On January 17-21, 2011, there will be a one-week conference at Luminy, France on recent advances in geometric analysis, with the title `Geometric Analysis'. Within the general field of geometric analysis, the conference will focus on microlocal analysis, spectral geometry, conformal invariants, analysis on non-compact manifolds including the study of index theory and de Rham cohomologies as well as wave propagation, and the study of geometric flows. These topics are chosen because of many recent advances and numerous open frontiers, and because in spite of many connections between them, there is sufficient distance between these topics that collaborative projects, benefitting from different points of view, are particularly fruitful. In particular, the conference is expected to facilitate productive interactions between researchers working on elliptic and parabolic (which are the traditional strengths of the field), respectively hyperbolic (such as wave propagation), problems in partial differential equations. The proposal serves to provide for the transportation to the conference, and accomodation there, for some participants, with emphasis on graduate students, postdoctoral fellows, and junior faculty from US universities.
Geometric analysis covers areas in which one studies phenomena, such as wave propagation or heat flow, on geometric backgrounds. One example of such a geometric background is general relativity, giving rise to a curved space-time of great physical interest. It also gives rise to `static' problems, without a `time' variable; the latter have received much of the attention as they are analytically more tractable. While often there is a physical motivation, recent work in the Ricci flow, which is an analogue of heat flow, shows that the field can also be used to answer questions of mathematical origin, in this case in topology (the study of shapes). This conference combines both aspects, and strives to encourage collaboration between researchers in different areas. It is also designed be ideal for doctoral students and young researchers as it will provide them with a host of techniques and problems relevant to their research. In particular, the speakers will be asked that the talks explain both the methods used and the motivations for the problem be it from a pure or an applied perspective, and there will be a discussion of new and open problems as well; this is expected to help young researchers just starting work in the field.
This project consisted of a conference in geometric analysis with 57 participants at the CIRM (Centre International de Rencontres Mathematiques) in Luminy, France, in January 2011. The purpose of the NSF funding of the project was to make the travel of US graduate students, postdocs and junior faculty to the conference possible. This was accomplished by funding the travel of three graduate students as well as that of three postdocs/junior faculty. The conference was held at the highest international level, with a list of speakers which included some of the most renowned researchers in the interplay between geometry and partial differential equations. The topics were chosen taking into account recent advances and open frontiers, and so that (in spite of many connections between them) there was sufficient distance between these topics that collaborative projects, benefitting from different points of view, could be particularly fruitful. One of the main goals of the conference was to present the state of the art of the various topics covered. The potential benefits for young researchers and researchers in nearby fields was central; a goal was that these researchers find important techniques and perspectives. Indeed, the lecturers were requested to explain both the methods used and the motivations for the problem be it from a pure or an applied perspective. The supported participants all found the conference helpful to their research and professional development. Among the comments they gave after the meeting was that the conference exposed them to many topics beyond their immediate research area, made it possible to talk to experts in their field who are not at their home institution, and also gave them an opportunity both to meet collaborators for existing projects and start new collaborative projects with other participants.
