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Geometric Structures on Manifolds and Graphs

Research Project

Project/Area Number 12640073
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOKAYAMA 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)
石川 佳弘  岡山大学, 大学院・自然科学研究科, 助手 (50294400)
吉岡 巌  岡山大学, 理学部, 助手 (70033199)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsrandom walks / nilpotent groups / Inverse spectral Problem / Gromov-Hausdorff convergence / スペクトル / 逆問題 / 安定性 / 酔歩 / ハイゼンベルグ群 / ハーパー作用素 / 半古典近似
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.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] 勝田 篤: "Stability and Reconstruction in Gel'fand inverse spectral problem"Springer Lecture Note in Math.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 勝田 篤: "BC-method and stability of Gel'fand inverse spectral problem"数理解析研究所講究録. 1208. 24-35 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 酒井 隆: "Warped products, Ricci curvature and distance functions"Proc. of the Fifth Pacfic Rim Geometry. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 田村 英男: "Asymptotic behavior of scattering amplitudes in magnetic fields at large separation"J.Math.Soc. Japan. 53. 645-668 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 島川 和久: "Configuration spaces with partially summable labels and homology theories"Math.J. Okayama Univ.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Katsuda, Y.V. Kurylev and M. Lassas: "Stability and Reconstruction Gel'fand Inverse Boundary Spectral Problem""New geometric and analytic methods in inverse problems", Springer Lecture Note in Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Katsuda: "BC-method and stability of Gel'fand inverse spectral problem"Suuriken Koukyuuroku. 1208. 24-35 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Sakai: "Warped products, Ricci curvature and distance functions"Tohoku Math. Publications. 20. 163-172 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Tamura and H. T. Ito: "Asymptotic behavior of scattering amplitudes in magnetic fields at large separation"J. Math. Soc. Japan. 53. 645-668 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhisa Shimakawa: "Configuration spaces with partially summable labels and homology theories"Math. J. Okayama Univ.. 43 (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 勝田 篤: "Stability and Reconstruction in Gel'fand inverse spectral problem"Springer Lecture Note in Math.. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 勝田 篤: "BC-method and stability of Gel'fand inverse spectral problem"数理解析研究所講究録. 1208. 24-35 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 酒井 隆: "Warped products, Ricci curvature and distance functions"Proc. of the Fifth Pacfic Rim Geometry. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 田村 英男: "Asymptotic behavior of scattering amplitudes in magnetic fields at large separation"J. Math. Soc. Japan. 53. 645-668 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 島川 和久: "Configuration spaces with partially summable labels and homology theories"Math. J. Okayama Univ.. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 酒井隆: "Warped product, Ricci curvature and distance function"Tohoku Math. Pub1.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] 田村英男: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Ann.H.Poincare. (to apper).

    • Related Report
      2000 Annual Research Report
  • [Publications] 池田章: "Spectral Zeta functions for compact symetric spaces of rank one"Kodai Math.J.. 23. 345-357 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 石川佳弘: "The generalized Whittaker functions for $SU(2,1))$ and the Fourier expansion of automorphic forms"Proc.Japan.Acad.. 76. 56-60 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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