Free harmonic analysis and applications

点击翻译按钮获取中文摘要

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.