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GEOMETRY OF NUMBERS AND CODING THEORY

Research Project

Project/Area Number 12640101
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTohoku University

Principal Investigator

UCHIDA Koji  Tohoku University, Graduate School of Medicine, Information Sciences, Professor, 大学院・情報科学研究科, 教授 (20004294)

Co-Investigator(Kenkyū-buntansha) ASOU Tohl  Tohoku University, Graduate School of Medicine, Information Sciences, Professor, 大学院・情報科学研究科, 助教授 (00111352)
IMAI Hideo  Tohoku University, Graduate School of Medicine, Information Sciences, Assoc. Professor, 大学院・情報科学研究科, 助教授 (10093668)
URAKAWA Hajime  Tohoku University, Graduate School of Medicine, Information Sciences, Professor, 大学院・情報科学研究科, 教授 (50022679)
SHIMOKAWA Koya  Tohoku University, Graduate School of Medicine, Information Sciences, Assis. Professor, 大学院・情報科学研究科, 助手 (60312633)
TAYA Hisao  Tohoku University, Graduate School of Medicine, Information Sciences, Assis. Professor, 大学院・情報科学研究科, 助手 (40257241)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2000: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsHarmonic morphism of graph / Discrete Laplacian / Inhomogeneous Yang-Mills equation / Einstein-Wey1 geometry / Totally real field / Lambda invariant / Dehn surgery / Heegaard splitting / CR manifold / pseudoharmonic map / Yang-Mills theory / Einstein-Weyl geometry / Greenberg's conjecture / P-class number / Heegaard splitting / Dehn surgery / 無限グラフ / スペクトル / グリーン核 / 熱核 / 岩澤不変量 / Z_P拡大 / Greenberg予想
Research Abstract

Urakawa defined the harmonic morphisms of graphs, and gave good estimate of the Green kernel of an infinite tree. He found good estimate of the spectrum of the discrete Laplacian of an infinite graphs. He built Hermitian connections of a vector bundle on a CR manifold, then showed existence and uniqueness of the solution of inhomogeneous Yang-Mills equation. He proved that CR-maps of pseudo convex CR manifolds are pseudoharmonic if and only if they are pseudohermitian. He also introduced the notion of stability of the maps, and proved pseudoharmonic maps into negatively curved Riemannian manifolds are stable. He developed the Yang-Mills theory without the equation Dh=0 for connections in a vector bundle over a Riemannian manifold, and applied this theory to Einstein-Wey1 geometry and to affine differential geometry.
Taya found the formula representing p-class numbers of intermediate fields of the cyclic Zpextension by the values of p-adic zeta functions, assuming Leopoldt conjecture. He also showed there exist infinitely many real quadratic fields in which the prime.3 splits such that lambda invariants are 0. He estimated the density of such fields.
Shimokawa considered Dehn surgeries on strongly invertible knots yielding lens spaces. He found conditions for a graph in the disc to contain some characteristic subgraphs. He showed any Heegaard splitting of trivial arcs in a compression body is standard. He introduced the notion of Heegaard splittings of the pair (M, T), where M is a compact orientable 3-manifold and T is a 1-submanifold.

Report

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

    (38 results)

All Other

All Publications (38 results)

