Workshop

This workshop is to be held at the International House on the UC Berkeley campus, at 2299 Piedmont Avenue. On site registration for the workshop will be at the International House, starting at 8:30 AM Monday and ending at 3:30 PM Monday. The field of nonlinear dispersive equations has experienced a striking evolution over the last fifteen years, gaining a place of its own within partial differential equations. During that time many new ideas and techniques emerged, enabling one to work on problems which until not long ago seemed untouchable. These include the systematic use of dispersive and Strichartz estimates for linear dispersive equations (used in the study of semilinear problems), the vector fields method for the linear and nonlinear wave equation, estimates for bilinear and multilinear wave interactions, the use of wave packet methods (for both linear and bilinear interactions), and the increasingly preeminent role played by the geometrical aspects of nonlinear dispersive equations. These ideas and methods have led to solutions to many problems. On the other hand, they have opened the way for the study of other unsolved problems, and in the process have spawned numerous open questions. It is difficult to predict where the field will be three years from now, but what is clear is that it is experiencing a very strong growth which is likely to continue for years to come. The evolution process for this field has its origin in two ways of quantitatively measuring dispersion. One comes from harmonic analysis, which is used to establish certain dispersive ($L^p$) estimates for solutions to linear equations. The second has geometrical roots, namely in the analysis of vector fields generating the Lorentz group associated to the linear wave equation. Ideas in harmonic analysis became even more important in the process of understanding the structure of nonlinearities, which is tied to the nonlinear wave interaction. Furthermore, there is now a two way interaction between harmonic analysis and the analysis of dispersive equations. Connections with number theory have also been established. On the other hand, one must also emphasize the geometrical aspects of these problems, which are slowly gaining a preeminent role in the theory. These arise in two ways. Many semilinear dispersive equations arise in geometrical contexts (e.g wave-maps, Yang-Mills) and the geometry of the target space for the solutions appears to determine certain properties of the equation. On the other hand, in both nonlinear problems and problems arising in nonflat geometries (e.g. on manifolds with various curvature properties) one needs to control the geometry of the underlying space-time and its effect on the corresponding equations. Geometry can also play a role in problems involving perturbations of ones in Minkowski space. For instance the presence or absence of trapped geodesics plays an important role in nonlinear equations involving obstacles or compactly supported metric perturbations. Geometry plays an important role in the study of nonlinear dispersive equations in several ways. One can begin with linear effects, caused by the geometry of the underlying space time. Sometimes one also needs to consider the geometry of the target space (manifold) which usually produces semilinear effects. Finally, the most difficult is the nonlinear case, where the geometry of the underlying space time itself depends on the solutions to the equation. While at this point we are far from having a good understanding of the geometric aspects of nonlinear dispersive equations, this is currently a hot topic and we anticipate that considerable progress will be made until the program begins. This workshop will bring together leading researchers in the field of nonlinear dispersive equations, especially ones whose work uses both analytic and geometric techniques. Its goal will be to follow several different threads, and introduce different problems and techniques to a wide spectrum of mathematicians. ----------------------------------- Preliminary Schedule of Speakers (updated 1 Nov. 2005) Monday, 11/28 9:30-9:45 Welcome to MSRI (at I-HOUSE) 9:45-10:45 A. Soffer: Dispersive Decay, Methods and Applications 11:00-12:00 C. Zeng: Energy estimates of free boundary problems of the Euler equation 12:00-2:00 Lunch Break 2:00-3:00 J.M. Delort: Almost global solutions for Hamiltonian Klein-Gordon equations on Zoll manifolds 3:00 -3:30 Tea Break 3:30-4:00 Travel to 60 Evans Hall, UCB 4:10-5:10 C. Kenig (at 60 Evans Hall, UCB): Some recent progress on the global well-posedness of dispersive equations Tuesday, 11/29 9:45-10:45 F. Planchon: Existence and uniquess for the Benjamin-Ono equation 11:00-12:00 L. Vega 12:00-2:00 Lunch Break 2:00-3:00 J. Shatah 3:00-3:30 Tea Break 3:30-4:30 S. Klainerman: A break-down criterion in General Relativity Wednesday, 11/30 9:45-10:45 X. Zhang 11:00-12:00 M. Weinstein 12:00-2:00 Lunch Break 2:00-3:00 M. Keel 3:00-3:30 Tea Break 3:30-4:30 J. Colliander Thursday, 12/01 9:45-10:45 J. Metcalfe: Elastic waves in exterior domains 11:00-12:00 H. Koch: Absence of positive eigenvalues for rough Schroedinger operators 12:00-2:00 Lunch Break 2:00-3:00 W. Craig: On the Boltzmann equation: global solutions in one dimensional geometry 3:00-3:30 Tea Break 3:30-4:30 T. Sideris: Incompressible elastodynamics Friday, 12/02 9:15-10:15 P. Raphael: Construction of a blow up solution for a L2 supercritical NLS equation 10:30-11:30 J. Sterbenz: Regularity for High Dimensional Gauge Fields ------------------------------------- This workshop will be held at International House Berkeley.