Abstract: This will be a survey talk on the method of geometric quantization in symplectic geometry, which attempts to associate to each symplectic manifold, viewed as a model for classical mechanics, a "quantum" Hilbert space in a way that is consistent with quantum mechanics. For simplicity I will stick to the case of Kaehler manifolds. In particular I will discuss the Kostant-Souriau approach via prequantum line/circle bundles and polarizations; the Poisson algebra of functions and Hamiltonian group actions; and the role of index theory for Dirac operators. Time permitting, I will also mention the relationship between geometric quantization and Berezin-Toeplitz quantization.

Abstract: A manifold is aspherical if its universal cover is contractible. There are only a few known techniques for constructing closed aspherical manifolds. I will give an overview of these techniques, and explain how they can be used to produce some interesting examples.