This is a US-Polish mathematical research project to explore a new connection between classical probability and non-commutative harmonic analysis. The principal investigator is Dr. Wlodek Bryc from the University of Cincinnati. His Polish collaborator is Dr. Marek Bozejko from Wroclaw University. The need for their collaboration arose from an unexpected occurrence of the same class of processes in their research, indicating an unexplored link between certain models of classical probability and those of deformed quantum theory. The main goal of this collaboration is to advance the understanding of non-commutative processes which have classical versions, and to use classical probability as a guide and motivation in defining appropriate generalizations of the concept of a "non-commutative Brownian motion". The researches will construct new classes of non-commutative processes and will study the classical versions of already known non-commutative processes. To facilitate the exchange of expertise, the researchers will rely on mutual visits, email contacts, internet posting at http://math.uc.edu/free-gauss/, participation in common meetings and workshops. The US participants will attend summer workshops hosted by the Polish affiliates, and likewise, the Polish partners will visit the US for seminars, which will both provide unique training for US graduate students and foster the rapid exchange of ideas.
By bringing together leading experts in the U.S. and Central/Eastern Europe, the project will open and advance a new layer of connections between classical probability and non-commutative harmonic analysis. In the past, cross boundary research and exchange of ideas has had great success and was critical for major advances in non-commutative and classical probability. Over the last decade, significant progress in the understanding of free group von Neumann algebras arose from Voiculescu's ingenious introduction of free probability and free convolution. Likewise, the connection between probability and non-commutative harmonic analysis has attained increasing importance, as concepts which originally arose in free probability are now becoming powerful tools in classical probability, as in the works of Dembo, Guionnet and Zeitouni. The works of Buchholz, Pisier and Xu on non-commutative martingales have opened new directions of research, and the researchers hope for a similar impact from this project.