Geometric Structures on Manifolds and Graphs

本文研究了有限图的幂零覆盖上的随机游动的渐近性态和广义Gel'fand逆谱问题的稳定性，作为前人研究的继续。第一个项目：热核和随机游动的渐近性态是概率论和整体分析中的热点。在这几个研究中，我们关注的是具有群作用对称性的无限图。本课题的目的是理解交换群的非交换形式的研究，特别是利用交换群的理论（傅立叶分析）的研究（例如Kotani，Shirai和Sunada的结果）。我们的策略是一个组合的表示理论的幂零李群的嵌入离散幂零群，半经典分析，陈的理论的迭代积分。我们需要几个领域的知识。目前，我们在簇图的覆盖情况下得到了一些结果，对于其他图还需要进一步的研究。值得注意的是，已有Alexopoulos，Ishiwata等人的工作，我们认为我们的方法在获得详细信息和应用其他问题，如双曲动力系统中闭轨道分布的可能性方面具有优点，后者是与Y. V. Kurylev（Loughborgh Univ.）和M. Lassas（赫尔辛基大学）在几年内。Gelfend逆问题是：能否从拉普拉斯算子的边界谱数据信息重建有边界流形上的Riemannian度量。我们写了一份调查文件的稳定性，增加了几个反例没有假设的有界几何。除了上述工作的曲率和拓扑Sakai，散射理论下magetic领域的田村，拓扑学的配置空间的shimakawa和p-Laplacian的图竹内。

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.