Master Class on Microlocal Methods in Symplectic Geometry
The University of Copenhagen
October 5 - 9, 2009
Organizer: Ryszard Nest
- Boris Tsygan , Northwestern University, Evanston and Hebrew University, Jerusalem
Symplectic manifolds arise in two different ways. On the one hand, the phase space of a classical mechanical system has a symplectic structure. On the other, a smooth complex projective variety is always a symplectic manifold. There are two methods of studying symplectic manifolds, both with their roots in mathematical physics. One comes from quantum mechanics and studies the manifold by quantizing the classical system of which it is the phase space; mathematically, this corresponds to microlocal methods in linear PDEs. The other treats the manifold as a space in which a string evolves. While the theories are quite different, there are similarities that have been noticed long time ago. For example, Lagrangian submanifolds play crucial role in both approaches. We will review recent works relating the two approaches (Bressler-Soibelman, Kapustin-Witten, Nadler-Zaslow, Tamarkin, and myself).
October 5 (Mon) 15-17, aud. 8, A review of microlocal analysis
I will review the notions of the wave front of a distribution, the singular support of a D-module and of a sheaf. I will also outline the standard quantization construction which is a passage from a function on a cotangent bundle to an operator.
October 6 (Tue) 13-15, aud. 7, Deformation quantization of symplectic manifolds
I will explain what deformation quantization is, and how the quantization construction of lecture 1 gives rise to it. I will describe the Fedosov construction of deformation quantization.
October 7 (Wed) 14-16, aud. 7, The Fukaya-Floer cohomology and the sheaf-theoretic microlocal methods
I will review the definition and basic properties of the Fukaya-Floer cohomology. Then I will outline the results of Nadler-Zaslow and of Tamarkin relating the Fukaya-Floer theory of the cotangent bundle of the manifold to the category of sheaves on this manifold. I will finish by discussing an idea how to apply Tamarkin's methods to more general symplectic manifolds, in particular in the case of the torus.
October 9 (Fri) 10-12, aud. 9, Oscillatory modules
I will explain how the category of modules over deformation quantization algebra can be modified to become more close to the Fukaya category of the symplectic manifold. Again, I will consider the example of the torus.
The lectures will be held at the Department of Mathematics at the University of Copenhagen.