Abstract
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.