Operator Algebras, Random Matrices, and Quantum Expander Graphs

Below is a brief outline of the contents of each of the lecture series.

Serban Belinschi.

Title: Fundamentals of Free Probability
Abstract: Topics will include the following.
1. definition and two models, L(F_n) and the full Fock space;
2. Uffe Haagerup's R-transform (with proof);
3. freeness with amalgamation and amplification to matrices;
4. free noncommutative functions and transforms, linearization;
5. analytic description of freeness via subordination.

Charles Bordenave.

Title: Operator norms of random matrices
Abstract: The use of operator norms is ubiquitous in applications of spectral methods. In this course, we will discuss some of the techniques developed over the years to get sharp estimates on norms of random matrices. The plan is the following :
Lecture 1: illustration of the resolvent method for Gaussian unitary matrices.
Lecture 2: the expected high trace method for Wigner matrices.
Lecture 3: strong convergence of unitary representations and the linearization trick.
Lecture 4: strong convergence of random unitary and permutation matrices.

Cécilia Lancien.

Title: Quantum expanders - Random constructions & Applications
Abstract: The goal of this series of lectures will be to understand what quantum expanders are, what they are useful for, and how they can be constructed. In the first lecture, we will start by recalling the definition and main properties of classical expander graphs. We will then explain how quantum analogues of these objects can be defined. In the second lecture, we will show that, both classically and quantumly, random constructions typically provide examples of optimal expanders. We will spend some time going through the proofs in the quantum case, and see that they rely on a spectral analysis for random matrix models with a tensor product structure. In the third lecture, we will present implications of these results in terms of typical decay of correlations in 1D many-body quantum systems with local interactions.

David Pérez García.

Title: Random techniques in nonlocal games
Abstract: Nonlocal games constitute a central tool in quantum information theory, particularly in cryptographic and verification tasks. Recently, they have also emerged as an important new tool in operator algebras. This lecture will focus on how to estimate different norms for random matrices and tensors in order to obtain examples of nonlocal games with desirable extremal properties. The structure of the course will be the following:
1. An introduction to nonlocal games and their associated tensors.
2. Estimating norms of random matrices and tensors.
3. Random tensors as a source of nonlocal games with extremal properties.

Adam Skalski.

Title: Haagerup property for groups and operator algebras
Abstract: Topics will include the following.
1. Haagerup property for discrete groups: origin, examples and equivalent characterisations.
2. von Neumann algebras and C*-algebras of discrete groups: construction, some basic properties, Herz-Schur multipliers.
3. Haagerup property of von Neumann algebras: finite case, general case, relative case.
4. Maximal Haagerup subgroups and subalgebras: examples and open questions.

• Serban Belinschi, Université Paul Sabatier, Toulouse & Queen's University, Kingston.
• Charles Bordenave, Aix-Marseille Université.
• Cécilia Lancien, Institut Fourier, Grenoble.
• David Pérez García, Universidad Complutense, Madrid.
• Adam Skalski, Institute of Mathematics, Polish Academy of Science.

A tentative program can be found here.

Please register here. 

Deadline for registering for financial support: May 15, 2024.
Deadline for registering for the master class: June 1, 2024.

Magdalena Musat, musat@math.ku.dk

Pieter Spaas, pisp@math.ku.dk