Scientific landscape: The principal subject of this conference is the moduli space M(S,G)
of representations in a Lie group G of the fundamental group of a
surface S. Roughly speaking, in the previous decades, these moduli spaces have shown two distinct but parrallel behaviour, depending on the group G.
When G is compact, for example SU(2), the moduli space is compact and
the mapping class group acts ergodicallly. A choice of a complex structure on S
identifies M(S,G) with a space of holomorphic objects. This space is
the classical phase space for gauge theory in dimension 3, and its
quantisation has led to advances in several different mathematical subjects:
topological invariants of 3-manifolds, representations of quantum groups,
symplectic geometry and algebraic geometry.
The work of Thurston in the late 1970's underscored the importance of
hyperbolic geometry in 3-dimensional topology. In particular,
through uniformisation, Teichmüller space identifies as a connected
component of M(S,G) when G=SL(2,R) which acts by
orientation-preserving isometries of the hyperbolic plane.
The groups SU(2) and SL(2,R) are
at opposite sides of the spectrum --- as the two real forms of SL(2,C),
which, itself, acts by isometries on the hyperbolic 3-space.
From the viewpoint of moduli spaces, Teichmüller space is fundamentally
different than M(S,G) when G is compact:
it is is contractible,
the mapping class group acts properly,
the quotient has a holomorphic interpretation as
the Riemann moduli space .
Furthermore Teichmüller space itself
enjoys a rich geometry revealed by Thurston and his disciples.
From the viewpoint of physicists, Teichmüller space plays a prominent
role as the phase space for 2+1-dimensional quantum gravity.
Recently, and partially motivated by physicists, new
research perspectives and techniques have led to connections of the general topic of the conference with new areas
of mathematics. It is the purpose of this workshop to clarify
and disseminate some of the advances of the last few years resulting from
Focus of the conference:
Specifically, the conference will focus on the following topics:
The construction of Hitchin's connection over Teichmüller space related to M(S,G) when G is compact,
conformal field theory, and finite dimensional unitary representations of the mapping class group.
Positive and tropical geometry of
higher Teichmüller-Thurston spaces in the sense of
Fock and Goncharov, and infinite dimensional unitary representations of the mapping class group arising through cluster algebra.
Translation surfaces and moduli spaces of holomorphic quadratic differentials.
Hyperbolic geometry in low dimensions and 3-dimensional Lorentzian conformal geometry.
More than to report on the very last recent developments, the purpose of this
conference is to bring together researchers in different fields - complex geometry,
dynamics, cluster algebra, hyperbolic geometry -
to explore connections between these topics and explain to others the background of their subjects. Therefore, talks will be elementary and introductory.
We have not yet fixed the detailed program for the conference which will depend on the interests of the
participants, in the spirit of the last workshop at
the American Institute of Mathematics.
The format of a typical day is the following
Begin the day with 2 minicourses of 2-3 hours,
based on the themes above; so far J. Andersen, A. Goncharov and P. Hubert (in the order of apparition)
have agreed to expose these introductory materials
In the afternoon after 5 p.m, after a long pause devoted to informal discussions, break into smaller working groups devoted either to develop, explain the material
of the morning lectures or to present other aspects of representations of surface groups.
End each day with a summary of the day's proceedings,
and develop a plan for the next day.
The structure will be flexible enough to encourage
informal discussion outside of the lectures and
Here is a more precise Program
Lodging: The centre will accommodate -- food and lodging -- most, if not all, participants. The CIRM in Luminy is a restored Provencal mansion with a very good library and conference centre. It is located above the
gorgeous calanques close to Marseille and
Cassis. Do not forget to bring your swimsuit,
although, due to cold sources, bathing requires a little bit of courage. There are also beautiful hikes in the cliffs, but hiking shoes might be recommended.
How to get there: Marseilles - in French Marseille - has an international airport -- Code : MRS -- as well as fast train connections to Paris. Click here for more information
Support: We expect to cover local expenses of most participants. Thanks to our NSF support, we may hope to cover also some travel expenses for some of the US
Young searchers will be encouraged.
Here are some references
An overview of the subject by the organizers of the previous AIM conference.
A course on representations of surface groups with emphasis on the geometric structure. Still incomplete: no bilbliography
The webpage of A. WEINSTEIN where you find information about geometric quantization
and in particular his textbook with S. BATES.
The wiki maintained by G. WILKIN whose
user name is "surface" and password is "groups".
We expect about 70 participants.
The following list of participants maintained on the CIRM site updates those who have completed their registration through the registration form above. Those who are on this list should feel satisfied about this amazing administrative achievement, those not on this list should worry and ask Labourie what is going on ...