Recent Advances on de Giorgi's Conjecture and the Study of Entire Solutions of Nonlinear Scalar Equations: Interaction of PDEs and Differential Geometry (10w5036)

Objectives

This five-day workshop will provide a forum for the dissemination of current advances in the study of De Giorgi's Conjecture, and entire solutions for the
Allen-Cahn and Nonlinear Schrodinger equations. The focus will be on the interaction of the theories of minimal and constant mean curvature surfaces and partial differential equations.

Our aim is to bring together two groups of mathematicians working on partial differential equations: one specialized on De Giorgi conjecture, Allen-Cahn equation (AC), Nonlinear Schrodinger equation (NLS), and another specialized on Differential Geometry (DG) (of minimal surfaces and constant mean curvature(CMC) surfaces).

In recent years there have been many important developments in the study of De Giorgi's Conjecture and entire solutions of Allen-Cahn equations and Nonlinear Schrodinger equations. The old connection between Allen-Cahn equations and minimal surfaces, and new connection between nonlinear Schrodinger equation and constant mean curvature (CMC) surfaces, have to be understood and developed. There is a strong need for researchers in semilinear elliptic equations to be exposed to new techniques of minimal surfaces and CMC surfaces.

There are two main aims of this workshop. A primary goal is to provide a forum for the interaction between researchers in nonlinear PDEs, especially in the study of AC and NLS, and specialists in minimal and CMC surfaces. This interaction will stimulate new mathematical ideas, and also expose the mathematical community to new problems that await a mathematical understanding. The second main focus of the workshop is to expose a limited number of postdoctoral fellows and advanced graduate students to current problems associated with Allen-Cahn equation and NLS and to show different mathematical approaches to study simple-looking PDE problems.

To illustrate the importance and relevance of this research area, over the past three years there have been many conferences devoted to the study of De Giorgi conjecture and NLS. A sample of these include a one-week workshop in BIRS in August 2007 (organized by Esposito, Pacard and Tarantello), a one-week workshop in October 2007 (organized by Y. Du, Dancer and Zhao), a one-week conference in ICTP, Italy in June 2008 on nonlinear dynamics for PDEs related to materials science (organized by J. Li, ICTP and G.Tian, Princeton), a one-week conference on variational method and nonlinear analysis at POSTECH in Oct. 2008 (organized by P. Rabinowitz, Univ. Wisconsin and B. Byeon, POSTECH), a thematic program at PIMS in 2009 (organized by O. Druet, R. Roksi, N. Ghoussoub), a one-week conefrence in CIRM organzied by Malchiodi, Pacard and Terracini, also in 2009.

We are not aware of any workshops in the study of De Giorgi Conjecture that are currently proposed for the summer of 2010. Our intention is to invite many of the main participants of these previous meetings to our workshop, with the view of highlighting the advances made on open problems. A key feature of the workshop is that we intend to also invite certain participants who work on
differential geometries of minimal surfaces and constant mean curvature surfaces, among them: D. Hoffman (Univ. Massachusetts), N. Korevaar (Univ. Utah), R. Kusner (Univ. Massachusetts), N. Kapouleas (Brown Univ.), W. Meeks III (Rice Univ.), Mazzeo (Stanford), Pollack (Univ. Washington),F. Pacard (Univ. Paris 12), L. Simon (Stanford), Xujia Wang (ANU-Australia).