The research: Probability, Stochastic Processes, and Functional Analysis

The researchers working in this area are scientific descendants of the legendary Lvov School of Mathematics (Banach, Mazur, Orlicz). The high reputation of the scientific group in this field in Warsaw was established by Kwapien, Pelczynski and Zabczyk.

Nowadays this big research group consists of a number of established scientists (Bojdecki, Latala, Oleszkiewicz, Stettner) and growing number of younger researchers (Adamczak, Bednorz, Osekowski, Talarczyk). Current research interests of the group concern theoretical aspects of probability theory and stochastic processes as well as applications of probabilistic methods in functional and harmonic analysis, convex geometry, combinatorics, theoretical computer science, and financial mathematics.

Particular research interests and plans of this research group include: limit theorems and the study of asymptotic behaviour of random vectors, random multilinear forms and stochastic processes, random matrix theory and its applications, Markov chains and Monte Carlo methods, stochastic inequalities with particular emphasis on martingale inequalities and their connections with Fourier and harmonic analysis, chaining methods in the study of boundedness and regularity of stochastic processes.

Other objectives of the current research in this area involve the construction and study of random objects and their properties in combinatorics, functional analysis and convex geometry, random particle systems and applications of stochastic calculus in mathematical finance, discrete harmonic analysis, and its connection with learning theory and theoretical computer science.

In a recent landmark work, Mossel (Berkeley), O’Donnell (Carnegie-Mellon) and Oleszkiewicz established a new invariance principle and used it for proofs of two conjectures: “The Majority is Stablest” from theoretical computer science and “It Ain’t Over Till It’s Over” from social choice theory. The invariance principle makes it possible to transfer many problems from product probability space setting into the Gaussian space framework, which proved to be useful in a number of applications both in computer science and in mathematics.