1. This is a series of three talks about a recent method which allows one to describe a derived category of coherent sheaves on a smooth (algebraically) symplectic manifold in algebrac terms, in the spirit of McKay equivalence and of recent work by T. Bridgeland and M. Van den Bergh in dim 3. The method involves reduction to positive characteristic and applying a Fedosov-type geometric quantization procedure there. The resulting algebraic structure in char 0 is very similar to G. Lusztig's quantum Frobenius map. In the first talk, we will describe the results and give a general outline of the argument.
2. In the second talk of the series, we will describe a geometric quantization procedure in positive characteristic and its ingredients such a notion of a restricted Poisson algebra (the Poisson generalization of a restricted Lie algebra).
3. In the third talk, we will show how to apply the quantization in positive characteristic to construct the desired equivalence of derived categories.