Its main theme is quantum integrability. Its program will be organized

around four main lecturers delivering 3 lectures of 1.30h each:

- AdS-CFT:
*Niklas Beisert* - Semi-Classical Methods:
*Yves Colin de Verdière* - Combinatorics:
*Philippe Di Francesco* - Correlation Functions:
*Tetsuji Miwa*

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

discussions will be available.

9h30-11h | 11h30-13h | 14h30-16h | 16h30-17h30 | ||||

Monday, 15 | Introduction | Miwa | Beisert | Seminar | |||

Tuesday, 16 | Colin de Verdière | Beisert | Miwa | Seminar | |||

Wednesday, 17 | Di Francesco | Colin de Verdière | Free | Free | |||

Thursday, 18 | Beisert | Miwa | Di Francesco | Seminar | |||

Friday, 19 | Colin de Verdière | Di Francesco | Leave | Leave |

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.

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

References:

. Dimassi-Sjöstrand, Spectral asymptotics in the semi-classical limit. 1999

. Evans-Zworski, Lectures on semi-classical analysis,

London mathematical society lecture note series (268).

http://math.berkeley.edu/~zworski/

. San Vu Ngoc, Symplectic Technics for semi-classical integrable

systemes, http://www-fourier.ujf-grenoble/~svungoc/

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.

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).