Titlebook: Noncommutative Differential Geometry and Its Applications to Physics; Proceedings of the W Yoshiaki Maeda,Hitoshi Moriyoshi,Satoshi Watamur

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Remarks on the Characteristic Classes Associated with the Group of Fourier Integral Operators,ar symplectic group. Analogously we also introduce secondary characteristic classes associated with Lie group FIO(N) of invertible Fourier integral operators on a manifold . ([.,., .] and [.,.]). As an application we find a nontrivial cycle in FIO(..) as a lift of the geodesic flow of a sphere.

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Singular Systems of Exponential Functions,enomenas different from formal deformation quantization by studying the convergence of the parameter in computing the product of the exponential functions of the quadratic form. In the case of deformation quantization with the positive real parameters, the associativity for the Moyal product fails f

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On Geometry of Non-Abelian Duality,m groups, using a non-abelian version of Poincaré duality. A continuous classical version (Poisson-Lie T-duality) have some simple mechanical analogs. Finally a connection between Poisson-Lie T-duality and Courant algebroids is presented, which conjecturaly gives a unified picture.

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,Weyl Calculus and Wigner Transform on the Poincaré Disk,ect to the Schr?dinger representation of the Heisenberg group, and the Wigner transform is defined to be the Fourier transform of the matrix coefficients of this representation. As in [.], [.], the Weyl calculus and the Wigner transform are closely related to each other. In this paper we will define

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,*-Algebraic Deformation and Index Theory,In this note we explain how .*-algebraic deformation of symplectic manifolds yields a down to earth proof of Atiyah-Singer type of index formulae.