  • [Publications] H.Urakawa: "A discrete analogue of the harmonic marphism"Harmonic morphisms, harmonic maps &.,2000. 97-108 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphisms and Green kernel comparison theorems"Glasgow Math. J.. 42. 319-334 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Urakawa(S.Dragomir): "On the inhomogeneous Yang. Mills equation d^*_DR^D=f"Interd. Inform. Sci.. 6. 41-52 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Urakawa: "The spectrum of an infinite graph"Canad. J. Math.. 52. 1057-1084 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.taya: "Remarks on Iwasawa λ-invariants"Alg. Number Th. and Diophantine Analysis. 453-465 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Taya: "Iwasawa invariants and class numbers of quadratic fields for the prime 3"Rroc. AMS. 128. 1285-1292 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Shimokawa(M.Hirasawa): "Dehn surgeries on strongly invertible knats which yield lens spaces"Rroc. AMS. 128. 3445-3451 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Urakawa(E.Barletta): "Psendobarmonic maps from nonnegative CR manifolds to Riemannian manifolds"Indiana Univ. Math. J.. 50. 719-746 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Urakawa(S.Dragomir): "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Taya: "On padic zeta functions and class groups of Zp erdenssions of certain totally real field"Advanced Studies in Pure Math.. 30. 401-414 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Shimokawa(C.Hayashi): "A condition for a graph in the disc to contain the half hemispleere of the dodecahedron"Interd. Inform. Sci.. 7. 5-16 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Shimokawa(C.Hayashi): "Heegaard splittings of trivial arcs in compression badies"J. of knat theory and its Ramifications. 10. 71-87 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Shimokawa(C.Hayashi): "Thin position of a pair(3-manifold, 1-subrmanifolop)"Pacific J. of Math.. 197. 301-324 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] UraKawa, Hajime: "A discrete analogue of the harmonic morphism"Harmonic morphisms, harmonic maps, and related topics. 97-108 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] UraKawa, Hajime: "A discrete analogue of the harmonic morphism and Green kernel comparison theorems"Glasgow Math. J.. 42. 319-334 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] UraKawa, Hajime: "The spectrum of an infinite graph"Canad. J. Math.. 52. 1057-1084 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Dragomir, Sorin and Urakawa, Hajime: "On the inhomogeneous Yang-Mills equation d^*_DR^D=f"Interd. Inform. Sci. 6. 41-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taya, Hisao: "Remarks on Iwasawa λ-invariants"Algebraic Number Theory and Diophantine Analysis. 453-465 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taya, Hisao: "Iwasawa invariants and class numbers of quadratic fields for the prime 3"Proc. AMS. 128. 1285-1292 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hirasawa, Mikami and Shimokawa, Koya: "Dehn surgeries on strongly invertible knots which yield lens spaces"Proc. AMS. 128. 3445-3451 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Barletta, Elisabetta, Dragomir, Sorin and Urakawa Hajime: "Pseudoharmonic maps from nonnegative CR manifolds to Riemannian manifolds"Indiana Univ. Math. J.. 50. 719-746 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Dragomir, Sorin, Ichiyama, Toshiyuki and Urakawa, Hajime: "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taya, Hisao: "On p-adic zeta functions and class groups of Z_Pextensions of certain totally real fields"Advanced Studies in Pure Math.. 30. 401-414 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hayashi, Chuichiro and Shimokawa, Koya: "A condition for a graph in the disc to contain the half hemisphere of the dodecahedron - Application to Dehn surgery theory"Interd. Inform. Sci.. 7. 5-16 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hayashi, Chuichiro and Shimokawa, Koya: "Heegaard splittings of trivial arcs in compression bodies"J. of Knot Theory and its Ramifications. 10. 71-87 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hayashi, Chuichiro and Shimokawa, Koya: "Thin position of a pair (3-manifold, 1-submanifold)"Pacific J. of Math.. 197. 301-324 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Urakawa (E.Barletta): "Pseudoharmonic maps from nonnegative CR manifolds to Riemannian manifolds"Indiana Univ. Math.J. 50・2. 719-746 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Urakawa (S.Draganir): "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Taya: "On p-adic zeta functions and class groups of Zp-extensions of certain totally real fields"Advanced Studies in Pure Math. 30. 401-414 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimokawa (C.Hayashi): "A condition for a graph in the disc to contain the half hemisphere of the dodecahedron-Application to Dehn surgery theory"Interd. Inform Sci.. 7・1. 5-16 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimokawa (C.Hayashi): "Heegaard splittings of trivial arcs in compression bodies"J.of Knot Theory and its Ramifications. 10-1. 71-87 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimokawa (C.Hayashi): "Thin position of a pair (3-manifold, 1-submanifold)"Pacific J. of Math. 197-2. 301-324 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism"Research Notes Math.. 413. 97-108 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa(S.Dragomir): "On the inhomogeneous Yang. Mills equation d_D*R_D=f"Interdisciplinary Information Sciences. 6・1. 41-52 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa: "A discrete analogue of the harmonic morphism and Green kernel comparison theorems"Glasgow Math.J.. 42. 319-334 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Urakawa: "The spectrum of an infinite graph"Canadian J.Math.. 52. 1057-1084 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hisao Taya: "Iwasawa invariants and class numbers of quadratic fields for the prime 3"Proc.AMS. 128. 1285-1292 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hisao Taya: "Remarks on Iwasawa λ-invariants"Algebraic Number Theory and Diophantine Analysis (Proc.Int.Conf.in Grag, Austria). 453-465 (2000)

    • Related Report
      2000 Annual Research Report

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

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