Microlocal analysis and pseudo-differential operators

1.微分方程（内田，岩崎，杉本）（1）利用热方程将局部指标定理推广到有边流形和Koeler流形。（2）通过对一般微分方程组的强伪凸性条件的表述，给出了Hoermander唯一性定理的几何证明。（3）通过关联动力系统的简单系统变换，研究了色散方程的光滑性。2.伪微分算子（Ashino）的数值分析与应用（1）奇异值分解和小波分析在图像分析中的应用。3.概率论（Kotani，Isozaki）（1）通过推广偏差定理，得到了一个包含三个随机变量的定理，并将其应用于2维随机游动的半直线的命中时间分布的研究。(2)A得到了由Burgers方程产生的团簇在随机初值下的分布的最终答案。（3）通过引入一维空间中对应于Lyapounov指数的绿色函数的指数，研究了具有随机势的高维Schroedinger算子.对于一维算子，利用Sato理论构造了KdV流.（4）从鞅和Krein谱理论的角度研究了一维扩散过程。

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.