The PI's research is concentrated at the interface of functional analysis, probability theory (commutative and non-commutative), and combinatorics. In previous work, the PI has developed the machinery of polynomial families associated to Free Probability theory. He now proposes to use this machinery to obtain a number of applications in free probability and related fields. Specific projects include the investigation of von Neumann algebras coming from special tracial states on polynomials, the study of the free Fisher information of these states, as well as their connection with multi-matrix models in the theory of random matrices. He will also investigate operator algebras arising from Gaussian pairs of states in the two-state free probability theory, for which polynomial families provide extra structure. Conversely, the PI plans to use probabilistic techniques to investigate combinatorial properties of polynomial families, such as their linearization coefficients. He will also explore the issue of whether further classes of polynomial families have probabilistic interpretations.
It is a fundamental property of matrices that they may not commute. Since the beginning of quantum mechanics, probability theory of non-commuting objects has been an important field of research. In the 1980s, Voiculescu started the investigation of free probability theory, a theory of this type which also has numerous (sometimes spectacular) applications to operator algebras and the theory of random matrices, itself playing an increasingly important role in physics and signal processing. On the other hand, polynomials are ubiquitous in mathematics, although polynomials in variables which do not commute are less familiar. This proposal applies fundamental techniques of non-commutative polynomials, combined with methods from free probability, to the study of operator algebra and random matrices. Parts of this project are well-suited for undergraduate research, which encourages interest in mathematics among students. The PI will continue to organize a seminar, which provides opportunities for young researchers to disseminate their work and meet colleagues. Finally, the PI will organize a conference on topics covered in the proposal, with the goal of bringing together researchers from different fields of mathematics, resulting in mutually beneficial interactions.
