Microlocal analysis and pseudo-differential operators

1.微分方程（Uchida, Iwasaki, Sugimoto）(1)应用热方程将局部指标定理推广到有边界流形和Koeler流形。(2)通过构造一般微分方程组的强伪凸条件，从几何上证明了Hoermander的唯一性定理。(3)考虑了色散方程的平滑性，通过关联动力系统进行简单系统的变换。(1)将奇异值分解和小波分析应用于图像分析。(2)应用小波对系统进行识别。概率论（Kotani, Isozaki）(1)通过对偏差定理的推广，得到了一个包含三个随机变量的定理，并将其应用于研究2-dim半线上的命中时间分布。随机漫步。(2)得到了初始值随机的Burgers方程生成的聚类分布的最终解。(3)通过引入对应于1-dim中的Lyapounov指数的Green函数的指数，研究了具有随机势的高维薛定谔算子。1-dim。算子，利用佐藤理论构造了kdv流。从鞅和Krein谱理论的角度研究了扩散过程。

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.