Introduction to the Microlocal project

The Microlocal project originates in the discovery by Tamarkin, a few years ago, of striking and unexpected applications of the microlocal theory of sheaves to symplectic geometry. Over the last three decades, symplectic geometry has demonstrated a remarkable ability to establish deep connections with other fields in mathematics and advanced topics in theoretical physics. New variational techniques, the theory of generating functions, the theory of pseudoholomorphic curves and Floer theory have provided fruitful interfaces with the theory of dynamical systems, algebraic geometry, gauge theory and the theory of mirror symmetry. The relations that the microlocal theory of sheaves creates between symplectic geometry and abstract homological algebra are in the same vein and there is no doubt that they will become the focus of intense research activities in the coming years. They offer already now a radically new approach to rigidity phenomena by giving original proofs of some of Arnold's conjectures which stimulated the rise of symplectic geometry. The primary aim of the project is to give to the concerned French mathematicians the possibility to play a leading role in these future developments.

The microlocal theory of sheaves, due to Kashiwara-Schapira, associates to any sheaf on a manifold its microsupport which is a conic subset in the cotangent space (where conic means that, away from the zero-section, it is a cone over a subset of the sphere cotangent bundle). The fundamental involutivity theorem of Kashiwara-Schapira says that this microsupport is coisotropic for the canonical symplectic structure of the cotangent. The microlocal theory of sheaves has many applications, notably to the study of linear partial differential equations, to representation theory and singularity theory. Kashiwara-Schapira have also been studying for several years its applications to the deformation quantization of complex symplectic and Poisson manifolds. The key discovery of Tamarkin is that, in a number of interesting cases, a given Lagrangian manifold can be realized as the microsupport of a sheaf ---~called a \emph{quantization} of the manifold~--- and that the homological properties of this sheaf explain the strong geometric rigidity of the Lagrangian manifold. This discovery has been confirmed and generalized by Guillermou, Kashiwara and Schapira. It shows that the microlocal theory of sheaves has the remarkable feature to provide a bridge in both ways between algebra and geometry. In addition to this, microlocal analysis appeared recently in works by Gayet-Welschinger on the topology of random nodal sets of pseudodifferential operators.

The overall purpose of the participants in this project is to explore widely the perspectives opened by these discoveries, to analyze carefully the geometry of singular supports and to undertake a systematic study of the applications of microlocal sheaf theory to symplectic topology. Three domains of applicability will be considered: cotangent spaces, general symplectic manifolds and complex symplectic manifolds. The mathematics involved will mostly be geometry in the first case (with some probabilistic aspects), algebra in the second and analysis in the third.