BIMSA-Tsinghua Quantum Symmetry Seminar

The standard invariant plays a major role in subfactor theory. In this talk, I will discuss a 2-categorical generalization of an axiomatization of the standard invariant and further discuss some algebraic structures associated to it.

References: arXiv:2211.03822, arXiv:2302.04921

• Video

In this talk, I will present a sequence of graphs which have trivial (classical) automorphism group and non-trivial quantum automorphism group, which we believe to be the first known examples of any kind of classical space with this property. The construction is inspired by solution groups of binary linear systems, as defined by Cleve, Liu and Slofstra in 2015 in relation to certain non-local games in quantum information theory.

Reference: This talk is based on joint work with David E. Roberson (Technical University of Denmark) and Simon Schmidt (Ruhr University Bochum, Germany), arXiv:2311.04889.

Speaker Intro:

Josse van Dobben de Bruyn is a postdoctoral researcher at the Technical University of Denmark (DTU), working on interactions between graph theory, quantum groups, and quantum information theory. He obtained his PhD in algebraic combinatorics from Delft University of Technology in the Netherlands in 2023, supervised by Prof. dr. Dion Gijswijt.

A bounded linear operator \(A\) is said to be quadratic if there is a polynomial \(p\) of degree \(2\) such that \(p(A)=0\). Square zero operators, involutions, and idempotents are all typical quadratic operators.

We will give characterizations of matrices could be expressed as commutators of two square zero matrices, and explain some related results about limits of commutators of two square zero operators acting on a complex, separable Hilbert space \(\mathcal{H}\).

We will also study the norm-closure of the set \(\mathfrak{C}_{\mathfrak{E}}\) of bounded linear operators acting on \(\mathcal{H}\) which may be expressed as the commutator of two idempotent operators. In particular, biquasitriangular operators belong to the norm-closure of \(\mathfrak{C}_{\mathfrak{E}}\) are fully charateriezed.

This talk is based on joint papers with Laurent Marcoux and Heydar Radjavi.

The free probability theory is a probability theory of noncommutative random variables, where usual independence is replaced by free independence. It was initially designed to study longstanding problems about von Neumann algebras of free groups. It turns out to be an extremely powerful framework to study the universality laws in random matrix theory due to the groundbreaking work of Voiculescu. These limiting laws are encoded in abstract operators, called free random variables.

Brown measure is a sort of spectral measure for free random variables, not necessarily normal. I will report some recent progress on the Brown measure of the sum \(X+Y\) of two free random variables \(X\) and \(Y\), where \(Y\) has certain symmetry or explicit R-transform. The procedure relies on Hermitian reduction and subordination functions. The Brown measure results can predict the limit eigenvalue distribution of various full rank deformed random matrix models. The talk is based on my work on Brown measure of elliptic operators and joint works with Hari Bercovici, Serban Belinschi and Zhi Yin.

In this talk, I will describe the construction known as ‘Zesting of Braided Fusion Categories’, a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. I will also present our work on classifying and constructing all possible braided zesting data for modular categories associated with quantum groups at roots of unity. We have produced closed formulas, based on the root system of the associated Lie algebra, for the modular data of these new modular categories.

Reference: This talk is based on the preprint https://arxiv.org/abs/2311.17255, a joint work with Giovanny Mora and Eric C. Rowell.

• YouTube
• Slides