Speakers and abstracts
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.
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.
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.
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.
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.