2001 Fiscal Year Final Research Report Summary
Geometric Structures on Manifolds and Graphs
| Project/Area Number |
12640073
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | OKAYAMA UNIVERSITY |
|---|
Principal Investigator |
KATSUDA Atsushi Okayama University, Faculty of Science, Associate Professor, 理学部, 助教授 (60183779)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SIMAKAWA Kazuhisa Okayama University, Faculty of Science, Professor, 理学部, 教授 (70109081)
TAMURA Hideo Okayama University, Faculty of Science, Professor, 理学部, 教授 (30022734)
SAKAI Takashi Okayama University, Faculty of Science, Professor, 理学部, 教授 (70005809)
TAKEUCHI Hiroshi Shikoku University, Faculty of Managements and Information, Professor, 経営情報学部, 教授 (20197271)
IKEDA Akira Okayama University, Faculty of Education, Professor, 教育学部, 教授 (30093363)
|
|---|
| Project Period (FY) |
2000 – 2001
|
| Keywords | random walks / nilpotent groups / Inverse spectral Problem / Gromov-Hausdorff convergence |
|---|
| Research Abstract |
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.
|
