IMS Graduate Summer School in Logic
National University of Singapore, Institute of Mathematical Sciences
Model theory of tracial von Neumann algebras
Lecturer: Isaac Goldbring
July 16-20, 2022
Abstract:
Tracial von Neumann algebras are a widely studied class of objects belonging to the mathematical area of operator algebras and have numerous interactions with a variety of other areas of mathematics, including representation theory, group theory, ergodic theory, and quantum physics, to name a few. With the advent of continuous logic, they became amenable to being studied through the model-theoretic lens, as was first carried out by Farah, Hart, and Sherman.
In this lecture series, we will introduce these mathematical objects and explain the logic appropriate for studying them. We will then move on to some applications of model-theoretic ideas to the study of tracial von Neumann algebras, including the model-theoretic take on the recent negative resolution of the famous Connes Embedding Problem as well as how model-theoretic techniques can be used to shed light on some problems concerning embeddings into ultrapowers. We will conclude the course by discussing a recent application of model-theoretic ideas by David Jekel to free probability.
The course will be fairly self-contained and only prior exposure to basic (classical) model theory will be assumed.
Tentative Schedule (updated June 27)
- Lecture 1
- Introduction to tracial von Neumann algebras
- We introduce the class of tracial von Neumann algebras, giving a variety of examples. We also discuss the important subclass of II_1 factors as well as introduce the tracial ultraproduct construction.
- Lecture 2
- Tracial von Neumann algebras as metric structures
- We introduce the appropriate continuous logic used for studying tracial von Neumann algebras and show that, in the right continuous language, they form an elementary class. We then move on to discussing the problem of identifying many different first-order theories of II_1 factors.
- Lecture 3
- Model theory and the Connes Embedding Problem
- We discuss the famous Connes Embedding Problem (CEP) and explain how model-theoretic techniques can be used to simplify the recent negative resolution of the CEP from a landmark result in quantum complexity theory known as MIP*=RE.
- Lecture 4
- Applications to problems around embeddings with factorial commutants
- We discuss two recent applications of model theory to von Neumann algebra theory: progress on Popa's Factorial Relative Commutant Problem and a new characterization of the hyperfinite II_1 factor in terms of embeddings into ultrapowers.
- Lecture 5
- Model theory and free probability
- We discuss recent results of David Jekel applying model-theoretic ideas to prove theorems about free entropy with applications to embeddings in matrix ultraproducts.
References/Suggested Reading