NSF-Free Probability, Polynomial Families, and Applications
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abstract

Free probability is a relatively young theory (started in the 1980s), with numerous powerful results and applications to the study of operator algebras, random matrices, and other fields. On the other hand, the study of polynomials goes back to the very beginnings of algebra, and orthogonal polynomials in particular are used throughout mathematics, physics, and engineering. Perhaps surprisingly, there are numerous relations between these two fields of study, through Fock space representations, martingale properties of polynomials, stochastic integrals, and special functions. Both fields have also developed numerous generalizations and extensions: operator-valued free probability theory and other non-commutative probability theories on one hand; and on the other, operator-valued orthogonal polynomials, polynomials of matrix argument, polynomials in multiple non-commuting variables, etc. This project will explore, through a series of specific problems, the objects mentioned above, emphasizing the connection between the two fields in the project title. It will also involve applications to other fields, notably random matrices. 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 (which itself plays 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 project will continue the study of the mutual interaction between these two subjects, applying techniques from each one to the study of the other. As part of the project, the PI will continue organizing seminars and conferences bringing together researchers and students from different fields. He will also continue his work with undergraduate students on research topics forming part of this project.