Relations of geometric structure of manifolds and graphs, spectre, asymptotic analysis and their applications

我们从几何的角度研究反问题。一个有边界的黎曼流形的边界距离表示是表示流形中给定点到边界上点的距离的函数集合。我们研究了如果该流形满足某些先验几何界限，则该表示是否以稳定的方式确定黎曼流形的问题。给出了一组离散的近似边界距离函数，构造了一个有限度量空间来逼近Gromov-Hausdorff拓扑中的流形，在应用中，边界距离表示出现在许多反问题中，其中测量对象的边界。例如，对于热传导率未知的热方程，边界测量确定对应于该传导率的黎曼度量的边界距离表示。Gel'fand反问题，它问一个具有边界的黎曼流形是否可以由Laplacian的特征值和特征函数的边界值确定，首先确定边界距离表示，其次确定内部的黎曼度量。麦克斯韦方程和狄拉克方程存在类似的问题，除了合作上述工作外，每个研究者都研究了自己的工作。清原研究的行为测地线的刘维流形，田村研究散射下两个螺线管磁场，岛川研究拓扑结构的配置空间，酒井研究isodiadic不等式，吉冈研究g-功能，田中研究半群的运营商。竹内研究p-调和函数图。

We studied inverse problems from geometric view points. A boundary distance representation of a Riemannian manifold with boundary is the set of functions which represent the distance from the given point in a manifold to points in the boundary. We study the question whether this representation determines the Riemannian manifolds in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold in the Gromov-Hausdorff topology.In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigations. For example, for the heat equation with unknown heat conductivity, the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity. The Gel'fand inverse problem, which asks whether a Riemannian manifold with boundaries can be determined by the eigenvalues and boundary values of the eigenfunctions of the Laplacian, first the boundary distance representation is determined and second they determines the Riemannian metric of the interior. Analogus problems exists for Maxwell and Dirac equations.Besides collaborating the above works, each investigator studied own works. Kiyohara studied behavior of geodesics on the Liouville manifolds, Tamura studied scattering under two solenoidal magnetic fields, Shimakawa studied topology of configuration spaces, Sakai studied isodiadic inequlities, Yoshioka studies g-fuctions, Tanaka studied semigroups of operators. Takeuchi studied p-harmonic functions on graphs.