A Study of Bi-Free Probability

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Operator Algebras is the area of mathematics where collections of continuous linear maps (known as operators) between vector spaces are studied. One important subarea of Operator Algebras is a non-commutative probability theory introduced by Voiculescu in the 1980s known as free probability. By studying free independence, which is a notion of independence for random variables that need not commute, several advancements in Operator Algebras were made. Additionally free probability has many applications and relations to other areas, such as combinatorics, representation theory of large groups, mathematical physics, quantum information theory, random matrix theory, and wireless communications.