Quantization of Poisson manifolds and noncommutative geometry

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In a joint project with R.Nest of the University of Copenhagen and I.Peter of the University of Munster the pricipal investigator showed that under a topological condition every closed symplectic manifold has a strict quantization. Strict quantization is an analytic deformation theory. An algebraic deformation theory (existence of deformation quantization) has been known since 80's.The aim of the project is to show existence of strict quantizations for Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The project is divided into three steps. The first step is to re-examine the existence proof of strict quantization for symplectic manifolds, in order to have a deep understanding of mechanism of existence. In particular, re-examination of the proof by B.Fedosov, which played a crucial role in our proof, of existence of deformation qu … More antization is an important step. The second step is to understand the existence proof of deformation quantization for Poisson manifolds and to rewrite it from the viewpoint of Fedosov. The last step involves actual construction of strict quantization.Through quite a few discussions with Nest, the mechanism of existence became fairly clear, and we obtained a refined version of our result. Thanks to a recent appearance of a simpler proof of existence of deformation quantization for Poisson manifolds than Kontsevich's, we have a prospect to achieve the second step.While working on the project discussed above, in a joint project with C.L.Olsen of the State University of New York at Buffalo, the principal investigator worked on the cases that are not covered by the results with Nest and Peter. In particular, we showed that the 2-sphere with a specific Poisson structure has a strict quantization. In the process to construct strict quantization we obtained new "noncommutative 2-spheres". These C^*-algebras are new examples of noncommutative Poisson manifolds.As explained above, unfortunately we could not achieve the goal of the project, i.e. the existence of strict quantizations for poisson manifolds. We certainly intend to continue working on the project. We will hopefully complete the project within a year or so. Less