
Program:
19 May, Monday, Morning session: Aula Dini*
10:00  10:30 
Registration and opening of the School(together with coffee  break) 
10:30  11:15 
Lecture: Israel Michael Sigal 
11:30  12:45 
Lecture: Israel Michael Sigal 
19 May, Monday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***
14:30  15:15 
Lecture: Andrew Comech 
15:15  15:45 
Coffee  break and discussions 
15:45  16:30 
Lecture: Andrew Comech 
16:30  17:00 
Discussions: Define working groups of common interests, plan of the work. 
17:00  18:00 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto), Lecture Room (Room 2, ground floor, Palazzo Puteano) and Meeting Room (Room 3, ground floor, Palazzo Puteano). 
20 May, Tuesday, Morning session: Aula Dini*
9:30  10:15 
Lecture: Claude Zuily 
10:15  10:45 
Coffee  break and discussions 
10:45  11:30 
Lecture: Claude Zuily 
11:30  12:30 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto) 
20 May, Tuesday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***
14:30  15:15 
Lecture: Israel Michael Sigal 
15:15  15:45 
Coffee  break and discussions 
15:45  16:30 
Lecture: Andrew Comech 
16:30  18:00 
Working group (lecturer and instructor Andrew Comech). Aula Dini( Palazzo del Castelletto). Discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . 
21 May, Wednesday, Morning session: Aula Dini*
9:30  10:15 
Lecture: Alim Sukhtayev 
10:15  10:45 
Coffee  break and discussions 
10:45  11:30 
Lecture: Alim Sukhtayev 
11:30  12:30 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto) 
21 May, Wednesday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***
14:30  15:15 
Lecture: Israel Michael Sigal 
15:15  15:45 
Coffee  break and discussions 
15:45  16:30 
Lecture: Andrew Comech 
16:30  18:00 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto), Lecture Room (Room 2, ground floor, Palazzo Puteano) and Meeting Room (Room 3, ground floor, Palazzo Puteano). 
22 May, Thursday, Morning session: Aula Dini*
9:00  9:45 
Lecture: Claude Zuily 
9:45  10:15 
Coffee  break and discussions 
10:15  11:00 
Lecture: Claude Zuily 
11:15  12:00 
Lecture: Claude Zuily 
12:00  12:30 
Short discussions on the lectures 
22 May, Thursday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*, Lecture room** and Meeting room***
14:30  15:15 
Lecture: Dmitry Pelinovsky 
15:15  15:45 
Coffee  break and discussions 
15:45  16:30 
Lecture: Dmitry Pelinovsky 
16:30  18:00 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto), Lecture Room (Room 2, ground floor, Palazzo Puteano) and Meeting Room (Room 3, ground floor, Palazzo Puteano). 
Special session starting at 20:00 
Dinner for Participants 
23 May, Friday, Morning session: Aula Dini*
9:30  10:15 
Lecture: Dmitry Pelinovsky 
10:15  10:45 
Coffee  break and discussions 
10:45  11:30 
Lecture: Dmitry Pelinovsky 
11:30  12:30 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto) 
23 May, Friday, Afernoon session: lectures in Aula Dini, working in groups Aula Dini*
14:30  15:15 
Lecture: Andrew Comech 
15:15  15:45 
Coffee  break and discussions 
15:45  16:30 
Lecture: Andrew Comech 
16:30  17:30 
Working in groups: discussions on the lectures, possible open problems, ideas to solve them, talks and poster presentations . Places for discussionsAula Dini( Palazzo del Castelletto) 
17:30  18:00 
Closing of the School and plan for the work during the Workshop 
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 GinzburgLandau theory of superconductivity. This theory is based on the GinzburgLandau 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 minicourse 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. Higherorder 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 VakhitovKolokolov 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 BerthierGeorgescu. Unique continuation principle for the Dirac operator.
Claude
Zuily (Universite ParisSud, 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 CraigSulemZaharov 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.

