Combinatorial Methods in Free Probability

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Intellectual merit: Anshelevich's research lies at the interface between functional analysis, probability, and combinatorics. He will use combinatorial methods to obtain new results in free probability theory. For example, he will develop a theory of free Appell polynomials, and use it to obtain new results about limiting behavior of averages of stationary correlated random sequences. He will also use methods from functional analysis to obtain new results in probability theory. For example, he will investigate the chaos decomposition for general Levy processes using the machinery of Fock spaces. Finally, he will use methods from non-commutative probability to obtain new results in combinatorics and special functions theory. For example, he will investigate the linearization coefficients for the q-analogs of the Meixner orthogonal polynomials. Matrices do not commute. The importance of this fact in our interpretation of nature was realized with the invention of quantum mechanics. That theory has also used probabilistic ideas and language since its inception. Non-commutative probability is the probability theory of non-commuting objects, such as matrices and operators. In the 1980s Voiculescu introduced free probability theory, which is a particular type of such a non-commutative theory. Its origins lie in the, currently quite abstract, theory of von Neumann algebras of free groups, but also in the theory of random matrices, directly applicable in an number of areas of physics and signal processing. Anshelevich's expertise lies primarily in functional analysis, thus one of the consequences of this research will be to gain a deeper understanding of a number of fields with potential connections to free probability. Also, some of the combinatorial problems considered are well-suited for undergraduate research, and will be used to encourage interest in mathematics among students.