| ## Centre Emile Borel |

I. Introduction to Superrigidity

II. Orbit Equivalence in Ergodic Theory

III. Measurable Group theory

IV. Actions on manifolds

The general theme of these lectures is the "rigidity" that a structure of a group can impose on its actions in various settings: linear representations, actions on manifolds, orbit structures of measurable actions etc. Our goal is to describe the general landscape of the questions, results and (some of the) ideas which appear in this area.

**Paul Baum:**
*Cosheaf homology*

Given a local system on a topological space X
(i.e. a representation of the fundamental groupoid of X)
both the homology and the cohomology of X with coefficients in the local system are defined. This talk considers what happens when the defining conditions for a local system are relaxed to obtain sheaves and cosheaves. With sheaves one can do cohomology and with cosheaves homology. Cosheaves arise naturally from group actions on topolgical spaces, and enter into Chern character problems relevant to the Baum-Connes conjecture. Recent results of C. Voigt relate cosheaf homology to Bredon homology and to periodic cyclic homology.

**Moulay-Tahar Benameur:**
*Homotopy invariance of Higher signatures in Haefliger cohomology*

We shall explain in this talk some recent results obtained in
collaboration with James Heitsch and related with index theory for
foliations. We shall focus on the definition of the leafwise signature
in Haefliger cohomology and sketch a new geometric proof of the leafwise
homotopy invariance of higher signatures with coefficients in leafwise
flat bundles.

**Paulo Carrillo:** *Analytical indices for Lie groupoids*

For a Lie groupoid there is an analytic index morphism taking
values in the K-theory of the C*-algebra associated to the groupoid. This
is a good invariant, but extracting numerical invariants from it, with the
existent tools, is very difficult. In this talk, we will explain how to
define another analytic index morphisms associated to the Lie
groupoid. These ones take values in some groups that allow us to do
pairings with cyclic cocycles. We obtain some abstract index formulas.

**Claire Debord:**
*Poincaré Duality for stratified pseudomanifolds*

We associate to a stratified pseudomanifold X a
differentiable groupoid T^{S}X which plays the role of the tangent
space to X. We construct a Dirac element. Thanks to a recursive
process on the depth of the stratification together with the stability
of the constructions which are involved we show that the Dirac element
induces a K-duality between the C*-algebras C*(T^{S}X) and C(X).

**Alexander Gorokhovsky:**
*Deformation quantization of gerbes on etale groupoids*

This is a joint work with P.Bressler, R. Nest and B. Tsygan.
We will discuss problems in which formal deformations of etale groupoids
and gerbes on them arise and
give an explicit description of the differential graded Lie algebra
which controls this deformation theory.
Deformation quantization of gerbes on etale groupoids,

**Eli Hawkins:**
*A groupoid approach to quantization*

I define a notion of "polarization" for Lie groupoids. Using
polarized symplectic groupoids, I present a general strategy for
constructing C*-algebras that quantize Poisson manifolds. This
unifies previous constructions including classical geometric
quantization of a symplectic manifold and the C*-algebra of a source-
connected Lie groupoid.

**Adrian Ioana:**
*Orbit inequivalent actions for groups containing a copy of F_2*

I will prove that any countable discrete group G which contains
a copy F_2 admits uncountably many non orbit equivalent actions.

**Steven Hurder:**
*Index theory and LS category for Riemannian foliations*

**Alex Kumjian:**
*Fell bundles associated to groupoid morphisms*

Given a continuous open surjective morphism $\pi :G \to H$ of \'etale
groupoids with amenable kernel, we construct a Fell bundle $E$
over $H$ and prove that its C*-algebra $C^*_r(E)$ is isomorphic to
$C^*_r(G)$. This is related to results of Fell concerning C*-algebraic
bundles over groups. The case $H=X$, a locally compact space, was treated
by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to
a crossed product, the C*-algebra of a Fell bundle arising from an action
of the groupoid $H$ on a C*-bundle over $H^0$. We apply the theory to
groupoid morphisms obtained from extensions of dynamical systems and from
morphisms of directed graphs with the path lifting property.
This is joint work with Valentin Deaconu and Birant Ramazan.

**Klaas Landsman and Rogier Bos:**
*Continuous representations of Lie groupoids*

The notion of a representation of a locally compact
groupoid G with Haar system on a measurable field of Hilbert spaces
has been developed by Jean Renault and has the advantage of yielding
a connection with the representation theory of the associated C*-
algebra C*(G). However, in the case of Lie groupoids it is worth
studying representations on continuous fields of C*-algebras (i.e.
on Hilbert C*-modules over C_0(M), where M is the base space of G).
Using ideas from geometric quantization, we present a method to
construct such representations, as well as an associated
"quantization commutes with reduction" theorem in case that G is proper.

