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Earth Mantle Convection
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Martin Kronbichler

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:

  • https://arxiv.org/abs/1504.05919
  • https://arxiv.org/abs/2108.0631
  • https://arxiv.org/abs/2208.11286

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:

  • https://arxiv.org/abs/2104.02662
  • https://arxiv.org/abs/2108.06312

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:TBA

Date:April 21, 2023
Time:4:00pm
Location:BLOC 306
Speaker:Michael Anshelevich, TAMU
Title:TBA

Date:April 28, 2023
Time:4:00pm
Location:BLOC 306
Speaker:Michael Anshelevich, TAMU
Title:TBA