We will review recent works by Tamarkin and by myself on constructing invariants of symplectic manifolds using microlocal methods. Traditionally, geometric objects on manifolds are called microlocal if they can be made local not only on the manifold itself but on its cotangent bundle. Among them: distributions and their wave fronts; D-modules and their singular supports; asymptotic and algebraic versions of the above, described by deformation quantization of the cotangent bundle. (Note that deformation quantization, unlike distributions, D-modules, and sheaves, can be defined for any symplectic manifold). It had been suggested long time ago that the key constructions of symplectic geometry and topology, such as Floer cohomology and the Fukaya category, have some formal resemblance to microlocal constructions. I will describe how to construct a category of microlocal nature starting from a symplectic manifold, both by sheaf-theoretical methods of Tamarkin or by deformation quantization methods.
The talk will be accessible to non-experts. The main example of a symplectic manifold will be the plane, and no prerequisites from symplectic geometry or from deformation quantization will be needed.