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.
Programme
The school will consist of 4 introductory courses:
 Sergei Gukov (Caltech): From Higgs bundles to knots and 3manifolds  Expand Abstract
Lecture notes: http://arxiv.org/abs/1211.6075.
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 3manifolds. Our goal in these lectures will be to review this relation, based on ChernSimons 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.
 Nigel Hitchin (Oxford): Spectral curves  Expand Abstract
Lecture slides
 Matrices and spinning tops
This will be an introduction to the algebraic curves and line bundles corresponding to conjugacy problems for ntuples of matrices and their use in solving equations of spinning top type, notably Nahm's equations.
 Spectral curves for monopoles and harmonic maps
This talk will focus on the way that spectral curves appear in the study of monopoles in ℝ^{3} and harmonic maps from a 2torus into a Lie group.
 Spectral curves for Higgs bundles
The lecture will describe spectral curves for a general Higgs bundle, which is a different context from the previous talks: the curve gives information about the moduli space but does not solve the equations.
 Spectral data and real forms
When we specialize to the moduli space of Higgs bundles corresponding to representations of a surface group into a real form, then the spectral data takes a particular form. This sometimes necessitates looking at nongeneric fibres of the Higgs bundle integrable system.
 François Labourie (Orsay): Cyclic surfaces and Hitchin components  Expand Abstract
Lecture notes
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: HitchinSimpson 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 ρequivariant 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.
 Peter Topping (Warwick): The Teichmueller harmonic map flow and minimal immersions  Expand Abstract
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 socalled Teichmueller harmonic map flow, which is a geometric flow that tries to find branched minimal immersions.
Likely topics that we will cover include:
 The harmonic map energy; its properties for maps from surfaces, and links with minimal immersions.
 The harmonic map flow of EellsSampson, and its theory in two dimensions. Bubbling.
 Uniformisation/geometrisation of surfaces.
 Teichmueller space from a differential geometric point of view.
 Teichmueller harmonic map flow  definitions and basic theory, asymptotics, singularities.
 Refined understanding of the space of holomorphic quadratic differentials, and applications.
Schedule

Monday 
Tuesday 
Wednesday 
Thursday 
Friday 
09:00  10:30 
12:00  Shuttle bus 
13:00  13:15  Registration 

Breakfast 
Breakfast 
Breakfast 
Breakfast 
10:30  11:30 
F. Labourie 
P. Topping 
10:30  11:30  F. Labourie 
11:30  11:50  Break 
11:50  12:50  N. Hitchin 
12:50  14:30  Lunch 
14:30  15:30  P. Topping 
15:30  15:45  Break 
15:45  16:45  N. Hitchin 
16:45  17:15  Break 
17:15  18:15  P. Topping 

N. Hitchin 
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 
18:00 

Spanish Wine 

20:00 
Dinner 
Dinner 
Social Dinner 
Dinner 
List of participants
Click here to view an updated list of the registered participants
Venue and accommodation
The meeting will take place at a Residence and Conference Hall called La Cristalera. This is about 2 Km from Miraflores de la Sierra, a small village located 49 Km North of Madrid.
Residencia La Cristalera
Address: Carretera a Rascafría, Km 10. 28792, Miraflores de la Sierra, Spain
Phone: +34 91497 6599 / 6598
Fax: +34 918444464
Email: residencia.cristalera()uam.es
Map
Road map
Participants will be accommodated in this hall of residence. Breakfast, lunch and dinner will be provided at La Cristalera for all the participants.
How to get to the venue of the school
Click here to obtain information on how to get to Miraflores de la Sierra and to the venue of the school.
Computer facilities
There are no computers in the venue of the school, but you may want to bring your laptop, since there is WiFi connectivity available throughout in Residencia La Cristalera.
Registration and financial support
The online application form is now closed. The application deadline was 31st March, 2014.
Organizing committee
Luis ÁlvarezCónsul (ICMAT)
Tomás L. Gómez (ICMAT)
Poster
Click here to download a school poster in A3 format (17Mb pdf file).
Sponsors
ICMAT Severo Ochoa Programme
GEAR
MODULI