In the second half of the 20th century, Kolmogorov made the striking discovery that quasi-periodic motion is robust and prevalent from a probabilistic (or measure theoretical) sense, in a number of physical situations, including the quasi-integrable Hamiltonian systems. Indeed, KAM Theory (after Kolmogorov-Arnol’d-Moser) established the persistence of most of the invariant quasi- periodic tori of quasi-integrable Hamiltonian systems.
The proof is based on an inductive scheme of successive conjugations of the perturbed system to the integrable one on that part of the phase space where Diophantine frequencies occur. Through this scheme, the a priori possibly damaging effect of small divisors on the conjugacy at each step of the induction, is tamed out by the quadratic speed of convergence of the scheme.
Since this fundamental discovery the quadratic conjugation scheme of KAM theory was applied in numerous settings to prove stability and rigidity results in dynamics.
The main objective of the school is to give a panorama of some of the most important results and applications of the classical KAM theory as well as some of the new directions of research in various areas of dynamics that pushed the theory way beyond its classical limits.
Topics and speakers
This summer school will consist of 6 mini-courses of 5 hours each, accessible to PhD students and young researchers, as well as presentations of recent results in the domain.
Mini-courses:
- Modern developments of classical KAM theory (Luigi Chierchia, Univ. Roma 3) -Hamiltonian PDEs and KAM in infinite dimension (Michela Procesi, Univ. Roma 3) -Quasi-periodic Schrödinger operators (Raphaël Krikorian, Ecole Polytechnique) -Rigidity of higher rank algebraic actions (Danijela Damjanovic, KTH Stockholm, and Bassam Fayad, Univ. Maryland) -Coexistence between KAM and instability (Abed Bounemoura, CNRS, CEREMADE) -Liouvillean constructions and inverse KAM theory (Masha Saprykina, KTH Stockholm)
Talks: Artur Avila (Univ. Zürich), Dario Bambusi (Univ. Milan), Pierre Berger (CNRS, IMJ-PRG), Massimiliano Berti (SISSA, Trieste), Spencer Durham (Univ. Maryland), David Fisher (Rice University), Lingrui Ge (Beijing University), Benoît Grébert (Univ. Nantes), Marcel Guardia (Univ. Politecnica de Catalunya), Svetlana Jitomirskaya (UC Irvine), Konstantin Khanin (Univ. Toronto), Philipp Kunde (Jagiellonian University in Krakow), Jessica Massetti (Univ. Roma 3), Carlos Matheus (CNRS, CMLS), Jaime Paradela Diaz (Univ. Maryland), Laurent Stolovitch (CNRS, LJAD), Frank Trujillo (Univ. Zürich), Dmitry Turaev (Imperial College London), Zhenqi Wang (Michigan State University), Jiangong You (Nanjing University), Qi Zhou (Nankai University)
Scientific committee: Dmitry Dolgopyat (Univ. Maryland), Hakan Eliasson (Univ. Paris Cité), Svetlana Jitomirskaya (Univ. California Irvine), Federico Rodriguez-Hertz (Penn State Univ.), Tere Seara (Univ. Politecnica de Catalunya)
Organizing committee: Pierre Berger (CNRS, IMJ-PRG), Claire Chavaudret (Univ. Paris Cité), Bassam Fayad (Univ. Maryland), Jacques Féjoz (Univ. Paris Dauphine), Claire Lavollay (Univ. Paris Cité), Patrice Le Calvez (Sorbonne Univ.)
Thanks to the local organizers Skander Charfi, Maxime Chatal, Odylo Costa and Victor Maeght.
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