The fifth Takagi Lectures
October 4, 2008 (Sat) 12:10--13:10
October 5, 2008 (Sun) 10:50--11:50
Graduate School of Mathematical Sciences
The University of Tokyo

Holonomic D-modules and positive characteristic
Maxim Kontsevich
(Institut des Hautes Études Scientifiques)


I will talk about a new picture, based on reduction to positive characteristic, which relates holonomic modules over quantum algebras (e.g. algebras of differential operators, or quantum tori), with Lagrangian submanifolds in the corresponding symplectic manifold. In the case of one variable everything can be made explicit and absolutely elementary. In particular, we get a new class of "transcendental" expressions (a sort of determinants), hypothetically related with crystalline cohomology. Also, the new correspondence seems to be related to integrable systems and a higher-dimensional generalization of the Langlands duality in the functional field case.