Weyl quantization and an index theorem for pseudodifferential operators

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The aim of this project is to show an Atiyah-Singer-type index formula for pseudodiffer- ential operators on open manifolds, particularly on simply connected hyperbolic manifolds. The project is divided into two steps. The first is to isolate a class of pseudodifferential operators that have FredhoIm indices. The second is to actually prove the index theorem.In the first year of this research grant, we aimed at completing the first step. A prelim- inary study strongly indicated a difficulty in doing so, due to the lack of general spectral theory on those manifolds. This suggested us to study the geometry of hyperbolic mani- folds. Having had discussions with researchers of various fields in order to get information pertinent to our project, we managed to isolate a class of pseudodifferential operators that possibly have Fredholm indices and hoped to complete the first step. However, it was post- poned till the second year of the grant to prove that the class we isolated is the right on … More e. This goal turned out too ambitious due to technical obstacles, and unfortunately we could not comp1ete the first step of the project.While working on the objective discussed above, we worked at the same time on devel- oping basic tools needed for the proof of the index theorem. One of them is the notion of C^*-algebraic deformation quantizations of symplectic manifolds. In this direction, a significant progress has been made. In a joint paper with R.Nest of the University of Copenhagen (Paper 3 in Item 11 of this report) we studied the closed Riemannian sur- faces, which are primary examples of symplectic manifolds, and showed the existence of C^*-algebraic deformation quantizations. Generalizing this result, in a joint project with R.Nest and I.Peter of Muenster University, under a certain topological condition, we showed the existence of C^*-algebraic deformation quantization for any symplectic mani- fold. We plan further investigations into this direction. In particular, it is the up-coming project to study C^*-algebraic deformation quantization for Poisson manifolds, which are generalization of symplectic manifolds.As for the initial target of the research, that is, an index formula for pseudodifferential operators, we certainly intend to continue working on it. We will hopefully complete the project within a year or so. Less