Program:

19 May, Monday, Morning session: Aula Dini*

19 May, Monday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***

20 May, Tuesday, Morning session: Aula Dini*

20 May, Tuesday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***

21 May, Wednesday, Morning session: Aula Dini*

21 May, Wednesday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***

22 May, Thursday, Morning session: Aula Dini*

22 May, Thursday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***

23 May, Friday, Morning session: Aula Dini*

23 May, Friday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*

Israel Michael Sigal ( Dept. of Mathematics, University of Toronto, Canada):

Soliton dynamics. I Singularity formation in the mean curvature flow

Abstract. Since its extensive investigation in 1965, the concept of soliton took deep roots in mathematics and had profound impact on such disparate fields as Differential Equations, Geometry and Topology. In this lecture I will discuss the solitons in context of mean curvature flow. This flow appears naturally in the motion of interfaces in material science, physics and biology and found its applications in topological classification of surfaces, among many other things. I will discuss recent results on formation of singularities under this flow. Here the notion of the soliton (shrinker or expander) plays a central role and understanding its stability is a key problem.

Soliton dynamics. II and III Vortices, vortex lattices and theta functions

In this talk, the solitons will appear as vortices and vortex lattices in the Ginzburg-Landau theory of superconductivity. This theory is based on the Ginzburg-Landau equations, a pair of coupled nonlinear equations for the macroscopic wave function (order parameter) and magnetic potential. (These equations appear also in abelean Higgs model with unknowns called the Higgs and gauge fields and, in general, serve as a paradigm for the description of a large class of phenomena in physics, material science and biology.) In this talk I will review recent rigorous results on the key solutions of these equations - the magnetic vortices and magnetic vortex lattices, their existence and stability.

Dmitry Pelinovsky (McMaster University, Canada. )

Analysis of nonlinear evolution equations with the help of integrability

Abstract: The mini-course will consists of three lectures:

1. Energy method in orbital stability of solitary waves

2. Backlund transformation and stability of a solitary wave in L^2

3. Inverse scattering and stability of multiple solitary waves in L^2

4. Higher-order conserved quantities and global existence in Sobolev spaces of higher regularity

Andrew Comech (Texas A&M University, USA):

On linear stability of solitary waves in nonlinear Dirac equation

Abstract: Solitary waves and Vakhitov-Kolokolov stability condition for NLS and for nonlinear Dirac. Jost solutions, Evanc functions, numerical results. Existence of solitary waves in nonlinear Dirac equation. Limiting absorption principle. Agmon's Appendices A and B. Carleman estimates by Berthier--Georgescu. Unique continuation principle for the Dirac operator.

Claude Zuily (Universite Paris-Sud, Orsay, France)

Cauchy theory for the gravity water wave system with rough data

Abstract: The goal of this course is to sketch the proof of a recent result, obtained in collaboration with Thomas Alazard and Nicolas Burq, which concerns the local existence of a solution for the Cauchy problem for the gravity water wave system in any dimension, when the data belong to a Sobolev space of low regularity. This course can be followed by phd students. Here are the main sections which will be developed. 1. Description of the equations. The Craig-Sulem-Zaharov formulation.The main result. 2. The Dirichlet- Neumann operator and its paralinearization. 3. The reformulation of the equations and the energy estimates. 4. End of the proof. 5. Description of several extensions and open problems.

Aula Dini*, you can find in Palazzo del Castelletto, Via del Castelletto 11, Pisa. The classroom is on the ground floor of the Palazzo del Castelletto, which is right opposite the Science Collection of the Library of the Scuola Normale Superiore.

Conference rooms in the Palazzo Puteano:
Lecture Room** (Room 2, ground floor)
Meeting Room*** (Room 3, ground floor)

These rooms are on the ground floor of the Palazzo Puteano, just past the main entrance. The lecture room has a traditional chalkboard and a videoprojector. The room can seat up to 30 people.