EVENT DETAILS AND ABSTRACT

Title: Noncommutative martingales, Poincare inequalities, and mixed q-Gaussian algebras
Speaker: Qiang Zeng
Speaker Info: Harvard University
Brief Description:
Special Note: Qiang will be a Boas starting next year.
Abstract:

In noncommutative probability theory, operators acting on Hilbert spaces are regarded as random variables in an operator algebra. This framework provides an umbrella for classical probability, random matrices, etc.

In this talk, I will first introduce the Burkholder--Rosenthal inequalities for noncommutative martingales, and give the optimal order of constants. Together with Bakry--Emery's Gamma_2 condition, these inequalities yield Lp Poincare inequalities and concentration inequalities, following ideas from classical diffusion theory. Finally, I will introduce the so-called mixed q-Gaussian algebras as a unified model of various Gaussian systems. Hypercontractivity and Lp Poincare inequalities are deduced from mixed spin matrix models by Speicher's CLT. Examples and applications of the abstract theory will be given. Based on joint works with Marius Junge.