Geometric Structures on Manifolds and Graphs

流形和图上的几何结构

基本信息

批准号：
12640073
负责人：
KATSUDA Atsushi
金额：
$ 2.24万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

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.

我们已经研究了，随机步行的渐近行为在有限图的nilpotent覆盖范围内以及广义凝胶频谱问题的稳定性作为先前研究的延续。第一个项目：热仁和随机步行的渐近性和随机步行的渐近性对概率理论感兴趣和全球分析。在几项研究中，我们关注的是，在无限的图表上，群体对动作的对称性进行了对称。该项目是针对以前研究的非交通性版本的理解，尤其是使用阿伯利亚群体理论（即傅立叶分析）进行的研究），例如。 Kotani，Shirai和Sunada的结果）。我们的策略是通过嵌入离散的nilpotent群体，半古典分析，Chen的迭代积分理论来结合Nilpotent Lie组的表示理论。我们需要了解几个领域。在这一刻，我们在花束图的覆盖率并需要进一步研究其他图的情况下获得了一些结果。应该注意的是，有些作品Alexopoulos，Ishiwata等人。我们认为，我们的方法具有获取详细信息并应用其他一些问题的可能性，例如双曲线动力系统中闭合轨道的分布。后者是与Y.V.的关节合作。 Kurylev（Loughborgh Univ。）和M. Lassas（Helsinki Univ。）多年。凝胶反面问题是folloings；一个人可以根据laplacian的OUDCOLARY SPECTRAL数据的信息重建歧管上的Riemannian度量。我们撰写了一份调查文件，以解决此问题的稳定性，并在不假定有限的几何形状的情况下添加了几个反示例。上面，Sakai的曲率和拓扑作品是tamura下的曲率和拓扑作品，Tamura在岩浆领域下的散射理论，构型的构造空间，由配置空间进行。 Shimakawa和P-Laplacian在Takeuchi的图表上。