Recent trends in geometric and nonlinear analysis (12w5100)
Geometric analysis is traditionally and by definition at the intersection of pdes and differential geometry. This specifity is one of the richnesses of the field. Indeed, geometric analysis has always taken advantage of the trends either in analysis or in geometry, and viceversa, its own progresses have been a source of new developments and techniques in its two natural constitutive fields. It is in this spirit that we present this proposal. We are convinced that periodic meetings between experts of some fields that are not straightforwardly related (at least at first glance) are benefic for the geometric analysis community. We do not claim to cover all this abundant field: we have selected a few themes that are very active and novative today and that we would like to have presented during the meeting. Of course, due to the delay between the proposal and the meeting itself, we might modify the structure to follow the most recent trends.
Conformal geometry: There has been a recent regain of interest for the conformally covariant differential operators defined via the ambient metric of Fefferman-Graham. Amongst the consequences are some nice interpretations of the conformal volume and quite a few rigidity theorems. On the other hand, the blowing-up behavior of conformal metrics with constant scalar curvature is now well understood. The meeting would be an opportunity to deliver a status report on the present situation and the next directions to tackle.
General relativity: Amongst the abundance of subfields of this huge domain, we have chosen some selected topics that fit quite well with the other themes of the meeting. The first one is the stability of the spaces and the apriori estimates required for this: despite this is an old subject of investigation, the proofs are still very long and technical and there are still fundamental advances on the subject (for instance on trapped surfaces). Therefore, we would appreciate to have some experts lecturing on these works to a large audience. The second theme we want to consider is the structure and symmetries of solutions to the Einstein equation: this equation is so specific that some natural surfaces happen to be generic (like "ground states" in pdes), and it is a natural theme of investigation to see them as potential general asymptotics.
Geometric evolution equations: The brilliant analysis of the Ricci flow by Perelman is now well understood by the specialists of the domain. In the recent past years, many experts have assimilated this analysis to apply it to various evolution problems with some great success. Here again, the understanding is now so much advanced that it is probably a good period to have some new points of view explained to a wide audience to stimulate applications to other fields.
Singularities and peaks: The geometric invariances often generate the formation of peaks at some specific localizations, and it is an active field in nonlinear analysis to constructs peaks with various properties. Geometers and analysts have been understood for decades that there is a crucial interaction to develop here and have been meeting in this idea: some nice consequences of this interaction gave rise to nice examples of glueing of Kahler manifolds (see below) and to the construction of CMC manifolds with various ends and topological properties. Very recently, there has been new constructions of some tower solutions with infinite energy: it seems important to present some of these recent solutions to the geometric community and to hope for some new applications.
Complex structures and Kahler geometry: This subject is a very central theme in our proposal. Indeed, it arises naturally as an important field of application of some of the topics mentioned here. The geometric flows have here an application via the Kahler-Ricci flow and other variants: it is a natural direction, but the underlying difficulties delayed the mathematical production that is now mature and impressive. Beside the geometric flows, Kahler geometry has also recently benefited from progress in the singular solutions to elliptic pdes: as for CMC submanifolds, the analysis techniques have been at the origin of geometric developements and the construction of new complex manifolds.