Quantization of Lagrangians via reduction mod p

Let $X$ be a smooth complex variety and let $L$ be a smooth Lagrangian subvariety of $T^*X$. In this talk we shall explain a "quantization correspondence” which attaches, to any such $L$ satisfying some cohomological conditions, a $D$-module on $X$. The existence of such a correspondence has a number of nice consequences, e.g. for auto-equivalences of the Weyl algebra. Though the correspondence is rather abstract (it based on reducing $D$-modules to finite characteristic), in the talk I shall focus on the case $X=A^1$ which is relatively concrete and where the key constructions can be seen rather explicitly.