Organizers: W. Craig, S. Kuksin
Abstract:
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 wellposedness 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 shortcourse 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)
Schedule:
Tuesday, May 11

2:10  3:00 
Michael Weinstein (Bell Labs and Columbia) 
Wednesday, May 12

11:00  12:00 
Bill Kath (Northwestern) 
Thursday, May 13

10:30  11:30 
Bill Kath (Northwestern) 
11:30  12:00 
break 
12:00  1:00 
Michael Weinstein (Bell Labs and Columbia) 
Friday, May 14

10:30  11:30 
Jim Colliander (Toronto) 
11:30  12:00 
break 
12:00  1:00 
Walter Craig (McMaster) 
Saturday, May 15

10:30  11:30 
V. E. Zakharov (Moscow & Arizona) 
11:30  12:00 
break 
12:00  1:00 
V. E. Zakharov (Moscow & Arizona) 