Related papers: Notes on Free Probability Theory

We review recent developments in the use of von Neumann algebras to analyze the entanglement structure of quantum gravity and the emergence of spacetime in the semi-classical limit. Von Neumann algebras provide a natural framework for…

This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of…

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von…

These are extended notes for my talk at the ICMP 2003 in Lisbon. Our goal here is to demonstrate how natural and fundamental random partitions are from many different points of view. We discuss various natural measures on partitions, their…

This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and…

This lecture note surveys the gamma matrices in general dimensions with arbitrary signatures, the study of which is essential to understand the supersymmetry in the corresponding spacetime. The contents supplement the lecture presented by…

These are notes from my mini-course at ICRA13, Sao Paulo, 2008

The present notes are based on a course on Cherednik algebras given by the first author at MIT in the Fall of 2009. Their goal is to give an introduction to Cherednik algebras, and to review the web of connections between them and other…

We study the $t$-deformation of gaussian von Neumann algebras. They appear as example in the theories of Interacting Fock spaces and conditionally free products. When the number of generators is fixed, it is proved that if $t$ sufficiently…

This text is based on a series of three expository lectures on a variety of topics related to "thin orbits," as delivered at Durham University's Easter School on "Dynamics and Analytic Number Theory" in April 2014. The first lecture reviews…

This paper is a collection of lecture notes on modified gravity. Various modified gravity models formulated within the Riemannian formalism are discussed.

These notes are based on the lecture courses given at the Ruhr-Universit{\"a}t-Bochum (03--08.02.1997) and at the Universit{\'e} Paul Sabatier (Toulouse, 08-12.01.1996).

These are the lecture notes for a short course on geometric quantization given by the author at the XVIII Modave Summer School on Mathematical Physics, Sep 5 - Sep 9.

These notes form an introduction to Lie algebras and group theory. Most of the material can be found in many works by various authors given in the list of references. The reader is referred to such works for more detail.

Notes from a course given at Oujda university, Morocco, october 2002 - march 2003 within the support of a fellowship from the Agence Universitaire de la Francophonie. These notes present a brief introduction to Connes' non commutative…

This is the draft of lecture notes for Phd students in Sichuan University. In this notes we expand Li-Ruan's paper with much more detailed explanations and calculations.

This note was originated many years ago as my reaction to questions of several people how free strongly homotopy algebras can be described and what can be said about the structure of the universal enveloping A(m)-algebra of an L(m)-algebra,…

This is a lecture notes for a mini-course in Department of Mathematics, Ghent University, 14 Mar.-25 Mar. 2023.

These are the lecture notes of a "Nachdiplomvorlesung" course taught at ETH Zurich in the Spring of 2013. They appeared in the EMS series Zurich Lectures in Advanced Mathematics.

We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number…