Symplectic Geometry and Physics

Symplectic Geometry and Physics Workshop II: Chaotic Dynamics and Transport

May 19 - 23, 2003

Schedule and Presentations

Organizing Committee:

Valentin Afraimovich (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)

Introduction

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. 

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. In the early days of the study of chaos, ergodic theory provided an adequate support for the kinetic approach. But realistic physical models have typically different coexisting dynamical behavior such as stability islands, hyperbolic orbits and chaotic trajectories. This diversity of dynamical behavior makes the study more vivid and interesting on one hand and admittedly less universal than initially anticipated on the other. Many outstanding results have been obtained by many physicists and applied mathematicians such as V. Turaev, A. Neishtadt, V.V. Kozlov, V. Afraimovich, V. Rom-Kedar, G. Zaslavsky, Lai-Sang Young, M. Courbage, M. Wojtkovsky, C. Liverany (Italy), M. Shlesinger, and G. Haller.

Today new ideas and methods are needed for creating an adequate description of kinetics and transport of real dynamics. These new ideas are especially needed in oceanographic and atmospheric sciences, condensed matter, accelerator and plasma physics. 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 time 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.

Applications to applied sciences: 

The proposed workshop is designed to further connections between symplectic dynamics and its most feasible applications. Symplecticity of the dynamics assumes only Hamiltonian systems, although weakly dissipative systems, close to the Hamiltonian ones, also will be included. The connection is based on the following steps: 

1. Detailed investigation and classification of the phase space topology, bifurcations, and singular objects (stable/unstable manifolds, cantori, boundaries between islands and chaotic sea, etc.). The new symplectic invariants and additional structures on Lagrangian cycles (the Fukaya category with weights coming from the action of the braid group on objects) could be very useful. 

2. Application of these properties to the transport in chaotic sea and along stochastic layers and webs. 

3. Control of the transport. The last two items are at the heart of applications. Here are some applications that will be covered during the Chaotic Dynamics workshop: 

• Charged particle transport in contemporary plasma devices (tokamak, spheromak, stellarator). The focus will be on description prognosis, and control of the anomalous transport in plasmas and the plasma edge. These subjects are considered as the most important for the future fusion program. They will be covered in a lecture series by B. Carreras and G. M. Zaslavsky. 
• Passive particle dynamics in incompressible flows. This set of problems is related to the mixing properties in fluids and is already applied to the ocean and atmospheric physics. Passive particle dynamics is the Hamiltonian one, and normal/anomalous transport properties are extremely important for such phenomena as pollutive dispersion, plankton distribution, and turbulent mixing. Those main contributors to this topic will be G. Haller, V. Rom-Kedar, G.M. Zaslavsky, and J. Weiss. 
• General problems of geophysics such as wave dynamics, diagnostic of turbulent flows, underwater acoustics, advection of the vortex-field. This part of the workshop is fairly broad and is important for analysis of the surface and internal behavior of waves in the ocean and their possible influence on the climate as well as climate diagnostics. This part of the workshop will be covered by M. Ghil, A. Vulpiani, S. Prants, E. Ott, V. Kozlov, L.Lerman, V. Kaloshin, De la Llave, V. Rom-Kedar, V. Afraimovich, M. Courbage, C. Liverani, N. Chernov, and G. Haller.
• Celestial mechanics (dynamics of asteroids and comets, stability of the Solar system, and related issues). These items will include the results of A. Neishtadt and D. Treshchev.

Speakers

Related Seminars

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