DS10 - Défi de tous les savoirs

Symplectic topology, microlocal sheaf theory and quantization – MICROLOCAL


Here are two results which perfectly reflect the diversity of the works done in the Microlocal project.

Proof of the three cusps conjecture.

By analyzing constructible sheaves in dimensions 1 and 2, Stéphane Guillermou proved Arnold's three cusps conjecture, namely that, in the projectivized cotangent bundle of the real projective plane, the front of every generic curve in the Legendrian isotopy class of the fiber has at least three cusps.

Percolation of random nodal lines

With Vincent Beffara, Demien Gayet proved a Russo-Seymour-Welsch percolation theorem for nodal domains and nodal lines associated to a natural (infinite dimensional) space of real analytic functions on the real plane. More precisely, let U be a connected bounded open set with smooth boundary in the plane and a, a' two disjoint arcs of positive lengths in the boundary of U. The authors prove that there exists a positive constant c such that, for any positive scale s, with probability at least c, the positive locus of f(s.) in the closure of U has a connected component intersecting both a and a', where f is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For s large enough, the same conclusion holds for the zero locus.



Submission summary

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.

Project coordination

Emmanuel Giroux (Unité de mathématiques pures et appliquées -- ENS de Lyon)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


UMR 8553 CNRS ENS Département de Mathématiques et Applications de l'ENS
UMPA Unité de mathématiques pures et appliquées -- ENS de Lyon

Help of the ANR 436,800 euros
Beginning and duration of the scientific project: - 48 Months

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