High Dimensional Probability
High Dimensional Probability has its roots in the investigation of limit theorems for random vectors and regularity of stochastic processes. It was initially motivated by the study of necessary and sufficient conditions for the boundedness and continuity of trajectories of Gaussian processes and extension of classical limit theorems, such as laws of large numbers, laws of the iterated logarithm and central limit theorems, to Hilbert and Banach space-valued random variables and empirical processes.
It resulted in the creation of powerful new tools - the methods of high dimensional probability and especially its offshoot, the concentration of measure phenomenon, were found to have a number of applications in various areas of mathematics, as well as statistics and computer science. These include random matrix theory, convex geometry, asymptotic geometric analysis, nonparametric statistics, empirical process theory, statistical learning theory, compressed sensing, strong and weak approximations, distribution function estimation in high dimensions, combinatorial optimization, random graph theory, as well as information and coding theory.
The aim of this workshop is to bring together leading experts in high dimensional probability and a number of related areas to discuss the recent progress in the subject as well as to present the major open problems and questions. We want to deepen contacts between several different communities with common research interests focusing on stochastic inequalities, empirical processes, strong approximations, Gaussian and related chaos processes of higher order, Markov processes, concentration of measure techniques and applications of these methods to a wide range of problems in other areas of mathematics, statistics and computer science. We would also like to foster and develop interest in this area of research among new researchers and recent Ph.D.'s. There are many exciting open problems in the area that may be formulated in a way that can be understood by graduate students. We hope that they will attract attention of young people taking part in this workshop.
Particular areas of focus and interest for the meeting include:
- Application of generic chaining techniques to study the regularity of stochastic processes and lower and upper bounds on norms of random vectors and matrices
- Relation between various isoperimetric and concentration inequalities as well as their applications to convex geometry, statistics and computer science
- Applications of modern empirical process and strong approximation methods to problems of machine learning and inference in high- and infinite-dimensional statistical models
- Interactions between information-theoretic inequalities, convex geometry and high-dimensional probability
- Stein’s method and its use in high-dimensional probability
- Nonasymptotic random matrix theory and applications to quantum information theory
- Interactions between high dimensional probability and statistical physics
- Super-conconcentration phenomena in high dimension: new tools, examples and open problems
- Application of Itô calculus to convex geometry
- Identification of major problems and areas of potentially high impact for applications and use in other areas of mathematics, statistics, and computer science