## Salomon Bochner Lectures in Mathematics, 2003-2004

**Rice University**

Department of Mathematics

# Nicolas Monod

University of Chicago

###
February 16, 2004 - February 27, 2004

Tuesday, February 17

4 PM - 5 PM Herman Brown 227

**From Geometric Group Theory to Measure Equivalence.
**
Whilst geometric group theory has become a very classical topic
during the two last decades, its relative, Measure Equivalence, is
a recent focus of interest. Geometric group theory proposes to consider
abstract groups as geometric objects, and to study algebraic properties
from the point of view of large-scale geometry. Measure equivalence, on
the other hand, aims at studying abstract groups through ergodic-theoretic
techniques. We will give an introduction to this programme.

Wednesday, February 18

4 PM - 5 PM Herman Brown 227

**Equivalence Relations on Probability Spaces.**
The structure of (measurable, countable) equivalence relations on
probability spaces has certain analogies with the theory of factors in
operator algebras. It is typically difficult to say much about the
"quotient" structure, that is, about the orbit space. It turn out that
there is a direct connection between these questions and measure
equivalence; we will present recent results in this direction, using
geometric tools.

Friday, February 20

4 PM - 5 PM Herman Brown 227

**Metric Spaces of Negative Curvature and Boundaries.**
We will present the basic notions of negative curvature for general
metric
spaces: Gromov-hyperbolicity, CAT(-1) geometry. The boundary at infinity
of such spaces is particularly useful for the understanding of groups of
isometries. On the other hand, we will consider random walks and explain
how the associated Poisson boundary relates to the boundary of the metric
space.

Tuesday, February 24

4 PM - 5 PM Herman Brown 227

**Bounded Cohomology.**
We will give a short and application-oriented introduction to
bounded
cohomology, focusing on the tools that can be used in rigidity questions.

Tursday, February 26

4 PM - 5 PM Herman Brown 227

**Superrigidity for Actions on Negatively Curved
Spaces.**
We will present a cohomological viewpoint for group actions on
negatively
curved spaces. Amongst the applications, we will (1) prove superrigidity
statements for actions of lattices on such spaces; (2) indicate how to use
this viewpoint for the ergodic-theoretical questions addressed in the
first lectures.

There will be a tea preceding each lecture
at 3:30PM in the Mathematics Commons Room, Herman Brown 438.

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