**Jean-Marie Lescure:**
*An index theorem for conical pseudomanifolds*

We define an analytical index map and a topological index map for
conical pseudomanifolds. These constructions generalize the analogous
constructions used by Atiyah and Singer in the proof of their index
theorem for a smooth, compact manifold M. A main ingredient is a
non-commutative algebra that plays in our setting the role of
C_0(T*M). We prove a Thom isomorphism between non-commutative
algebras which gives a new example of wrong way functoriality in
K-theory. We then give a new proof of the Atiyah-Singer index
theorem using deformation groupoids and show how it generalizes to
conical pseudomanifolds. We thus prove a topological index theorem for
conical pseudomanifolds.

**Ieke Moerdijk:**
*Subgroupoids of Lie groupoids*

**Bertrand Monthubert:**
*Boutet de Monvel's Calculus and Groupoids*

Can Boutet de Monvel's algebra on a compact manifold with
boundary be obtained as the algebra Psi^0(G)
of pseudodifferential operators on some Lie groupoid G?
If it could, the kernel {\mathcal G} of the principal symbol homomorphism
would be isomorphic to the groupoid C*-algebra C*(G).
While the answer to the above question remains open, we exhibit a groupoid
G such that C*(G) possesses an ideal I isomorphic to
{\mathcal G}.
In fact, we prove first that {\mathcal G}\simeq\Psi\otimes{\mathcal K}
with the C*-algebra
\Psi generated by the zero order pseudodifferential operators on the
boundary and the
algebra $\mathcal K$ of compact operators.
As both \Psi\otimes {\mathcal K} and I are extensions of
C(S*Y)\otimes {\mathcal{K}} by {\mathcal{K}} (S*Y is
the co-sphere bundle over the boundary) we infer from
a theorem by Voiculescu that both are isomorphic.

**Victor Nistor:**
*A topological index theorem for
manifolds with corners*

We show that the analytic and
topological index for the groupoid G(M)
associated to a compact manifolds with
corners coincide. When the faces of M are
contractible, these indices are isomorphisms
from K^*(TM) to K_*(G(M)) := K_*(C^*G(M)).
This is joint work with Bertrand Monthubert.

**Alan L. T. Paterson:**
*The E-theoretic descent functor for groupoids*

The descent functor enables us to go from equivariant asymptotic
morphisms to asymptotic morphisms of crossed product C*-algebras.
It is important in a number of contexts, in particular for the
Baum-Connes conjecture and the topological index. The functor
for locally compact groups was established in their memoir by
Guentner, Higson and Trout, and the talk will discuss what can
be said in the groupoid case. (Earlier work on this was done
by R. Popescu in his thesis.) There seem to be technical
difficulties with establishing in complete generality the descent
functor for groupoids, but we prove its existence under certain
conditions which apply in a number of cases that arise in practice.

**Mikael Pichot:**
*The space of triangle buildings*

**Paolo Piazza:**
*Foliated rho-invariants*

I will present some ongoing work in collaboration with Moulay Benameur
about the definition of a foliated rho invariant and the proof of some
of its stability properties. This invariant generalizes to measured
foliations the classical Cheeger-Gromov rho-invariant on Galois coverings.
I will first recall work of Keswani and Piazza-Schick dealing with the
homotopy invariance of the Cheeger-Gromov rho invariant on Galois
coverings under a Baum-Connes assumption on the maximal
group C*-algebra of the covering group; I will then move on and
explain how these results can be generalized to measured foliations,
assuming the bijectivity of the Baum-Connes map for foliations.
I shall also explain how index theoretic considerations play a crucial
role throughout.

**Sorin Popa:**
*On the superrigidity of group actions in ergodic theory and
von Neumann algebras*

I will explain how the unlikely combination
of deformation and rigidity properties of a measure preserving
action of a group on a probability space can make it recognizable by
merely knowing the orbit equivalence class of the action,
or just its von Neumann algebra.

**Ian Putnam:**
*A homology theory and C*-algebras for chaotic dynamical systems*

Smale spaces were defined by David Ruelle as an abstract
approach to the chaotic dynamical systems in Smale's program. He also
described how operator algebras may be constructed from such systems. We
will describe a kind of homology theory for these systems and how it may
be used to compute the K-theory of the C*-algebras.

**Anton Savin:**
*Homotopy classification of elliptic operators on manifolds with corners*

This is joint work with V.E. Nazaikinskii and B.Yu. Sternin.
We compute the group of stable homotopy classes of elliptic operators
on manifolds with corners. It turns out that this group is isomorphic
to the analytic K-homology group of a certain explicitly constructed
C*-algebra.

**Stefaan Vaes:**
*Computations of automorphisms and finite index
subfactors of certain II_1 factors*

We present the first concrete examples of II_1 factors without
non-trivial finite index subfactors. We also present relatively
easy examples of II_1 factors with trivial outer automorphism group.

**Stéphane Vassout:**
*Uniform non amenability and the first l^2 Betti number*

In this talk, I will expose a recent work in collaboration with
Mikael Pichot. We define uniform Cheeeger isoperimetric constant for
a given finitely generated r-discrete measured groupoid of type
II_1, and show that it is bounded below by the first Betti number.
In particular we derive an invariant for equivalence relations and
define a notion of ergodic uniform non amenability for finitely
generated groups which is weaker than the usual one.