Issue No 11


Workshop CMI Symposium and EuroWorkshop on Hamiltonian Systems
23 May to 1 June 2001

Scientific Organising Committee:
Dario Bambusi (Milan),
Walter Craig (McMaster),
Sergei Kuksin (Heriot-Watt),
Eugene Wayne (Boston)

Supported by:
The Clay Mathematical Institute and the European Commission
(Framework 5)

This workshop built on an ICMS meeting on the same theme in 1999 and a similar format was followed: formal presentations in the mornings leaving the majority of the afternoons free for impromptu working seminars and discussions. Once more the organisers found that this format and the intimate atmosphere of 14 India Street made for a highly successful combination. This success was demonstrated when two lectures, by Eliasson and Groves, reported research which was initiated during discussions begun at the 1999 meeting.

Exceptionally, one day of the meeting was set aside for a series of colloquium lectures dedicated to the lives and works of Michael Herman and Jürgen Moser who had died since the last meeting. Both had been invited in 1999; Herman had come and had made an indelible mark on the proceedings but Moser had been unable to attend due to ill health. Talks in their memory were delivered at the Royal Society of Edinburgh by H Eliasson, A Chenciner, J Mather and J-C Yoccoz, all of whom had had close research associations with either Moser or Herman. These lectures were open to the general scientific community.

Elmer Rees and Walter Craig at the Herman-Moser Colloquium
Elmer Rees (left) and Walter Craig at the Herman–Moser Colloquium.

Several broad themes emerged during the meeting. The first was an increasing interest in and understanding of the dynamics of Hamiltonian systems in the presence of continuous spectra, such as arise when considering partial differential equations on unbounded domains. The continuous spectrum in the linearisation of these equations can lead to dramatically different behaviours from those which have been studied in finite-dimensional Hamiltonian systems, or indeed in infinite-dimensional Hamiltonian systems whose linearisation has discrete spectrum (such as arise from partial differential equations on bounded domains). The lectures by Deift and Weinstein emphasised these differences.

A second theme, involving recent results in celestial mechanics, was represented by the talk of Chenciner who described new periodic orbits in the N-body problems. Also Giorgilli and Chierchia lectured on possible extensions KAM and Nekhoroshev theories to include models and systems actually encountered in celestial mechanics.

A third topic to emerge was the extension of ideas from the KAM or Nekhoroshev theory of finite-dimensional dynamical systems to the study the behaviour of solutions of partial differential equations. There is still no general KAM theory for infinite-dimensional systems with anything like the range of applicability of the analogous finite-dimensional theory. There were lectures which covered both extensions and improvements of the existing KAM and Nekhoroshev results for PDEs , and the relationships between various current approaches. Lectures falling into one of other of these categories included those of Wayne, Deift, Eliasson, Schenkel and Bambusi. A significant conclusion of the talk by Eliasson was that two main approaches to proving KAM-like results in Hamiltonian PDEs , advocated by Kuksin and Pöschel and by Craig, Wayne and Bourgain, are essentially equivalent from a mathematical point of view, although one may be easier to implement than the other in a given context. A number of other talks, including those of You, Abad, Plotnikov and Giorgilli, were concerned with further aspects of the KAM problem in finite dimensions.

Finally, a topic current now and two years ago was the use of ideas from Hamiltonian systems in fluid mechanics. Irrotational motion of an incompressible, inviscid fluid is an infinite dimensional Hamiltonian system, and a number of lectures investigated this aspect of fluid motions. From this perspective but in different contexts, Craig and Groves both considered the existence of travelling-waves on water surfaces, and Kuksin was interested in the statistical properties of solutions of PDEs describing fluid moving under the influence of random forces, with a view to obtaining a better understanding of turbulence phenomena.

Although it is hard to summarise what exactly went on in a myriad of improvised and spontaneous meetings and discussions (some more, some less, formally organised), there was no doubt about the high level of constructive interaction between participants outside the formal lecture programme. The afternoon sessions were all well attended, but two will serve to illustrate the utility of our workshop-type arrangements. The first was a discussion of Riemann-Hilbert theory and homogenisation, led by Deift. It started as an attempt to understand the geometrical significance of KAM ideas of Deift and Zhou for perturbations of the integrable nonlinear Schrödinger equations. But it developed into an attempt to understand the relationship of this work with other research on nearly integrable Hamiltonian systems (such as depicted in Wayne’s lecture) and to clarify connections with more classical ideas, such as homogenisation theory. A second noteworthy afternoon session was led by Plotnikov who explained a variant of KAM theory developed by his group in Novosibirsk. That group has had relatively little contact with other researchers in this area and there was fruitful exchange between Plotnikov and the large group of participants from Europe, North America and China working with the more traditional KAM approach. The Novosibirsk variant of the KAM and Nash-Moser machinery was used recently to give the first rigorous proof of the existence of standing waves on a fluid surface: a long-standing and difficult small-divisor problem in PDE theory.

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Abad, Juan – University of Texas at Austin
Bambusi, Dario – University of Milan
Buffoni, Boris – EPF Lausanne
Carr, Jack – Heriot-Watt University
Chenciner, Alain – IMCCE (ASD) & U. Paris VII
Cheng, Chong-Quing – Nanjiang University
Chierchia, Luigi – Università di ‘Roma Tre’
Craig, Walter – McMaster University
Deift, Percy – Courant Institute & Univ Pennsylvania
Denzler, Jochen – University of Notre Dame
Eliasson, Hakan – Université Paris VI
Giorgilli, Antonio – University of Milan
Grebert, Benoit – Université de Nantes
Groves, Mark – Loughborough University
James, Guillaume – Inst. National des Sciences Appliquées Kappeler, Thomas – University of Zürich
Kuksin, Sergei – Heriot-Watt University
Mastropietro, Vieri – Università Tor Vergata
Mather, John – Princeton University
Matthes, Daniel – TU Berlin
McLaughlin, David – Courant Inst. of Mathematical Sciences
Panayotaros, Panayotis – University of Colorado at Boulder
Plotnikov, Pavel – Lavrentyev Institute of Hydrodynamics
Pöschel, Jürgen – University of Stuttgart
Schenkel, Alain – University of Helsinki
Seiler, Ruedi – Technische Universitaet Berlin
Séré, Eric – Université Paris Dauphine
Toland, John – University of Bath
Treschev, Dmitriy – Moscow State University
Wayne, Gene – Boston University
Weinstein, Michael – Bell Labs
Yoccoz, Jean-Christophe – College de France
You, Jiangong – Nanjing University

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