06w5030 Schrödinger Evolution Equations

Arriving Saturday, April 22 and departing Thursday, April 27, 2006

Organizers: James Colliander (University of Toronto), Jared Wunsch (Northwestern University).

Information for Participants

Confirmed Participants

Schedule (PDF file)

Final Report (PDF file)

Objectives

The recent work on variable coefficient dispersive estimates and almost conservation laws and their applications has been carried out by several teams working in isolation from one another. A main objective is to bring together, among other allied researchers, the quintet (Colliander, Keel, Staffilani, Takaoka, Tao) who have been focusing on well-posedness and scattering for NLS; a diverse group from Paris (Burq, Carles, Gerard, Tzvetkov, Planchon, Robbiano, Zuily) whose work has been on well-posedness for NLS and associated multilinear estimates via semiclassical methods, as well as Strichartz estimates for variable-coefficient problems; several Japanese researchers (Chihara, Doi, Nakamura, Yajima) who have concentrated on propagation of singularities, dispersive smoothing effects, and parametrices for the linear problem; a group who have been studying fully nonlinear problems of Schr"odinger type (Kenig, Ponce, Vega).

These groups, while aware of each others' work, have had little chance to interact in an intensive, structured way. Further interaction between groups using microlocal and geometric techniques developed for linear PDE, those using harmonic analysis tools, and those using methods adapted for rough coefficients and highly nonlinear PDE, will allow the central problems in the field to be viewed more synergetically. The state of the art of research into NLS has become rather difficult for a single mathematician to grasp, as phenomena split into a wealth of special cases depending on choice of dimension, regularity, nonlinearity, and behaviour of coefficients. A workshop will offer participants a much-needed broad view of this growing field. A better, more systematic view of the fundamental well-posedness questions, as well as of more refined phenomenology such as scattering and structure of blowup solutions, is bound to emerge.