Winter school program
|Autumn 2014||Noncommutative month||Winter school
|Program / Schedule
||Location / Travel info
Abstracts of the courses
Quantum entanglement in high dimensions
I will describe mathematical aspects connected to the concept of quantum entanglement. I will also give some background on quantum information theory, where entanglement is considered as a resource for communication and computation. This will motivate the study of entanglement in high-dimensional quantum systems.
I will explain how concepts from functional analysis can be used to understand entanglement. Many tools from local theory of Banach spaces, such as concentration of measure (more specifically, Dvoretzky’s theorem), can be applied.
I will give two examples of such applications (1) a characterization of quantum systems for which entanglement is generic, in terms of the dimension of the environment (2) a proof (after Hastings) that entanglement can be used to increase the capacity of quantum channels to carry information.
Weingarten calculus and applications
In this series of lecture, we will address the problem of computing the integral of polynomial functions against Haar measures on compact groups or on compact quantum groups. After exploring in some detail the proofs of the main theorems of this theory — nowadays known as Weingarten calculus —, we will describe some applications of these techniques, to random matrix theory, free probability and quantum information theory.
Grothendieck Inequalities, Tensor products of operator spaces and Related Topics
We will start from the classical Grothendieck Theorem (GT) also called Grothendieck’s Inequality and will present the non-commutative mutations that this result has been through more recently. In particular we will discuss the operator space version of GT, using the recent striking new approach due to Regev and Vidick. We will also discuss the Connes-Kirchberg problem and related questions involving tensor products of C*-algebras and operator spaces.
Operator spaces and their applications to (quantum) group algebras
I will first give a brief introduction to operator spaces. Then I will show some application of operator space theory to abstract harmonic analysis and group C*-algebras. Finally, I show that some properties can be naturally generalized to the quantum group case.
Quantum symmetry groups and related topics
Groups first entered mathematics in their geometric guise, as collections of all symmetries of a given object, be it a finite set, a polygon, a metric space or a differential manifold. Original definitions of quantum groups, also in the analytic context, had rather algebraic character. In these lectures we describe several examples of quantum symmetry groups of a given quantum (or classical) space. The theory is based on the concept of actions of (compact) quantum groups on C*—algebras and viewing symmetry groups as universal objects acting on a given structure. Initiated by Wang in 1990s, in recent years it has been developing rapidly, exhibiting connections to combinatorics, free probability and noncommutative geometry. In these lectures we will present both older and newer research developments regarding quantum symmetry groups, discussing both the general theory and specific examples.
Detailed program (pdf)
Free probability and non-commutative symmetries
For non-commuting (random) variables there exist canonical notions of independences and symmetries. The most prominent examples are given by free independence and easy quantum groups (in particular, the quantum permutation and quantum orthogonal group). There is a lot of interplay between these structures (like de Finetti Theorems), with the lattice of non-crossing partitions featuring prominently. Also operator algebras related to these structures show interesting properties and problems. In my lectures I will give an introduction to these subjects.