The main goal of this school is to introduce graduate students, postdocs and senior researchers to several active fields of current research in the theory of moduli spaces and their interplay with geometry, topology and theoretical physics. The school is organized within the ICMAT Research Term on the Geometry and Physics of Moduli Spaces.
The school will take place at Residencia La Cristalera in Miraflores de la Sierra (Madrid, Spain). The school will start with a lunch on Monday 19th of May and will end by lunch time on Friday 23rd of May.
The school will consist of 4 introductory courses:
The generalized volume conjecture relates holomorphic Lagrangian submanifolds (or, put differently, (A,B,A) branes) in Hitchin moduli spaces with quantum group invariants of knots and 3-manifolds. Our goal in these lectures will be to review this relation, based on Chern-Simons gauge theory with complex gauge group, and to see how it explains known facts and predicts new ones.
In particular, since many quantum group invariants of knots can be categorified to homological invariants, one may wonder whether the generalized (or quantum) volume conjecture admits a natural categorification. I will argue that the answer to this question is "yes" and introduce a certain deformation of the holomorphic Lagrangian submanifold in the Hitchin moduli space that completely describes the "color behavior" of the HOMFLY homology. This deformation is strong enough to distinguish mutants, and its most interesting properties include relation to knot contact homology and knot Floer homology.
In this minicourse, I will first recall basic concepts about Lie (with a glimpse on root systems theory) symmetric spaces, harmonic mappings and Higgs bundles.
I will then state the major results of the subject: Hitchin-Simpson and Corlette result. I will explain the Hitchin fibration and the Hitchin section. I will mainly deal with the case of SL(n,ℝ) but if time permits, I will explain Kostant theory of principal SL2 and the construction of the Hitchin section in full generality.
After these very long and hopefully pedagogical preliminaries, I will finally come to a new result. Namely, as a generalisation of results for SL(3,ℝ) (proved independently by the author and Loftin) and for SL(2,ℝ)×SL(2,ℝ) (proved independently by the author and Schoen), I will explain that for any rank 2 semisimple split real group G and any Hitchin representation ρ in G, there exists a unique ρ-equivariant minimal surface in the corresponding symmetric space. As a corollary all the corresponding Hitchin components are parametrised, in a mapping class group invariant way, by a pair (J, Q), where J is a complex structure on the surface and Q a holomorphic differential. A weaker result subsists for general cyclic Higgs bundles as introduced and studied by Baraglia.
I hope in these lectures to give an introduction to some core topics in differential geometry and geometric analysis, and to combine them to give a tour of the so-called Teichmueller harmonic map flow, which is a geometric flow that tries to find branched minimal immersions.
Likely topics that we will cover include:
|09:00 - 10:30||
|10:30 - 11:30||F. Labourie||P. Topping||
|11:30 - 12:00||Break||Break||Break|
|12:00 - 13:00||F. Labourie||S. Gukov||N. Hitchin|
|13:00 - 15:00||Lunch||Lunch||Lunch||Lunch|
|15:00 - 16:00||F. Labourie||S. Gukov||P. Topping||
|16:00 - 16:30||Break||Break||Break|
|16:30 - 17:30||F. Labourie||S. Gukov||S. Gukov|
Click here to view an updated list of the registered participants
Participants will be accommodated in this hall of residence. Breakfast, lunch and dinner will be provided at La Cristalera for all the participants.
Click here to obtain information on how to get to Miraflores de la Sierra and to the venue of the school.
There are no computers in the venue of the school, but you may want to bring your laptop, since there is Wi-Fi connectivity available throughout in Residencia La Cristalera.
The online application form is now closed. The application deadline was 31st March, 2014.
Click here to download a school poster in A3 format (17Mb pdf file).