The main topics of this workshop will be Phase Space Topology and Transport and its applications to theory of elementary particles, oceanographic and atmospheric sciences, condensed matter, accelerator and plasma physics and other disciplines on the classical and quantum levels. It aims to further connections between symplectic dynamics and its most feasible applications.
Viewed from the perspective of more applied fields, such as fluid dynamics and soft condensed matter, the highly fruitful initial symbiosis between symplectic geometry and physics has not been sufficiently pursued by the mathematical community. In view of the many new techniques and the powerful mathematical machinery emerging in the works of contact and symplectic topologists, it makes a lot of sense to reexamine the roots of the subject and try to attack the problems of Phase Space Topology and Kinetics and Transport using the present arsenal of highly developed methods of symplectic topology. Many difficulties arise when one studies “realistic systems” even when such systems have only a few degrees of freedom. As an example, the phase space of a smooth Hamiltonian system is not ergodic in a global sense due to the presence of an infinite number of islands of stability: the mixing of trajectories is not uniform in phase space, the Gaussian nature of particle distribution is lost due to so called flights and trappings, related some times to Lagrangian Levy type of processes. Truly high-dimensional structures used in plasma physics give rise to other difficulties. For example Arnold diffusion may be viewed as a higher dimensional mechanism for a trap. Isolated fixed points with at least one hyperbolic direction give a set of examples of new mechanisms which cannot be studied with the old methods.
If at the beginning the contemporary ergodic theory of dynamical systems provided fairly satisfactory method for studying chaotic kinetics, it is clear now that today tools should be significantly expanded and focus on understanding deeper properties of the phase space using methods of modern symplectic and contact topology and geometry.
(Universidad Autonoma de San Luis Potosi, Mexico)
Vered Rom-Kedar (Weizmann Institute of Science)
Lai Sang Young (New York University/Courant Institute of Mathematical Sciences)
George Zaslavsky (New York University)