Dynamics in Geometric Dispersive Equations and the Effects of Trapping, Scattering and Weak Turbulence (14w5080)
Stephen Gustafson (University of British Columbia)
Jeremy Marzuola (University of North Carolina, Chapel Hill)
Daniel Tataru (University of California, Berkeley)
As this trend of the 'geometrization' of the mathematical study of nonlinear wave equations begins to accelerate, it seems the time is right for a workshop which gathers experts in the various geometric aspects of dispersive equations as well as dynamical systems to discuss recent advances, identify promising new directions, and generally survey the state of the field.
Geometry enters the PDE of Mathematical Physics in various ways. The idea of this workshop is focus on a few such classes of equations, the analysis of which exhibit some key common features while differing significantly in detail. The hope is for a cross-pollination of ideas leading to fruitful new approaches.
--Dispersive PDE on non-compact manifolds and exterior domains: Many physical evolution equations can naturally be posed on domains with non- trivial geometry, representing genuinely non-flat physical domains, the effect of impuritites in the system, or some other effect. Mathematically, the study of (for example) nonlinear Schroedinger or wave equations on manifolds is rich with interesting and challenging phenomena stemming from the interaction of the waves with the metric. Key analytical issues include linear dispersive estimates (and obstacles thereto) and eigenfunction estimates. This has quietly become one of the hottest areas in analysis, progressing quickly through application of techniques from harmonic and micro-local analysis.
--Nonlinear Dispersive PDE on compact manifolds: It is quite natural (in modeling water waves, for example) to pose nonlinear dispersive PDE on compact domains. Since dispersive effects are much more subtle in a compact setting which allows reflection/return of waves, the long-time behaviour of solutions is much more complex and mathematically challenging than in the non-compact setting. Dramatic recent results have opened an emerging research direction centered on the study of energy transfer between length scales and 'weak turbulence' phenomena. In particular, large Hamiltonian dynamical systems theory has played a major role in our understanding of dispersive equations on the torus. However, progress has also been made recently in critical equations on other types of manifolds, provided one understands the nature of the spectrum of the underlying Laplacian.
--Dispersive PDE with target geometry: A number of physical models -- for example gauge theories from particle physics, wave maps connected with relativity, and Schroedinger maps arising in ferromagnetism -- have geometrically non-trivial target spaces; that is, they describe the dynamics of maps into manifolds. In sub-critical and critical cases, great strides have been made recently toward understanding the role played by the geometry in the key issue of singularity formation or non-formation. However, the super-critical cases (which in some examples include the physically most important cases) are still largely an open book, for which a better understanding of (among other things) mechanisms of singularity formation remains a key objective.
This workshop would bring together mathematicians working at the forefront of each of these areas -- aiming for a mixture of junior and senior, and of 'pure' and 'applied' analysts -- in the sincere hope that the resulting interactions will lead to new collaborations, the identification and exploitation of new connections between problems, and ultimately to the development of new techniques for the mathematical understanding of physical systems.