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Abstract. We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin c quantization. We prove the analog of Kodaira vanishing for the Spin c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semiclassically. We also introduce a new quantization scheme, based on a rescaled Laplacian, for which we are able to prove strong semiclassical properties. The two quantizations are shown to be close semiclassically.

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We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by counting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.