The primary goal of this trimester is to provide a state of the art description of the subject of interaction between dynamics and number theory accessible both to young mathematicians wishing to do research in this area and to professional mathematicians from both disciplines interested in learning some of the most recent developments in the field.

In the framework of the research programme, it will take place the Clay Research Summer School at the Centro di Ricerca Matematica "Ennio De Giorgi" (Pisa, Italy) June 11th to July 6th 2007

The Clay Mathematics Institute will conduct its annual summer school for 2007 at the De Giorgi Mathematics Research Center in Pisa, Italy, from June 11th to July 6th 2007. The topic is "Homogeneous flows, moduli spaces, and arithmetic." Designed for graduate students and mathematicians within five years of their Ph.D., the program is an introduction to the theory of flows on homogeneous spaces and moduli spaces. These flows define concrete dynamical systems whose complex and subtle behavior leads to a rich theory, many interesting problems and conjectures, and a wealth of applications to number theory and geometry, e.g., equidistribution, diophantine approximation, rational billiards, and automorphic forms.

The school will consist of three weeks of foundational courses and one week of mini-courses focusing on more advanced topics. During the first three weeks there will be three long courses, of 14 lectures each, by Alex Eskin and Dmitry Kleinbock, Jean-Christophe Yoccoz, Manfred Einsiedler and Elon Lindenstrauss.

Organizing Committee: Manfred Einsiedler (Ohio State University), David Ellwood (Clay Mathematics Institute), Alex Eskin (University of Chicago), Dmitry Kleinboc (Brandeis University), Elon Lindenstrauss (Princeton University), Gregory Margulis (Yale University), Stefano Marmi (Scuola Normale Superiore Pisa), Peter Sarnak (Princeton University), Jean-Christophe Yoccoz: (College de France), Don Zagier (Max Planck Institut, Bonn and College de France).

The Erwin Schrodinger Institute (ESI) will host a special semester devoted to AMENABILITY during the period end of February 2007 - mid-July 2007.

The organizers are:
Anna Erschler (Orsay, France) anna.erschler@math.u-psud.fr,
Vadim A. Kaimanovich (Bremen, Germany) v.kaimanovich@iu-bremen.de,
Klaus Schmidt (Vienna, Austria) klaus.schmidt@univie.ac.at

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. Dye 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 K-theory, 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, non-elementary amenable groups;

○ self-similar groups and iterated monodromy groups of rational maps;

○ graphed equivalence relations and amenability; L2 cohomology;

○ amenable groupoids; topological amenability of boundary actions;

○ amenability at infinity; Baum--Connes and Novikov conjectures;

○ amenability and rigidity; bounded cohomology;

○ amenable algebras;

○ quasi-isometric classification of amenable groups, geometricity of various group properties;

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

○ generalizations of amenability: A-T-menabilty (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 one Midwest Dynamical Systems Conference next academic year. It will be held at Indiana U. Purdue U. Indianapolis, Friday-Sunday, October 13-15, 2006.

The meeting will follow the traditional format. The first talk will start at 3pm Friday and the last talk will end by 12:30 pm Sunday.

The webpage of the meeting is http://www.math.iupui.edu/~mmisiure/mwds06/index.html

Organizers:

Will Geller <wgeller@math.iupui.edu>
Bruce Kitchens <bkitchens@math.iupui.edu>
Michal Misiurewicz <mmisiure@math.iupui.edu>
Rodrigo Perez <rperez@math.iupui.edu>

Organizers:

    Mikhail Kapranov (Yale University, USA)
    Sergiy Kolyada (Institute of Mathematics, Ukraine)
    Yuri Manin (MPIM, Germany)
    Pieter Moree (MPIM, Germany)
    Leonid Potyagailo (Universite de Lille 1, France)

 Alexander (Sasha) Reznikov (1960-2003) was a brilliant mathematician who died unfortunately very early. This conference in his remembrance focuses on topics Sasha made a contribution to.

In particular:

1. Hyperbolic, Differential and Complex Geometry.
2. Geometric group theory.
3. Three dimensional topology.
4. Dynamical systems.