A dialogue between mathematicians and physicists has been one of the most important motivating influences in the evolution of the mathematical sciences, and nowhere is this more the case than in the theory of dynamical systems and partial differential equations. In this tradition, this short course that is being held during the program on PDE at the Fields Institute will have at its principal goal the stimulation of an interchange of ideas and a discussion of problems, between practicing mathematicians and practicing physicists. Topics under consideration vary enormously in their physical scales; from cosmology and string theory, to celestial mechanics, to fluid dynamics and material science, to photonics, superconductivity and quantum mechanics. The common denominator is the fact that the governing evolution equations possess similar mathematical characteristics, and their solutions often will entail similar features, albeit on vastly differing spatial and temporal scales. Indeed most of the conservative equations that arise in physics are in fact able to be posed as Hamiltonian dynamical systems, often possessing infinitely many degrees of freedom, and it is the class of Hamiltonian PDE which plays an increasingly central role.

In the last decade there have been a number of major advances in PDE which can be seen as extensions of the analytic theory of Hamiltonian dynamical systems to problems with infinitely many degrees of freedom. The focus of mathematical research in PDE has turned from the basic question of well-posedness for the inital value problem, to more complex questions of Hamiltonian dynamics in infinite dimensional phase space. Questions include the symplectic structure of the natural phase space, the persistence of invariant tori under perturbation, stability questions over exponential timescales, and the existence of energy cascades in nonlinear evolution PDE. A positive outcome of this short course is for mathematicians to understand the impact of these results to problems of immediate relevance to the physical sciences, as well as dissemination of this class of results among a broader scientific community.

The short-course on Hamiltonian PDE will occur during the same week as Jean Bourgain's Fields Institute Distinguished Lectures.

Confirmed Speakers:

Jim Colliander (Toronto)

Walter Craig (McMaster)

Bill Kath (Northwestern)

Michael Weinstein (Bell Labs and Columbia)

V. E. Zakharov (Moscow & Arizona)