This school is part of the  European Network ENIGMA.
Its main theme is quantum integrability.  Its program will be organized
around  four main lecturers delivering 3 lectures of 1.30h each:

In addition, seminars by participants will be organized and time for
discussions will be available.

Monday, 15 Introduction
Tuesday, 16 Colin de Verdière
Wednesday, 17 Di Francesco
Colin de Verdière
Thursday, 18 Beisert
Di Francesco
Friday, 19 Colin de Verdière
Di Francesco


Niklas Beisert

The study of Maldacena's AdS/CFT (string/gauge) correspondence has                                     
recently advanced a lot by making use of the apparent integrability                                    
of the models in the planar limit. Using integrable methods like the                                   
Bethe ansatz allows us to compute observables away from the natural                                    
(disjoint) perturbative regimes of the constituent gauge and string                                    
models and thus see the duality of the models explicitly.  
In my set of lectures I will start by giving an overview of the                                        
AdS/CFT duality. I will then introduce and review several aspects and                                  
tools of integrability in AdS/CFT such as: * The classical spectrum                                    
of string theory by means of spectral curves. * Spin chains and the                                    
Bethe ansatz. * Algebraic construction of the S-matrix.    
Lecture notes: Intro, AdsCFT, SU22, Curve.

Yves Colin de Verdière

An introduction to semi-classical analysis.

 I plan to cover some of the following topics:
-- microlocalization, pseudo-differential operators and Fourier
integral operators, star-products
-- Bohr-Sommerfeld quantization rules around critical levels
-- Semi-classical normal forms
-- Wigner measures and equipartition of waves

. Dimassi-Sjöstrand, Spectral asymptotics in the semi-classical limit. 1999
. Evans-Zworski, Lectures on semi-classical analysis,
London mathematical society lecture note series (268).
. San Vu Ngoc, Symplectic Technics for semi-classical integrable
systemes, http://www-fourier.ujf-grenoble/~svungoc/

Philippe Di Francesco

Borrowing examples from statistical physics, we address                                                         
various enumerative and algebraic combinatorial problems,                                                       
such as counting maps with marked points at fixed geodesic                                                      
distances, or exploring relations between alternating sign                                                      
matrices, plane partitions and the geometry of orbital                                                          
varieties, via a connection to lattice loop gas. We will                                                        
show how classical or quantum integrability                                                                     
inherited from the structure of the physical theories at hand                                                   
can be used to derive exact combinatorial results.     
Lecture notes: Combinatorics.

Tetsuji Miwa

Correlation functions of the XXZ model.    
The XXZ model is an exactly solvable model of quantum spin chain.                                                 
It contains a parameter Delta. When Delta=1, the model is called                                          
the XXX model, and Bethe invented the Bethe Ansatz method for                                                   
diagonalization of the XXX Hamiltonian. In this lecture I give a summary
of the recent developments by Boos, Jimbo Smirnov, Takeyama and myself
on the computation of  the correlation functions of the                                                                                 
XXZ model.

The basic ingredient is construction of certain   Grassmann
variables which act on the quantum space C^2 otimes ... otimes C^2
in terms of L operators. We obtain an algebraic formula for the
correlation functions. This is a generalization of                                              
the classical results that the nearest and the next nearest neighboor
correlation functions for  the XXX model is given  by using log2  and zeta(3).