Issue No 10


Workshop on Dynamical Systems
10–14 July

(A satellite meeting of the XIII International Congress in Mathematical Physics)

Konstantin Khanin (Isaac Newton Institute/Heriot-Watt University),
Robert MacKay (Warwick) and
Mark Pollicott (Manchester)

Scientific Committee:
S Aubry (France),
L Bunimovich (USA),
J-P Eckmann (Switzerland),
H Eliasson (Sweden),
M Herman (France),
A Kupiainen (Finland),
J Palis (Brazil),
D Ruelle (France),
L Shilnikov (Russia),
Ya Sinai (Russia),
M Viana (Brazil),
L-S Young (USA)

Supported by:
The Engineering and Physical Sciences Research Council, The European Science Foundation Programme on Probabilistic Methods in Non-Hyperbolic Dynamics (PRODYN), and The London Mathematical Society.

The topic of dynamical systems is a frontier area between mathematics and physics. It is closely connected with the theory of differential equations, ergodic theory, probability theory and other branches of mathematics. It also has many applications in Hamiltonian mechanics, nonlinear optics and theory of turbulence. The last two decades have seen a period of intensive development of the theory of dynamical systems in connection with the concept of dynamical chaos and the methods created have proved very useful for the analysis of nonlinear phenomena in nature and society.

The motivation of this Workshop was to examine the mathematical problems in dynamical systems, map out possible directions in the development of the theory in the near future and illustrate some applications in mathematical physics. The Organisers invited over 50 experts, including many from Europe who were supported by the European Science Foundation Programme.

The meeting had somewhat non-standard structure, designed to facilitate fruitful discussion and maximise the involvement of the participants. The bulk of the meeting was made up of sessions on specially selected topics, chosen to cover the most promising areas of the modern theory of dynamical systems. Each session started with a review talk summarising the current state of the subject. This was followed by discussion sessions organised and chaired by one of the key people in the field who, in advance of the meeting, asked other participants to prepare short presentations on the most important open problems in the area. In this way, there was some pre-planned structure to this part of the meeting, but not so much as to preclude further contributions from other participants.

Having in mind the connection with the XIII International Congress in Mathematical Physics, there were also daily invited lectures on different applications of dynamical systems in mathematical physics and biology. The final discussion session of the workshop was entirely dedicated to applications of dynamical systems.

With a meeting as large as this one, and one that had so many younger researchers, the organisers felt it important to give everyone the opportunity to present their current work. This was provided for in the form of two poster sessions.

Scientific Programme

Monday, 10th July
Session 1 - Homoclinic bifurcations Review talks by Dmitry Turaev and Enrique Pujals
Discussion, chaired by Jacob Palis
Session 2 - Spatially extended dynamics Review talks by Serge Aubry and Jean-Pierre Eckmann
Discussion, chaired by Yakov Pesin

Tuesday, 11 July
Session 3 - Renormalisations Review talk by Sandy Davie
Discussion, chaired by Hans Koch
Session 4 - Hamiltonian Dynamics. Review talks by Michel Herman and Rafael de la Llave
Discussion, chaired by Sergey Bolotin

Wednesday, 12th July
Session 5 - Complex Dynamics Review talk by Jean-Christophe Yoccoz
Discussion, chaired by Mikhail Lyubich
Session 6 - Applications/Posters Hakan Eliasson on applications to Anderson localization
Jaroslav Stark on applications to life sciences Poster Session

Thursday, 13th July
Session 7 - Nonuniform hyperbolicity Review talk by Marcelo Viana
Discussion, chaired by Vivaine Baladi
Session 8 - Applications/Posters Oleg Kozlovski on applications to the dynamo problem
Jacques Laskar on applications to celestial mechanics Poster Session

Friday, 14th July
Session 9 - Applications Albert Libchaber on applications to molecular biology
Discussion, chaired by Colin Sparrow
Guest Lecture: Martin Kruskal on Surreal Number.

More than 100 people took part in the Workshop. The Organisers believe that this was the strongest grouping of such experts in the UK for over a decade. The majority of participants were from the UK and Europe, but there were also many from the USA, the FSU, South America, India and Japan. Around 25 were research students or at postdoctoral level.

The participants mostly represented pure and applied mathematics. Several participants working in theoretical physics and mathematical biology helped to insure an interdisciplinary profile of the meeting, which was one of the aims.

The main aim of the meeting was to get together the leading experts in the area of dynamical systems and young researchers, and to discuss the most interesting directions and open problems. In order to avoid parallel sessions and at the same time give everybody the opportunity to express their views, the Organisers developed a new format with a focus on discussion sessions. In the opinion of most of the participants this experiment was successful. Many of the discussion sessions were very active, deep and fruitful.

A number of open problems were discussed by the participants. Below are formulated just a few of them that are of general interest to specialists in nonlinear dynamics and mathematical physics.
  1. How to study transient behaviour especially in large systems where the recurrent regime might not be reached in a practical time?
  2. Do any non-compact tori of Nonlinear Schroedinger equation survive perturbation?
  3. Is it true that any critical circle homeomorphism with irrational rotation number is C1conjugate to pure rotation?
  4. Is non-uniform hyperbolicity typical?
  5. Can we find more general conditions for existence and/or uniqueness of SRB measures? For example, does almost everywhere boundness of Lyapunov exponents away from zero implies generic finiteness?
  6. Is there a useful shadowing lemma for very general non-uniformly hyperbolic systems?
  7. Does there exist a C* area-preserving diffeomorphism of the 2-sphere such that all periodic points are hyperbolic?
A number of interesting problems were also discussed in connection with applications of dynamical systems. In particular, many problems were formulated in the talk on applications to molecular biology by Albert Libchaber.

