Introduction

Transformation groups are basic objects in differential geometry,
algebraic geometry, Lie theory and algebraic and geometric topology
etc. One of the basic reasons for their importance is that symmetries are described by groups. The quotients of smooth manifolds by group actions are usually not smooth manifolds. On the other hand, if the actions of the groups are proper, then the quotients are orbifolds.

V-manifolds were first introduced by Satake in 1956 in the context of
locally symmetric spaces and automorphic forms. V-manifolds were reintroduced and renamed orbifolds by Thurston near the end of 1978 in connection with the Thurston geometrization conjecture on the geometry of three dimensional manifolds. Basically, orbifolds are locally quotients of smooth manifolds by finite groups. Besides arising from transformation groups, many natural spaces in number theory and algebraic geometry are orbifolds, for example, the Deligne-Mumford compactification of the moduli space of curves is a compact orbifold.
Recently, orbifolds have also found striking applications in algebraic
geometry and string theory such as the McKay correspondence.

At this summer school, experts on transformation groups, orbifold theory and related subjects will give comprehensive introductions to these important subjects.

Time：June 26－28，2008

First, I will intorduce basic notations and examples on transformation grops, such as topological groups, G-spaces. Then, topics I would like to cover are as follows:
One is on equivariant cohomology. This involves Borel construction, principal bundle, localization theorem etc. The next one is on the calculation of equivariant cohomology. This contains
two theories--Davis-Januszkiewicz theory and GKM theory. Finally, I would like to discuss the equivariant classification up to homeomorphism (diffeomorphism), cobordism, homotopy etc.

The topics I will cover are at the intersection of surgery and
orbifolds:First I will discuss L^2 signature invariants, and the
flexibility theorem for manifolds with torsion in their fundamental
groups (this will overlap with some of the lectures of Davis, but goes
in a completely different direction).

Then I will move on to the classification of group actions (building
on work that Yan will be explaining about stratified spaces), if there
is time I will discuss replacement theorems, but in any case I will
close with our current knowledge of counterexamples to the equivariant
Borel conjecture.

I will give two lectures on "Morphism between orbifolds"
and two lectures on "Chen-Ruan cohomology of orbifold". If there
is further interest, I can discuss then.

Feng Luo

Title:Mapping class groups of surfaces

The mapping class group of a surface is the group of homotopy
classes of self-homeomorphisms of the surface. It plays a key role
in low-dimensional topology and geometry.
In our lectures, we will cover the following topics in the
subject. These include: the work of Dehn and Lickorich on
generating the mapping class group by Dehn twists, the work of
Dehn-Nielsen on identifying the mapping class group with the outer
automorphisms of the fundamental group. Also, if time permits, we
will prove Thurston's classification of the mapping class group
elements.

The lecture is aimed at graduate students with some background in
topology/geometry.

Basic reference:

1. Casson, Andrew; Bleiler, Steven A., Automorphisms of surfaces
after Nielsen and Thurston. London Mathematical Society Student
Texts, 9. Cambridge University Press, Cambridge, 1988. iv+105 pp.

2. Thurston, William P. On the geometry and dynamics of
diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19
(1988), no. 2, 417--431.

3. Travaux de Thurston sur les surfaces. (French) [The works of
Thurston on surfaces] Sinaire Orsay. Reprint of Travaux de
Thurston sur les surfaces, Soc. Math.\ France, Paris, 1979
[MR0568308 (82m:57003)]. Astisque No. 66-67 (1991). Soci
Mathatique de France, Paris, 1991. pp. 1--286.

4. Dehn, Max, Papers on group theory and topology. Translated from
the German and with introductions and an appendix by John
Stillwell. With an appendix by Otto Schreier. Springer-Verlag, New
York, 1987.

5. Lectures notes by Benson Farb.

(a) Terminology from transformation groups
(b) Orbifolds - first definitions
(c) 1 and 2 dimensional orbifolds
(d) Covering space theory for orbifolds
(e) Developability
(f) Euler characteristics of orbifolds
(g) Classification of 2-dimensional orbifolds

(a) Definition of groupoid
(b) Haeiger’s definition of an orbifold
(c) Classifying spaces

(a) Seifert fibered orbifolds
(b) Andreev’s Theorem
(c) Thurston’s Geometrization Theorem for orbifolds

(a) The basic construction
(b) The developing map
(c) Spherical and Euclidean groups
(d) Lanner’s classification of simplicial reflection groups
(e) Some hyperbolic examples
(f) Vinberg’s Theorem (no examples in high dimensions)

-- Basic definitions about orbifolds and transformation groups: the
orbifold fundamental
group, covering space theory for orbifolds, Euler characteristics of
orbifolds, orbifolds
of dimension 1, 2, 3.

-- Geometric reflection groups (spherical, Euclidean and hyperbolic):
Lanner's Theorem
and Andreev's Theorem

Abstract: The first two lectures will be the basics of the non-stratified
surgery theory, which can be applied to the manifolds with free group
actions. The third lecture will be a brief survey on some "classical"
equivariant surgery theory. In the last two lectures, I plan to talk about
the stratified surgery theory.

This is an introduction to the theory of Lie groupoids and
differentiable stacks. Topics covered include: Lie groupoids
and Lie algebroids, Morita equivalence, Hilsum-Skandalis morphisms,
nerve of groupoids and simiplical manifolds, cohomology, groupoid
extensions
and gerbes; and, if time permits, convolution algebras of groupoids,
cyclic homology of etale groupoids and Chern-Connes character of
twisted K-theory of orbifolds.

Title: Orbifolds and Group Cohomology

In these five lectures we will discuss basic constructions
for orbifolds and discuss their cohomology and K-theory.
Connections to cohomology and group actions will be a unifying
theme.

Topics to be covered include:
--definition and properties of orbifolds
--almost free actions
--orbifolds as groupoids
--cohomology of orbifolds
--calculations for toroidal orbifolds
--orbifold K-theory --periodic complexes and group actions

References: Adem, Leida & Ruan: "Orbifolds and Stringy Topology"
Adem & Davis: "Topics in Transformation Groups"

Title: Higher dimensional Thompson groups and the Haagerup property

Abstract: (joint work with Dan Farley) It is shown that Brin's higher
dimensional Thompson groups have the Haagerup property; that is, they
are a-T-menable in the sense of Gromov. This is accomplished by
showing that these groups have zipper actions, a condition that is
equivalent to acting properly on a space with walls. In turn, zipper
actions are constructed from geometric actions on finite products of
compact ultrametric spaces.