Erwin Schrödinger Institute, Vienna
Amenability
Special semester, February  July 2007
Contents:
Introduction
Workshops
Registration
Introduction
The notion of amenability is a natural generalization
of finiteness or compactness. It was introduced in 1929 by J. von Neumann
(following the work of Hausdorff, Banach and Tarski; in 1955 M. M. Day first
called it amenability). Amenable groups are those which
admit an invariant mean (rather than an invariant probability measure, which
is the case for finite or compact groups). This classical notion has been
generalized in many directions and currently plays an important role in various
areas, such as dynamical systems, von Neumann and C*algebras, operator Ktheory,
geometric group theory, random walks, etc.
The semester will be centered around several interconnected
research subjects at the crossroads of Analysis, Algebra, Geometry, Dynamics
and Probability. More specifically, we are going to discuss the following
topics:

groups of intermediate growth, nonelementary amenable
groups;

selfsimilar groups and iterated monodromy groups of
rational maps;

graphed equivalence relations and amenability; L^{2}
cohomology;

amenable groupoids; topological amenability of boundary
actions;

amenability at infinity; BaumConnes and Novikov conjectures;

amenability and rigidity; bounded cohomology;

amenable algebras;

quasiisometric classification of amenable groups, geometricity
of various group properties;

Dixmier's conjecture on characterization of amenability
in terms of unitarizable representations;

generalizations of amenability: ATmenabilty (Haagerup
property); groups without free subgroups; superamenability;

random walks and other probabilistic models on amenable
groups;

quantitative invariants of amenable groups: growth, isoperimetry,
asymptotic entropy, etc.;
There will be two separate main periods of activity in the
first (end of February  April) and in the second (May  July) parts of the
semester. During the first period there was a threeweek workshop
February 26  March 17, 2007
The second period of the programme
will last from May 1 to July 31. During this time there will be a workshop
Algebraic, geometric and probabilistic aspects of amenability
June 18  July 14, 2007
This workshop is partially supported by the Marie Curie
Conferences and Training Courses Programme MSCFCT2006045987 GROUPS.
In addition to talks on current research
the programme of this workshop will contain several minicourses (46 lectures
each) as well as some shorter lecture series which will provide a comprehensive
introduction to various aspects of amenability and report on recent advances
in the area. These courses should be accessible to mathematicians at all
levels (starting from advanced Ph.D. students).
The following minicourses are planned:
Alekos Kechris
(Caltech): Extreme amenability: Some new interactions between
combinatorics, logic and topological dynamics, June 1820, 2007
Abstract: This minicourse will provide an introduction to
the property of extreme amenability (or fixed point on compacta property)
of topological groups, which arises in the context of topological dynamics
and is related to asymptotic geometric analysis, especially concentration
of measure phenomena, and describe its connections with ideas from finite
combinatorics, particularly Ramsey theory, and logic.
Schedule of lectures for the week June 18  22, 2007
Nicolas Monod (Geneva):
Some topics on amenable actions, June 25  29, 2007
Abstract: In this series of lectures, we shall study some aspects of
the classical question of the existence of invariant means on a set under
a group action. We shall restrict ourselves to the "naked" setting where
no topology or measuretheory is involved. We shall on the one hand
address classical results, giving for instance a short proof of Tarski's
theorem on paradoxical decompositions. On the other hand, we present
a few new aspects that have been historically overlooked, regarding e.g.
amenable actions of nonamenable groups. We will also present a few
problems that are simple to formulate but appear to be unsolved.
Schedule of lectures for the week June 25  29, 2007
Dave Witte Morris
(Lethbridge): Some discrete groups that cannot act on 1dimensional manifolds, July 2  6, 2007
Abstract: We will see that it is easy to give an algebraic characterization of the amenable groups that have a nontrivial action on the real line. The course will then discuss relations between amenability, the Furstenberg boundary, and actions of lattices on 1dimensional manifolds (or other spaces). Many questions remain open.
Schedule of lectures for the week July 2  6, 2007
Volodymir Nekrashevych
(Texas A&M): Contracting selfsimilar groups, July
9  13, 2007
Abstract: Contracting selfsimilar groups appear naturally as iterated
monodromy groups of expanding selfcoverings of orbispaces. They include
many examples of nonelementary amenable groups. An open question is if they
all are amenable. We will discuss relation of contracting groups to dynamical
systems and their properties related to amenability (growth, absence of free
subgroups, amenability of the associated groupoids of germs, etc.).
Schedule of lectures for the week July 9  13, 2007
The ESI will offer a number of
Junior Fellowships to support the participation of young researchers in the
programme. Support for junior researchers consists of a daily allowance of 70 Euros per day (which
is sufficient to cover the accommodation in Vienna with the special
ESI rate; ESI does not provide travel support). Senior participants will be supported
at the rate of 80 Euros per day.
Applications for participation and financial support
should be sent to the organizers at aesi@esi.ac.at.
Back to table of contents.
Registration
Participation is free. In order to register for participation
in one of the workshops, please send an email to aesi@esi.ac.at. The ESI secretaries will
provide you with the practical information concerning accommodation etc.
in due time.
If you want to give a talk, please give us a (tentative) title, although
we can not guarantee that all the talks will be included in the
final programme.
Back to table of contents.
Last modified on May 2, 2007