DEFORMATION QUANTIZATION AND NONCOMMUTATIVE GEOMETRY

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We investigate deformation quantization from the view point of Weyl manifolds, representation of algebras, noncommutative Geometry and asymptotic analysis. Main results are the following four points. 1. Noncommutative contact algebra and noncommutative sphere : We intoduce the dass of deformation algebras of noncommutative contact algebras by extending the class of classical commutative algebra from Poisson algebras to contact algebras. We study examples of noncommutative contact algebras such as spheres. 2. Berezin representation of deformation quantization : Using the structure of noncommutative contact algebras, we obtain Berezin representation of deformation quantization. 3. Relation of Weyl manifold and deformation quantization on symplectic manifold : A classification of deformation quantization is given by considering the classification of Weyl manifolds. We obtain the complete invariant of Weyl manifold, which is called Poincare-Cartan invariant. The Poincare-Cartan class is proved to be just the class given by the curvature of Fedosov connection. 4. Asymptotic analysis and partial differential Equations : The basic analysis is studied which is considered important future investigation of deformation quantization. Mainly inverse problem is investigated. The asymptotic distribution of poles of the resolvent is analysed by Nakamura criterion for the Reyleigh wave boundary condition.