Workshop on Partial Differential Equations in Geometry and Physics

PDEs are among the most powerful tools in both geometry and physics. Fundamental geo- metric problems like the Poincare conjecture have been solved with PDEs, and the basic field equations of physics, like those of Maxwell or Einstein, are expressed in terms of PDEs. Usually, those PDEs involve some singularities. For instance, the boundary or the underlying space could be singular. Or the data or the PDE itself could have singularities. Most importantly, how- ever, solutions of nonlinear PDEs can by themselves develop singularities. Understanding these singularities then is crucial for the underlying geometric or physical problem.

In recent years, substantial advances have been made in understanding various PDEs and their singularities in problems from geometry and physics. We want to bring together some of the contributors from China and abroad to take account of what has been achieved, what methods and techniques are available, and most importantly, to set the stage for future advances. We therefore intentionally bring people with different knowledge and different backgrounds together, ranging from abstract and nonlinear analysis to geometry and mathematical physics, in order to explore new connections between analysis, geometry, and physics, in particular quantum field theory and related fields.

Many (but not all) of the PDEs that are important in geometry and physics arise from variation- al problems. One can therefore try to solve them either by minimizing or saddle point methods or by studying the corresponding heat flow. Comparing these approaches often yields additional insight.

More specifically, we shall address key topics of geometric analysis, like minimal hypersurfaces and submanifolds, harmonic maps or geometric optimization, study variational problems motivated by QFT, and also discuss specific types of singular behavior at the boundary or in the interior. A problem session towards the end of the workshop will not only take stock of the known problems in the field, but also try to identify new problems that emerge from the discussions during the workshop. We believe that the scheme of our workshop will be particularly beneficial for young mathematicians that want to get an overview of the state of the art, concerning both methods and topics, and learn about new research directions and challenging open problems.

Much of this will build upon successful cooperation between mathematicians from China and Germany, but it is important for us to also bring the different groups together and to involve some talented graduate students and postdocs. In particular, it is an explicit aim of the workshop to stimulate new collaborations.

Organizers

NameUniversity
Jurgen JostMax Planck Institute for Mathematics in the Sciences, Germany
Qun ChenWuhan University