Asger Tornquist at Carnegie Mellon University on March 3, 2012

Saturday, March 3, 2012

Registration and morning refreshments 8:45 - 9:30 AM in Wean Hall 6220

Lectures 9:30 AM - 12:30 PM and 2:30 - 6 PM in Wean Hall 8220 (or a larger room if necessary)

Asger Tornquist : "Set theory and von Neumann algebras"

Lecture notes from this workshop by Asger Tornquist and Martino Lupini

Workshop description

For the past decade or more, countable Borel equivalence relations and actions of countable groups on standard measure spaces has played a central role in descriptive set theory. At the same time, a veritable explosion of results in the theory of so-called II₁ factors has taken place in the field of von Neumann algebras, lead by Sorin Popa and his collaborators. Interestingly, it is the direct connection between countable, measure-preserving equivalence relations and certain II₁ factors through the so-called "group-measure space construction" that has been the driving engine behind many of the discoveries in von Neumann algebras, and so these discoveries have undeniable relevance to the concerns of descriptive set theorists.

The aim of this workshop is to introduce set theorists with an interest in Borel equivalence relations to von Neumann algebras, and in particular to the connection between equivalence relations and von Neumann algebras, and to some of the powerful techniques that the operator algebraic point of view gives rise to.

After introducing and discussing the basic notions in the morning lectures of the workshop, the plan is to spend the afternoon discussing some more advanced topics. Possible topics include (1) applications of descriptive set theoretic ideas to the classification of II₁ factors, and (2) Popa's cocycle superrigidity theorems.

Background material and references

The lectures will not require any familiarity with the theory of von Neumann algebras, but knowledge of the rudiments of functional analysis (e.g. Hilbert spaces, bounded operators on these, etc.) will be assumed.

The notion of orbit equivalence for measure preserving actions will play a central role, but it will not be assumed that workshop participants are familiar with this area. However, participants should at least know basic descriptive set theory, for example, parts of Kechris' textbook.

Those who want to prepare themselves may want to look at the following two references.

Greg Hjorth, "Countable Borel equivalence relations, Borel reducibility and orbit equivalence" [PDF]
Alexander Kechris and Benjamin Miller, "Topics in orbit equivalence", Lecture Notes in Mathematics, 1852, Springer-Verlag, Berlin, 2004

A gentle introduction to the general theme of the lectures can be found in:

Roman Sasyk and Asger Tornquist, "Borel reducibility and classification of von Neumann algebras", Bulletin of Symbolic Logic 15 (2009) 169-183

A less gentle, but still accessible reference is:

Sorin Popa, "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians, vol. I, 445-477, Eur. Math. Soc., Zurich, 2007 [available on Popa's webpage]

A classical general reference in the basics of the field of von Neumann algebras is:

Shoichiro Sakai, "C^*-algebras and W^*-algebras", Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. Springer-Verlag, New York-Heidelberg, 1971

Advanced references:

Sorin Popa, "On a class of type II₁ factors with Betti numbers invariants", Annals of Mathematics (2) 163 (2006) 809-899
Sorin Popa, "Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups", Inventiones Mathematicae 170 (2007) 243-295
Sorin Popa, "On the superrigidity of malleable actions with spectral gap", Journal of the American Mathematical Society 21 (2008) 981-1000

Participant travel support

Funds provided by the NSF will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and postdocs, and to faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to Ernest Schimmerling by email.

Your name, university affiliation, mailing address, phone number and email address
Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant
A brief statement about your interest in the workshop
An itemized estimate of your expected transportation expenses

Schedule

Workshop lectures run from 9:30 a.m. to 6 p.m. with morning refreshments before the workshop, starting at 8:45 a.m., several coffee breaks during the workshop, and a lunch break from 12:30 to 2:30 p.m.

Lodging

The most popular choice is Shadyside Inn. Other options are listed here under the neighborhoods of Shadyside and Oakland.
Note: There is a shortcut from the Shadyside Inn to CMU that is a pleasant walk on Warwick Terrace and Devon Road. Ask for directions at the registration desk.

Bus transportation to Pittsburgh

Check megabus.com with service between other cities and the Pittsburgh Convention Center downtown. City buses connect downtown with the area around CMU.

Transportation to and from the airport

The least expense option is the 28X Airport Flyer with frequent service between the airport and CMU for $3.25 (no change given).
The Shadyide Inn is less than 3/4 mile from CMU; you could walk, take a bus, or call the Shadyside Inn to pick you up.
(If you arrive early on Friday, you may want to meet others in the Mathematical Sciences Department lounge, 6220 Wean Hall.)

Taxis from the airport cost about $50 or more.

Another door-to-door option is SuperShuttle at $27 per person in a shared van (2011 price).

Parking at CMU

The East Campus Garage is free on weekends.

Appalachian set theory