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Anantharaman, Nalini, Ecole Normale Superieure
Arrowsmith, D.K., Queen Mary & Westfield College
Artuso, Roberto, Universit dell’Insubria, Como
Aubry, Serge, CEA Saclay
Baesens, Claude, University of Warwick
Baladi, Viviane, Université Paris-Sud
Balint, Peter, Technical Universit of Budepest
Bandtlow, Oscar, University of Glasgow
Beck, Christian, Queen Mary & Westfield College
Bolotin, Sergey, Moscow State University
Briggs, Keith, BT Research Labs
Brindley, John, University of Leeds
Broer, Henk, University of Groningen
Bullett, Shaun, Queen Mary & Westfield College
Bunimovich, Leonid, Georgia Institute of Technology
Buzzi, Jerome, Ecole Polytechnique
Carleson, Lennart, Kungl Tekniska Hogskolan
Carletti, Timoteo, Institut de Mécanique Céleste
Carr, Jack, Heriot-Watt University
Chandre, Christel, CEA Saclay
Chen, Yi-Chiuan, University of Warwick
Contreras, Gonzalo, CIMAT
Davie, A M, University of Edinburgh
de la Llave, Rafael, University of Texas at Austin
Delshams, Amadeu, Universitat Politecnica de Catalunya
Dolgopyat, Dima, Penn State University
Dullin, Hoger R, Loughborough University
Eckmann, Jean-Pierre, University of Geneva
Eliasson, Hakan, Université Paris VI
Evanno, Vincent, University of Manchester
Fernandez, Bastien, Centre de Physique Theorique
Ferreira, Flavio, Instituto Politecnico do Porto
Friesecke, Gero, University of Oxford
Gelfreich, Vassili, Mathematical Institute, FU Berlin
Gentile, Guido, Universita di Roma Tre
Giberti, Claudio, University of Insubria
Gielis, Guy, University of Cambridge
Glutsuk, Alexy, IHES
Goldsheid, Ilya, Queen Mary & Westfield College
Gutkin, Eugene
Hall, Toby, University of Liverpool
Hayashi, Shuhei, University of Tokyo
Herman, Michel, University Paris VII
Holland, Mark, University of Warwick
Iturriaga, Renato, Newton Institute
Jenkinson, Oliver, Erwin Schrodinger Institute
Kadelburg, Vesna, University of Cambridge
Kaimanovich, Vadim, University of Rennes
Kaushal, Radhey S, University of Delhi
Khanin, Kostya, INI, Cambridge
Khmelev, Dmitri, Heriot-Watt University
Kifer, Yuri, Hebrew University, Jerusalem
Koch, Hans, The University of Texas at Austin
Kosygin, Denis, Princeton University
Kozlovski, Oleg, University of Warwick
Kuksin, Sergei, Heriot-Watt University
Lakner, Mitja, University of Ljubljana
Laskar, Jacques, IMC
Ledrappier, Francois, CMAT Ecole Polytechnique
Lenci, Marco, SUNY at Stony Brook
Lerman, Lev, Research Inst for Appl Math & Cybernetics
Levin, Genadi, The Hebrew University of Jerusalem
Libchaber, Albert, The Rockefeller University
Lindahl, Karl-Olof, Vaxjo University, Sweden
Litvak-Hinenzon, Anna, The Weizmann Institute of Science
Liverani, Carlangelo, Universita di Roma “Tor Vergata”
Lopes Dias, Joao, University of Cambridge Luchinsky,
Dmitri, University of Lancaster
Luzzatto, Stefano, UMIST
Lyubich, Mikhail, SUNY at Stony Brook
MacKay, Robert, University of Warwick
Mestel, Ben, University of Exeter
Moussa, Pierre, CEA/SACLAY
Nicol, Matthew, University of Surrey
Nowicki, Tomasz, Warsaw University
Oseledets, Valery, Moscow State University
Pacifico, Maria Jose, Cidade Universitaria Rio de Janiero
Paleari, Simone, Universita’ di Milano
Palis, Jacob, IMPA, Rio de Janiero
Penrose, Oliver, Heriot-Watt University
Pesin, Yakov, INI, Cambridge
Petek, Peter, Universityof Ljubljana
Pinto, Alberto, Universidade do Porto
Pollicott, Mark, Manchester University
Pujals, Enrique, Federal University of Rio de Janiero
Ruelle, David, Institute des Hautes Etudes Scientifique
Schenkel, Alain, University of Helsinki
Schmeling, Joerg, Freie Universitaet Berlin
Sharp, Richard, University of Manchester
Simo, Carles, University of Barcelona
Skapin-Rugelj, Meta, University of Ljubljana
Slijepcevic, Sinisa, McKinsey & Co
Slomczynski, Wojciech, Jagiellonian University
Sparrow, Colin, University of Cambridge
Stark, Jaroslav, University College London
Stirling, James, University of Crete
Szasz, Domokos, Technical University, Budpest
Takens, Floris, Mathematisch Institut
Todd, Michael, University of Warwick
Toth, Imre Peter, Technical University of Budapest
Treschev, Dmitriy, Moscow State University
Turaev, Dmitry, Weierstrass Inst
Urbach, David J., University of Geneva
van Strien, Sebastian, Warwick University
Vanneste, Jacques, University of Edinburgh
Viana, Marcelo, IMPA, Rio de Janiero
Walkden, Charles Peter, Manchester University
Yoccoz, Jean-Christophe, College de France

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