Geometric Structures on Manifolds and Graphs

作为前人研究的延续，我们研究了有限图的幂零覆盖上随机漫步的渐近行为和广义Gel'fand逆谱问题的稳定性。第一个专题：热核和随机漫步的渐近性，对概率论和全局分析感兴趣。在这几项研究中，我们关注的是群作用对称的无限图。本项目旨在理解以往关于阿贝尔群研究的非交换版本，特别是利用阿贝尔群理论（即傅立叶分析）所做的研究。Kotani， Shirai和Sunada的结果)。我们的策略是将幂零李群的表示理论与离散幂零群的嵌入、半经典分析、陈的迭代积分理论相结合。我们需要几个领域的知识。此时，我们已经在覆盖花束图的情况下取得了一些成果，对于其他图还需要进一步的研究。需要注意的是，Alexopoulos， Ishiwata等人的一些作品。我们认为，我们的方法有可能获得详细的信息，并应用于其他一些问题，例如双曲动力系统中封闭轨道的分布。后者是与yv Kurylev（拉夫堡大学）和M. Lassas（赫尔辛基大学）在几年的联合工作。凝胶反问题如下；利用拉氏度规的边界谱数据信息，可以重构有边界流形上的黎曼度规。我们写了一篇关于这个问题的稳定性的调查论文，并在不假设有界几何的情况下增加了几个反例。除此之外，还有Sakai关于曲率和拓扑的著作，Tamura关于磁场下散射理论的著作，shimakawa关于构形空间拓扑的著作，Takeuchi关于图上的p-Laplacian的著作。

We have studied that asymptotic behavior of random walks on nilpotent coverings of finite graphs and the stability of the generalized Gel'fand inverse spectral problems as a continuation of previous researches.The first project: asymptotics of heat kernels and random walks are interested in probability theory and global analysis. Among the several researches, our concern is that on infinite graphs with the symmetry of the action by groups. This project is directed toward understandings of non-commutative version of the previous researches in the case of abelian groups, especially, researches done by using the theory of abelian groups, i.e. Fourier Analysis )e.g. results of Kotani, Shirai and Sunada). Our strategy is a combination of the representation theory of nilpotent Lie groups by an embedding of discrete nilpotent groups, semi-classical analysis, Chen's theory of the iterated integrals. We need to knowledge of several fields. In this moment, we have obtained some results in the case when the cover of the bouquet graph and need to further research for other graphs. It should be noticed that there are some works Alexopoulos, Ishiwata et al. We believe that our method has merit in the possibilities to obtain the detailed informations and apply some other problems, e.g. distribution of closed orbits in hyperbolic dynamical systems.The latter is the joint works with Y.V. Kurylev (Loughborgh Univ.) and M. Lassas (Helsinki Univ.) during several years. Gel'fend inverse problem is the folloings; Can one reconstruct the Riemannian metric on manifold with boundary from the information of the oundary spectral data of the Laplacian. We wrote a survey paper for the stability of this problem with adding several counter examples without assumption of bounded geometry.Besides the above, there are works on curvature and topology by Sakai, the scattering theory under magmetic fields by Tamura, tpology of configuration spaces by shimakawa and p-Laplacian on graphs by Takeuchi.