# JISD2014

Twelfth

## - VENUE - Contents - Schedule - Seminars - Communications- Posters -

The twelfth edition of the WORKSHOP ON INTERACTIONS BETWEEN DYNAMICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS (JISD2014) will be held in Barcelona, June 16 - 20, 2014, at the Universitat Politècnica de Catalunya (UPC)

Lecturers of courses in former JISD editions

There will be four main courses of six hours each, some seminars, communications, and posters. The courses will be taught within the Master of Science in Advanced Mathematics and Mathematical Engineering (MAMME) of the UPC Graduate School.

• #### Supported by the Clay Mathematics Institute, FME, UPC, SCM, RSME, SEMA.

Organizers
- Xavier Cabré
- Amadeu Delshams
- Maria del Mar González
- Tere M. Seara

Scientific Committee
- Rafael de la Llave

- Alfonso Sorrentino

- Luis Silvestre

- Jacques Fejoz

- Enrique Pujals

- Jean-Michel Roquejoffre

- Sandro Salsa

Poster

Registration fee: 200 euros
There will be some *financial support* available for this edition.
Deadline to apply for financial support: March 31st (results will be notified on April 11th). (*)
Deadline to register: May 23rd.
Deadline to apply to give a communication/present a poster: March 31st (results will be notified on April 11th). (*)

## Contents

Courses will be held in the room S02 of the FME building (Facultat de Matemàtiques i Estadística), at C/ Pau Gargallo, n. 5 Barcelona, 08028.

 Course Abstract Regularity results in free boundary problemsAlessio Figalli (Univ. of Texas Austin) Syllabus Free boundary problems arise naturally in a number of physical phenomena. These problems consists in a couple \$(u,K)\$, where the function \$u\$ solves some PDE inside a domain whose boundary \$K\$ is unknown. The main challenge in these problems is to understand the regularity both of the solution and of the free boundary. In these lectures I will describe several examples of free boundary problems and show some methods to obtain regularity both for u and for K. Renormalization and Rigidity in Dynamics Konstantin Khanin (Univ. of Toronto) Syllabus Renormalization is one of the main tools in the modern theoryof dynamical systems. It started in the late 70's  with the seminal work of M. Feigenbaum on sequences of period-doubling bifurcations. Later the mathematical theory was further developed by D. Sullivan, C. McMullen,M.Lyubich and A. Avila. However, despite a big progress in the last 20 years manyfundamental problems are remaining open. In this mini-course we shall discuss mostly the real analytic aspects of the renormalization method in the context of rigidity theory for circle maps with singularities, and nonlinear interval exchange transformations. Point systems with Coulomb interaction: from variational study to statistical mechanics Sylvia Serfaty (Univ.  Pierre et Marie Curie Paris 6) Syllabus I will describe works in which we perform a Gamma-convergence analysis of the energy governing Coulomb gases in all dimensions and log gases in dimension 1. Beyond the leading order "mean field limit", we are able to derive a limiting “renormalized” energy, corresponding to the total Coulomb energy of an infinite “jellium”, computed on the microscopic patterns formed by the points, and inspired by the analysis of Ginzburg-Landau vortices. This renormalized energy is conjectured to be minimized by certain crystalline configurations. Equidistribution of points and energy can be proven for ground states of the energy. We give applications to the statistical mechanics problem of considering states with temperature: we can deduce a next order expansion of the partition function and crystallization-type results. This is based on joint works with Etienne Sandier, Nicolas Rougerie, Simona Rota Nodari, and Thomas Leblé. Parabolic trajectories, collisions and regularization in the variational approach to the n-body problem. Susanna Terracini (Università di Torino) Syllabus In several recent papers, the symmetries of the \$n\$-body equation have been exploited in order to find new periodic solutions using a variational principle with symmetries in time and space. Such solutions are the natural generalization of the relative equilibrium motions, the well known periodic orbits for the classical problem. The variational approach to the search of periodic solutions consists in finding critical points of the associated action or Maupertuis functionals. Though the presence of singularities has to be held responsible of the hardest difficulties in finding such critical points, it is also the ultimate cause of their existence. An intriguing aspect of the variational approach to the periodic \$n\$-body problem is that it involves the study of a variety of issues: analytical, algebraic, topological and computational. Sharp level estimates on colliding trajectories are possible when central collisions are known and classified (as in the three-body problem), and so in this field the study of central configurations (relative equilibria) plays a key role. Other ideas and results which are basic for this line of research are suitable regularization theorems and the analysis of colliding trajectories by asymptotic or topological methods, classification theorems for the symmetry groups and related local/global variations for colliding trajectories. We shall focus on collision trajectories and regularization. A special attention will be devoted to the study of parabolic trajectories and their variational characterization.

The JISD 2014 are supported by:

EMS

Clay Math. Inst.

RSME

SCM

(*) For further details, please contact:

xavier.cabre at upc.edu, amadeu.delshams at upc.edu, mar.gonzalez at upc.edu, or tere.m-seara at upc.edu

12-06-2014 - RMC