Free harmonic analysis and applications

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This research proposal is in non-commutative probability, a branch of pure mathematics that is often regarded as a non-commutative parallelism of probability theory. The theory has its roots in the mathematical foundation of quantum mechanics and is closely related to another area of mathematics, called functional analysis, in which one studies topological vector spaces and functions between such spaces. Roughly, the non-commutativity here means that usual functions (or random variables), say, X and Y, are replaced by matrices or operators on vector spaces so that the familiar identity XY=YX no longer holds. This feature of (non-commutative) random variables makes it possible to introduce various notions of independence among them and study their probabilistic behaviour thereafter. The most famous example of such is the notion of free independence and the corresponding free probability, on which this proposal is based on. The applicant proposes to study the harmonic analysis aspect of free probability, as well as its extension and applications to related fields such as bi-free probability theory, free local limit theorems, and infinite ergodic theory. Voiculescu introduced in 2013 the notion of bi-freeness, which is a suitable notion of non-commutative independence for pairs of non-commutative random variables. This area is currently under a rapid development, and most treatments of bi-freeness and the underlying bi-free convolution are combinatorial to date. In contrast to the classical or free probability theories, the literature lacks a harmonic analysis approach to bi-free probability at this moment. The primary goal of this proposed research is to develop such a theory, with an emphasis on bi-freely infinitely divisible distributions. In a recent joint work with Hao-Wei Huang, the applicant obtained some results to show that the classical limit theory of infinitely divisible laws, due to Levy and Khintchine, has a perfect bi-free analogue in the framework of commuting pairs of variables. The applicant believes that his methodology and the approach through limit theorems can be used to treat the general question of bi-free harmonic analysis, without the commutativity assumption. It is quite important that the aforementioned results be further developed in order to understand the bi-freeness better and explore its connections with random matrix. A success in this program would put Canada at the frontier of this research line, and brings impact to harmonic analysis aspect of free probability. This proposal also contains several unanswered questions that are accessible to graduate students or postdoc fellows. The applicant will use the NSERC Discovery Grant to contribute to the training of HQP, especially at the level of PhD students and Postdoctoral Fellows.