Geometry of convex and discrete measures – GeMeCoD

To make progress on these themes, we shall use the means of the ANR project to organize thematical conferences thématiques for post-docs and young researchers, workgroups gathering researchers from different horizons, conferences and at last recruit post-docs. This scientific activity around these themes will generate a dynamic, give a visibility to these questions and promote the collaboaration of researchers each one with his own expertise.

After six months of activity, we already organized a Winter school, Springdays, a Summer conference, recruited a postdoc and several submitted articles and works are in progress on the project's themes.

We planned the organization of two thematic schools fro the year 2012/2013. In parallel, in groups of two, three or four, the researchers explore the differents aspects of the questions asked in the projet's objectives.

After six months of activity, already have several submitted articles and works in progress on the project's themes.

The aim of this project is to develop the geometries of convex, spherical, Gaussian and discrete measures spaces in view of increasing their mutual interactions towards applications to mathematical questions addressed by theoretical computer science. Our team is constituted of mathematicians, from functional analysis, convex geometry, harmonic analysis and probability. We simultaneously realized that we share a common interest in questions coming from computer sciences, that computer scientists increasingly use standard tools in our theories to prove important theorems and that we could do it by ourselves. This project asks for support for this common thematic evolution. The historical prototype of such interactions are the hypercontractivity/log-Sobolev inequalities which are valid both in the Gaussian and the discrete settings and became a very important tool for theoretical computer scientists, since their application in the subject by Kahn-Kalai-Linial in the late eighties. Many members of our team are experts in these inequalities and extended it in various directions. We want to develop interactions with discrete analysis in view of applications to theoretical computer sciences and push further this development. This cross-fertilization also appears for other questions, like the arising of discrete versions of the Brunn-Minkowski/Prékopa-Leindler inequalities and of the Kannan-Lovasz-Simonovits localization theorem. Such interactions also occur in the search for reducing the level of randomness in algorithmic constructions of embbedings of subspaces of L_p. In all these models, the symmetry plays also a crucial role that need to be better understood. In each of these areas, members of our team observed the very fast increase of the interactions with the discrete setting. Our team constitutes a French group of mathematicians with different backgrounds motivated in developing in France methods and tools for the study of the geometries of convex, spherical, Gaussian and discrete spaces that are useful for theoretical computer sciences.