Microlocal analysis and pseudo-differential operators

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1.Differential equations (Uchida, Iwasaki, Sugimoto)(1)The local index theorem was extended to manifolds with boundary and Koeler manifolds by applying heat equations.(2)Hoermander's uniqueness theorem was proved geometrically by formulating his strong pseudo-convex condition for general differential equation systems.(3)The smoothing property for dispersive equations was considered by transforming by simpler systems through associated dynamical systems.2.Numerical analysis and applications of pseudo-differential operators (Ashino)(1)Decomposition of singular values and wavelet analysis were applied to image analysis.(2)Wavelet was applied to identify the systems.3.Probability theory (Kotani, Isozaki)(1)By extending the deviation theorem, a theorem including three random variables was obtained and was applied to the investigation of the distribution of the hitting time to the half line of 2-dim. random walks.(2)A final answer for the distribution of clusters created by Burgers equations with random initial values was obtained.(3)Higher dimensional Schroedinger operators with random potentials was studied by introducing an exponents of the Green functions corresponding to the Lyapounov exponents in 1-dim.. For 1-dim. operators, the KdV-flow was constructed by employing Sato's theory.(4)1-dim. diffusion processes were studied from the point of view of martingales and Krein's spectral theory.