The PI's research is in the two closely related areas of asymptotic geometric analysis and convex geometry, and their applications. Classical convexity studies the geometry of convex bodies in Euclidean space of a fixed dimension. Asymptotic geometric analysis deals with geometric properties of finite dimensional convex bodies as the dimension grows to infinity. In her research the PI uses methods from both areas, as well as probabilistic tools and concentration phenomena, to get a better understanding of the structure of convex sets. The PI was able to prove a variety of results where such structural aspects of convex sets play a role: in approximation of convex bodies by polytopes; to establish a link between the so called order statistics (which are fundamental objects in statistics) and Orlicz norms; to determine the ``sizes" of certain (convex) sets that appear naturally in quantum information theory. A further focus of the PI's research is the development of affine invariants. The PI and her collaborators started the systematic study of (affine invariant) functionals associated with convex bodies and their corresponding inequalities. Among the most important such functionals are affine surface area and p-affine surface area . The affine isoperimetric inequalities related to them 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 3 and higher.
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 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.
PROJECTS OUTCOME REPORT Dr. Elisabeth Werner During the funding period of this proposal, Dr. Werner has worked on problems raised in the proposal and has either solved them or, at least, made significant progress in all of them. More than thirteen papers were published or have been accepted for publication as the outcome of the research undertaken. Research Activities: an overview. The PI's research is in high dimensional convex geometry. We want to mention just a few typical issues arising in convex geometry and point out their relevance for other areas. A classical problem in convexity is to reconstruct a convex shape from its sections. Research carried out there finds applications in medicine and biology where convex shapes occur naturally (e.g., organs). The understanding of surface structure of convex bodies finds applications in chemistry as this is relevant for arrangements of, e.g., molecules and crystals. Methods from convexity theory are essential in some areas of computer science, for instance in the development of geometric algorithms and in quantum computing (see more below). The study of convex shapes was one of the goals of this grant. High dimensional geometry deals with geometric properties of finite dimensional spaces as the dimension or other relevant free parameters increase to infinity. This is important as 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. As these dimensions increase, the difficulty computation go up rapidly, a phenomenon scientists and mathematicians call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases, 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". This was another purpose of the grant. Research Activities: details. During the proposal period, the PI and her collaborators and students carried out a systematic study of some of the most important affine invariant functionals on convex bodies, the ``p-affine surface areas". Understanding those, helps to get a better understanding of the boundary structure of a convex body which in turn is relevant for the above mentioned applications. In particular, the PI, together with her PhD students, established ``affine isoperimetric inequalities"which serve as detectors for extremal situations. Very recent developments open totally new directions. For one, the PI found new affine invariants and proved that those and the above mentioned p-affine surface areas are certain entropies from information theory. In a related direction, the PI, Artstein, Klartag and Schuett showed that some of the above affine isoperimetric inequalities establish further links between information theory and high dimensional convex geometry and tie in with the PI's research on quantum information theory and convex geometry. To date, the PI's research in that area culminated in Aubrun's, Szarek's and her analysis of the "Additivity conjecture for quantum channels" via methods from high dimensional geometric analysis. Outreach and Education Activities Dr. Werner lectured on topics related to the project at conferences, seminars and colloquia, both in the United States and abroad (Canada, China, France, Germany, Israel, Italy, Poland, Russia, Spain). E.g., Dr. Werner has given a series of lectures on ``Local Banach space theory and quantum channels" at the CIEM International Center for Mathematics meetings in Spain in September 2010 and has given a plenary talk ``Entangled states and maps" at the 2012 GPOTS meeting in Houston. She was also organizer of several conferences and workshops, e.g.: "Invariants in convex geometry and Banach space theory" at AIM, Palo Alto (August 2012) "Geometry of Entanglement" at Centre International de Rencontres Mathématiques, Marseille, France (January 2012) "Perspectives in High Dimensions", Cleveland, OH (August 2010) "Geometry of Quantum Entanglement" at MFO, Oberwolfach, Germany (December 2009) The events in Oberwolfach and in Palo Alto had as their primary mission, the training of students and the involvement of students in research. Moreover, the PI holds a weekly Analysis Seminar at Case Western Reserve University. It is a main purpose of this seminar to expose PhD and graduate students to the latest trends in her area of research. Three PhD students, Umut Caglar, Justin Jenkinson and Deping Ye and one post doc student, Pawel Wolff, were involved in activities related to the award.