Recent trends at the interface between information theory, probability, statistics and high-dimensional geometry raise new questions on the role and importance of entropy in the understanding of both asymptotic and finite range phenomena, in particular towards discrete structures and applications to areas such as quantum information, percolation and random matrices. Recent fruitful developments warrant imminent blossom of new ideas and results in this area in the near future.
Another central theme at the interface of High-Dimensional Probability and Convexity is the study of isoperimetric properties of log-concave measures (including isoperimetric-type questions related to convex bodies). This is not a mere curiosity: convexity and isoperimetry play an important role in the study of concentration of measure phenomena which is crucial for harnessing various high-dimensional effects in Probability as well as several applied fields.
The scope of the semester includes developing further connections between Information Theory and Convexity, in all of the different directions described above. The semester will aim to strengthen these powerful interdisciplinary methods, aiming towards challenging issues and applications, including the isoperimetry questions for log-concave measures, and super-concentration properties of models from random matrix theory and statistical mechanics.
The semester’s objective is to foster cross-fertilization between probability, analysis, information theory and convex geometry (continuous and discrete). By organizing two thematic pre-schools, we will ensure active participation by junior researchers, which should have a lasting effect on their careers, as well as influence the routes along which these research areas will be developed in the future. Our two workshops will further provide opportunities for many re- searchers from across the globe to interact and collaborate. We believe that close interactions between these disciplines are a necessary condition to address many mathematical challenges raised in the last decades, and that the timing is optimal to bring together all these powerful ideas which have been recently de- veloped in several communities. Therefore, our proposed semester program will have a strong and lasting impact on High-Dimensional Probability, Convexity and Analysis as a whole.
Timeline for the trimester:
April 19th to 23rd - Introductory school at CIRM
May 10 to 14 - Workshop 1 at IHP: Convexity and Information Theory.
June 14 to 18 - Introductory School: Asymptotic Geometry Analysis and Functional Inequalities.
July 5 to 9 - Workshop 2 at IHP: Asymptotic Geometry Analysis and Functional Inequalities.