Algebraic lagrangian geometry: from geometric quantization to mirror symmetry

Geometric formulation of Quantum Mechanics is a translation to the language of projective spaces: the Hilbert space is replaced by its projectivization which is real quantum phase space; selfadjoint operators are replaced by certain smooth functions whose Hamiltonian vector fields preserve the Kahler structure; the Schrodinger equation in this setup turns to be just the Hamilton equation and the probabilistic aspects of QM are governed by the Riemannian mmetric. Therefore one can extend standard QM to certain approapriate algebraic varieties not only projective spaces.
Concerning the quantization problem one can thus reformulate it, and we call this reformulation “algebro geometric quantization”. As a solution of the algebro geometric quantization problem we can take ALAG – programme, proposed by A. Tyurin and A. Gorodentsev in 1999. It is a programme indeed – starting with a simply connected compact symplectic manifolod with integer symplectic form one gets an infinite dimensional algebraic manifold which is called the moduli space of half weighted Bohr–Sommerfeld cycles of fixed topological type and volume. On the other hand, this moduli space could be exploited in mirror symmetry, f.e. based on the Floer cohomology one can construct a family of vector bundles on the moduli spaces.

^* (Format: 2 hours + break for tea + 2 hours.)