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Free Probability and Operators
Spring 2023
| Date: | January 27, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Ken Dykema, TAMU |
| Title: | On spectral and decomposable operators in finite von Neumann algebras. |
| Abstract: | We describe spectral and decomposable operators on Hilbert spaces and then, in the case of operators belonging to finite von Neumann algebras, relate these properties to the Haagerup-Schultz subspaces of the operators. Finally, we will examine certain such operators arising naturally in free probability theory. |
| Date: | February 3, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Ken Dykema, TAMU |
| Title: | On spectral and decomposable operators in finite von Neumann algebras. (cont.) |
| Abstract: | We describe spectral and decomposable operators on Hilbert spaces and then, in the case of operators belonging to finite von Neumann algebras, relate these properties to the Haagerup-Schultz subspaces of the operators. Finally, we will examine certain such operators arising naturally in free probability theory. |
| Date: | February 10, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Ken Dykema, TAMU |
| Title: | On spectral and decomposable operators in finite von Neumann algebras. (cont.) |
| Abstract: | We describe spectral and decomposable operators on Hilbert spaces and then, in the case of operators belonging to finite von Neumann algebras, relate these properties to the Haagerup-Schultz subspaces of the operators. Finally, we will examine certain such operators arising naturally in free probability theory. |
| Date: | February 17, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Michael Anshelevich, TAMU |
| Title: | Hermite trace polynomials |
| Abstract: | We consider the algebra of trace polynomials with the state induced by the GUE random matrices. In addition to the monomial basis, another natural basis for this algebra consists of the Hermite trace polynomials. They satisfy several properties familiar for the ordinary trace polynomials. The algebra structure can be extended from polynomials to a larger family of stochastic integrals. While the Hermite trace polynomials are not orthogonal, they can be modified to obtain several versions of the chaos decomposition. The simplest of these is related to the Hermite polynomials of matrix argument. This is joint work with David Buzinski. |
| Date: | February 27, 2023 |
| Time: | 09:00am |
| Location: | ZOOM |
| Speaker: | March Boedihardjo, ETH Zurich |
| Title: | Spectral norm and strong freeness |
| Abstract: | We give a non-asymptotic estimate for the spectral norm of a large class of random matrices that is sharp in many cases. We also obtain strong asymptotic freeness for certain sparse Gaussian matrices. Joint work with Afonso Bandeira and Ramon van Handel. References:
ZOOM LINK: https://tamu.zoom.us/j/97389162643 |
| Date: | March 3, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Ken Dykema, TAMU |
| Title: | On B-valued circular operators |
| Abstract: | We will briefly introduce B-valued circular operators, where B is a *-algebra. We will describe (and perhaps prove) some results about these, in the special case when B is a commutative C*-algebra and describe how they are relevant to the study of DT-operators. |
| Date: | March 6, 2023 |
| Time: | 09:00am |
| Location: | ZOOM |
| Speaker: | March Boedihardjo, ETH Zurich |
| Title: | Spectral norm and strong freeness: Proofs |
| Abstract: | I will begin by proving an estimate for a quantity introduced by Tropp. I will then give a combinatorial proof and an analytic proof of the main result in my previous talk. References:
ZOOM LINK: https://tamu.zoom.us/j/97389162643 |
| Date: | March 24, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Zhiyuan Yang, TAMU |
| Title: | Modular structure of Hilbert space and twisted Araki-Woods algebras |
| Abstract: | We discuss the construction of T-twisted Araki-Woods von Neumann algebras following the preprint https://arxiv.org/pdf/2212.02298.pdf by da Silva and Lechner, which is a generalization of the q-Araki-Woods algebras. We will begin with the basic properties of the modular operators of standard subspaces and its correspondence with the semigroup approach often used in q-Araki-Woods literature. Then we describe a sufficient and necessary condition (crossing symmetry and satisfying the Yang-Baxter equation) on the twist T for the vacuum vector to be separating. |
| Date: | March 31, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Sheng Yin, Baylor University |
| Title: | Non-commutative rational functions in random matrices and operators. |
| Abstract: | It is well-known that many random matrices have an asymptotical limit which is described by free probability. That is, for any noncommutative polynomial in these d independent random matrices converges to the same polynomial in d freely independent random variables that describe the limit distribution of each sequence of random matrices. In this talk, we will present a natural generalization of this convergence result. Namely, under suitable assumptions, we can enlarge our test function from noncommutative polynomial to noncommutative rational functions. It is based on a joint-work with Benoît Collins, Tobias Mai, Akihiro Miyagawa and Félix Parraud. |
| Date: | April 14, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Michael Anshelevich, TAMU |
| Title: | Types of noncommutative independence and asymptotics of random matrices. |
| Abstract: | We will discuss how the asymptotic joint distribution of two random matrices "in a general position" can be described using cyclic c-freeness and its particular cases; or, conversely, how different types of noncommutative independence can be asymptotically modeled by random matrices. In the first talk we will concentrate on algebraic independence theories; in the second and third talks, we will explain the connection with random matrices. The talks are based primarily on the articles arXiv:2207.06249 by Cébron and Gilliers; and arXiv:2205.01926 by Cébron, Dahlqvist, and Gabriel. |
| Date: | April 21, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Michael Anshelevich, TAMU |
| Title: | Types of noncommutative independence and asymptotics of random matrices. |
| Abstract: | We will discuss how the asymptotic joint distribution of two random matrices "in a general position" can be described using cyclic c-freeness and its particular cases; or, conversely, how different types of noncommutative independence can be asymptotically modeled by random matrices. In the first talk we will concentrate on algebraic independence theories; in the second and third talks, we will explain the connection with random matrices. The talks are based primarily on the articles arXiv:2207.06249 by Cébron and Gilliers; and arXiv:2205.01926 by Cébron, Dahlqvist, and Gabriel. |
| Date: | April 28, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306 |
| Speaker: | Michael Anshelevich, TAMU |
| Title: | Types of noncommutative independence and asymptotics of random matrices. |
| Abstract: | We will discuss how the asymptotic joint distribution of two random matrices "in a general position" can be described using cyclic c-freeness and its particular cases; or, conversely, how different types of noncommutative independence can be asymptotically modeled by random matrices. In the first talk we will concentrate on algebraic independence theories; in the second and third talks, we will explain the connection with random matrices. The talks are based primarily on the articles arXiv:2207.06249 by Cébron and Gilliers; and arXiv:2205.01926 by Cébron, Dahlqvist, and Gabriel. |
| Date: | May 12, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 306, ZOOM |
| Speaker: | Jurij Volcic, Drexel University |
| Title: | State polynomials: positivity and applications |
| Abstract: | State polynomials are polynomial expressions in operator variables and formal states of their products. This talk addresses state polynomials whose evaluations on states and tuples of operators on a Hilbert space are positive. The most well-known occurrence of such a positive state polynomial in disguise is the Cauchy-Schwarz inequality. Much more recently, positive state polynomials emerged as nonlinear Bell inequalities for quantum networks. The talk presents the state polynomial analog of Hilbert's 17th problem, and an algebraic certificate for constrained positivity. The latter is used to design a scheme for state polynomial optimization, which resolves a few questions from quantum information theory. ZOOM LINK: https://tamu.zoom.us/j/97716603515 |
