Symplectic topology, microlocal sheaf theory and quantization – MICROLOCAL

The Microlocal project presented here 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 subject 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 a microsupport which is a conic subset in the cotangent space - conic meaning 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 manifolds. The key discovery of Tamarkin is that a number of interesting Lagrangian manifolds can be realized as microsupports of sheaves, and that this quantization property explains their strong geometric rigidity. 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 directions between algebra and geometry.

The purpose of the participants in this project is to explore extensively the perspectives opened by these developments. Three domains of applicability of the microlocal theory of sheaves will be considered: cotangent spaces, general symplectic manifolds and complex symplectic manifolds. The mathematics involved will mostly be geometry in the
first case, algebra in the second and analysis in the third. Our project will thus give to mathematicians belonging to disjoint communities the opportunity and the motivation to work together with a common aim.