Representations of Surface Groups

1 -- 5 September 2008 in Luminy-France [Aim and scope] [Organisation] [Pratical Information] [Support and Registration] [Participants] [Program] [References] [Pictures] [Poster] Aim and scopeThe principal subject of this conference is the moduli spaceScientific landscape:M(S,G)of representations in a Lie groupGof the fundamental group of a surfaceS. Roughly speaking, in the previous decades, these moduli spaces have shown two distinct but parrallel behaviour, depending on the groupG.

- When
Gis compact, for exampleSU(2), the moduli space is compact and the mapping class group acts ergodicallly. A choice of a complex structure onSidentifiesM(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)whenG=SL(2,R)which acts by orientation-preserving isometries of the hyperbolic plane. The groupsSU(2)andSL(2,R)are at opposite sides of the spectrum --- as the two real forms ofSL(2,C), which, itself, acts by isometries on the hyperbolic 3-space. From the viewpoint of moduli spaces, Teichmüller space is fundamentally different thanM(S,G)whenGis compact: it is is contractible, the mapping class group acts properly, the quotient has a holomorphic interpretation as theRiemann 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 these developments.

Specifically, the conference will focus on the following topics:Focus of the conference:

- The construction of Hitchin's connection over Teichmüller space related to
M(S,G)whenGis compact, conformal field theory, and finite dimensional unitary representations of the mapping class group.

- Positive and tropical geometry of
higher Teichmüller-Thurston spacesin 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.

.

[Top] Organisation

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 followingFormat:

- 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.
The structure will be flexible enough to encourage informal discussion outside of the lectures and working groups.

- End each day with a summary of the day's proceedings, and develop a plan for the next day.

Here is a more precise ProgramThe organisers are W. Goldman , F. Labourie and O. García-Prada whose emails you can find easily by googling.Organisers:

[Top]

Practical informationThe 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.Lodging:Marseilles - in French Marseille - has an international airport -- Code : MRS -- as well as fast train connections to Paris. Click here for more informationHow to get there:information is here .More

[Top] Registration and 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 based participants. Young searchers will be encouraged.Support:All participants who have previously "applied for participation" should have received a mail from CIRM asking them to actually register before July 7th. As a reminder, here is a link to this new registration form Such a registration is mandatory.Registration:

[Top]

Here are some references 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".

[Top]

They are here PicturesWe acknowledge so far the support of the CIRM , the ANR "Représentations de groupes de surface, the Math Department in Orsay, the GDR Géométrie, Dynamique et Représentations des Groupes , the NSF .Supporting institutions:

[Top] Registered Participants

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 ...

[Top]