abstract

Abstract Award: DMS 1207917, Principal Investigator: Elisabeth M. Werner These research projects are in asymptotic geometric analysis and affine convex geometry. A main emphasis of her research is on affine invariant and high dimensional objects and phenomena and on links with other areas of mathematics and mathematical sciences (probability, statistics, optimization, information theory and quantum information theory). Much of the research in convex geometry in recent years has been directed to the study of affine and high dimensional aspects of convex bodies. The PI and her collaborators carried out a systematic study of some of the most important affine invariant functionals on convex bodies, the p-affine surface areas. They obtained extensions of those invariants to all convex bodies and for all p and established their corresponding affine isoperimetric inequalities (which are stronger than their Euclidean counterparts). There are numerous applications of these works. We only mention the PI's work, with Reisner and Schuett, on Mahler's conjecture. While not providing the solution (yet), their result is the strongest indication to date that the minimum is indeed attained for polytopes. Very recent developments open totally new directions. For one, the PI found new affine invariants and proved that those and the p-affine surface areas are certain entropies from information theory. In another direction, the PI, Artstein, Klartag and Schuett showed that affine isoperimetric inequality for log concave functions corresponds to a reverse log Sobolev inequality for entropy. These directions will be explored further. Contributions are also her work linking quantum information theory and high dimensional convex geometry, culminating in Aubrun's, Szarek's and her analysis of the "Additivity conjecture for Quantum channels" via asymptotic geometric analysis. The PI will continue to exploit the unique perspective given by asymptotic geometric analysis for problems in quantum information theory. 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 you want 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 us to predict its future motion from position and velocity we would need six separate numbers in all. If you want to track 100 distinct molecules of the air in the room then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase, 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". To study such high dimensional phenomena, is one purpose of this grant.