### Speakers and abstracts

**Yan Soibelman**

**Wall-crossing, Mirror Symmetry, Donaldson-Thomas Invariants and Integrable Systems**

Complex integrable system is, roughly speaking, a complex symplectic
manifold endowed with a map to another complex manifold such that
non-degenerate fibers are holomorphic Lagrangian compact complex tori
(e.g. abelian varieties).

My course will be mainly devoted to complex integrable systems
with central charge. This class contains, e.g. Hitchin integrable
systems (possibly with singularities) as well as Seiberg-Witten
integrable systems.

I am going to discuss integrable systems from the point of
view of Mirror Symmetry and theory of Donaldson-Thomas invariants
developed jointly with Maxim Kontsevich. In particular, I will explain
how the base of Hitchin integrable system encodes the number of special
Lagrangian submanifolds in a certain non-compact Calabi-Yau 3-fold.

I am going to explain how the formalism of wall-crossing
formulas introduced by Kontsevich and myself is related to WKB
asymptotics of differential equations with small parameter.

The unifying concept for the course will be the one of
wall-crossing structure.I am going to define it and illustrate in
examples which include representation theory of quivers, cluster
algebras, complex integrable systems and Mirror Symmetry.

**Sjamaar Reyer**

**Integrable Systems and Connections with Toric Geometry**

Starting with the Arnold-Liouville theorem I will survey the
structure results for integrable systems on symplectic manifolds due to
Duistermaat, Dazord-Delzant, and others, and explore their connections
with toric varieties and multiplicity-free actions of Lie groups.

**Vadim Kaloshin**

**KAM theory**

We present several versions of Kolmogorov-Arnol'd-Moser
Theorem. Then we sketch two different proofs. One is due to
Levi-Moser, another is due to Fejoz. Finally, we show applications of
KAM to several models nearly Integrable systems such as Newtonian N
body problem, Fermi-Pasta-Ulam model and others. Time permitting, we
discuss connections with solutions to Hamilton-Jacobi equation.

**Milen Yakimov**

**Integrable Systems and Poisson Lie Groups**

One the main techniques to construct integrable systems and to
analyze their dynamics is through Lie theory. The first part of the
mini-course will build up the fundamentals of the theory of Lie
bialgebras and Poisson Lie groups from a geometric and
representation-theoretic perspective. This will lead to a detailed
description of the geometry of the Poisson structures on simple Lie
groups coming from the Belavin-Drinfeld classification. The second part
of the mini-course will be an overview of the major applications of
these techniques to the construction and study of integrable
systems.

**Michael Hitrik**

**Introduction to Semiclassical Analysis**

The origins of the semiclassical analysis lie with the correspondence
principle, formulated by Bohr in 1923. It states that the behavior of
quantum mechanical systems should reduce to classical physics in the limit
of large quantum numbers. From the mathematical point of view,
semiclassical analysis can be viewed as the study of linear and non-linear
PDE depending on a small parameter, and as such, it has developed
tremendously in the last thirty years, following the introduction of
microlocal analysis, that is local analysis in phase space, simultaneously
in the space and Fourier transform variables. The purpose of this
mini-course is to provide an introduction to the subject of semiclassical
analysis and to attempt to illustrate some of its inner workings. I hope
to be able to discuss the following topics:

(i) Local symplectic geometry.

(ii) Construction of approximate solutions, the WKB method.

(iii) The method of stationary phase and semiclassical pseudodifferential
operators.

(iv) Local canonical transformations and

-Fourier integral operators.

(v) Spectra of Schr\"odinger operators in the semiclassical limit.