Abstract:
The main idea in the theory of geometric quantization (introduced by Kostant and Souriau) of a manifold X is to associate with X "quantum" Hilbert spaces defined by means of some canonical geometric constructions. In the presence of a group action, an important meta-principle is "reduction commutes with quantization". This principle describes how the quantization of the quotient relates to the quantization of the original manifold. We will consider the following aspects of quantization: (1) Berezin-Toeplitz quantization, (2) Holomorphic Morse inequalities, (3) Ergodic complex geometry, (4) Quantization and reduction. One of the tools employed will be Szegö and Bergman kernel expansions.