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We discuss collective coordinate quantization of the half-BPS geometries of Lin, Lunin and Maldacena (hep-th/0409174). The LLM geometries are parameterized by a single function u on a plane. We treat this function as a collective coordinate. We arrive at the collective coordinate action as well as path integral measure by considering D3 branes in an arbitrary LLM geometry. The resulting functional integral is shown, using known methods (hep-th/9309028), to be the classical limit of a functional integral for free fermions in a harmonic oscillator. The function u gets identified with the classical limit of the Wigner phase space distribution of the fermion theory which satisfies u * u = u. The calculation shows how configuration space of supergravity becomes a phase space (hence noncommutative) in the half-BPS sector. Our method sheds some new light on counting supersymmetric configurations in supergravity.

In these two lectures I will present on overview of topological string theory, starting with the basic concepts and continuing to the most recent developments. Aspects that will be covered are: closed and open strings, Gromov-Witten theory, matrix models, quantum crystals, and special holonomy manifolds. I will end with some more speculative thoughts about the non-perturbative definition of (topological) string theory and quantum geometry.