Asger Tornquist at Carnegie Mellon University on March 3, 2012
Appalachian set theory
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)
: "Set theory and von Neumann algebras"
Participants in this workshop
Lecture notes from this workshop by Asger Tornquist and Martino Lupini
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
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
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"
- 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
- Shoichiro Sakai,
"C*-algebras and W*-algebras",
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60.
Springer-Verlag, New York-Heidelberg, 1971
- 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
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.
The most popular choice is
Other options are listed
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
at $27 per person in a shared van (2011 price).
Parking at CMU
East Campus Garage is free on weekends.