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Development and applications of discrete geometric analysis

Research Project

Project/Area Number 21340039
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionMeiji University

Principal Investigator

SUNADA Toshikazu  明治大学, 理工学部, 教授 (20022741)

Co-Investigator(Kenkyū-buntansha) TATE Tatsuya  名古屋大学, 多元数理科学研究科, 准教授 (00317299)
HIGUCHI Yusuke  昭和大学, 教養部, 講師 (20286842)
AHARA Kazushi  明治大学, 理工学部, 准教授 (80247147)
Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥10,530,000 (Direct Cost: ¥8,100,000、Indirect Cost: ¥2,430,000)
Fiscal Year 2011: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2010: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2009: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Keywords離散幾何解析学 / 標準的実現 / 位相的結晶 / 離散的代数幾何学 / アーベル・ヤコビ写像 / 位相敵結晶理論 / 離散解析幾何学 / 量子ウォーク / 結晶格子
Research Abstract

The primarily purpose of this project was to provide a mathematical insight into the modern crystallography, a typical practical science that originated in the classification of the observed shapes of crystals. The tools we employed are adopted from algebraic topology, a field in pure mathematics cultivated during the first half of the last century. More specifically the elementary theory of covering spaces and homology is effectively used in the study of 3D networks associated with crystals. Further we formulate a minimum principle for crystals in the framework of discrete geometric analysis, which provides us with the concept of standard realizations, a canonical way to place a given crystal structure in space so as to produce the most symmetric microscopic shape. In spite of its pure-mathematical nature, this concept combined with homology theory turns out to fit with a systematic design and enumeration of crystal structures, an area of considerable scientific interest for many years. Meanwhile, standard realizations show up in asymptotic behaviors of random walks on topological crystals, the abstraction of crystal structures, and are closely related to a discrete analogue of Abel-Jacobi maps in algebraic geometry.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (19 results)

All 2012 2011 2010 2009

All Journal Article (13 results) (of which Peer Reviewed: 5 results) Presentation (4 results) Book (2 results)

  • [Journal Article] Lecture on topological crystallo-graphy2012

    • Author(s)
      T. Sunada
    • Journal Title

      Japan. J. Math

      Volume: 7 Pages: 1-39

    • NAID

      40019247536

    • Related Report
      2011 Final Research Report
  • [Journal Article] Asymptotic behavior of quantum walks on the line2012

    • Author(s)
      T. Sunada and T. Tate
    • Journal Title

      Journal of Functional Analysis

      Volume: 262 Pages: 2608-2645

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
  • [Journal Article] Lecture on topological crystallography2012

    • Author(s)
      T.Sunada
    • Journal Title

      Japan.J.Math.

      Volume: 7 Issue: 1 Pages: 1-39

    • DOI

      10.1007/s11537-012-1144-4

    • NAID

      40019247536

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the spectrum of a discrete Laplacian on with finitely supported potential2011

    • Author(s)
      Y. Higuchi, T. Matsumoto, O. Ogurisu
    • Journal Title

      Linear and Multilinear Algebra

      Volume: 59 Pages: 917-927

    • Related Report
      2011 Final Research Report
  • [Journal Article] On the spectrum of a discrete Laplacian on (¥bf Z) with finitely supported potential2011

    • Author(s)
      Yu.HIGUCHI, T.MATSUMOTO, O.OGURISU
    • Journal Title

      Linear Multilinear Algebra

      Volume: 59 Issue: 8 Pages: 917-927

    • DOI

      10.1080/03081087.2010.536981

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic Euler-Maclaurin formula over lattice polytopes2010

    • Author(s)
      T. Tate
    • Journal Title

      Journal of Functional Analysis

      Volume: 260 Pages: 501-540

    • Related Report
      2011 Final Research Report 2010 Annual Research Report
  • [Journal Article] A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions2010

    • Author(s)
      T. Tate
    • Journal Title

      Asymptotic Analysis

      Volume: 67 Pages: 101-123

    • Related Report
      2011 Final Research Report 2010 Annual Research Report
  • [Journal Article] Ray-Singer zeta functions for compact flat manifolds2009

    • Author(s)
      T. Sunada and H. Urakawa
    • Journal Title

      Contemporary Math

      Volume: 284 Pages: 287-297

    • Related Report
      2011 Final Research Report
  • [Journal Article] New metallic carbon crystal2009

    • Author(s)
      M. Itoh, M. Kotani, H. Naito, T. Sunada, Y. Kawazoe, and T. Adschiri
    • Journal Title

      Phys. Rev. Lett

      Volume: 102 Pages: 55703-4

    • Related Report
      2011 Final Research Report
  • [Journal Article] Spectral structure of the Laplacian on a covering graph2009

    • Author(s)
      Y. Higuchi and Y. Nomura
    • Journal Title

      European Journal of Combinatorics

      Volume: 30 Pages: 570-585

    • Related Report
      2011 Final Research Report
  • [Journal Article] Ray-Singer zeta functions for compact flat manifolds2009

    • Author(s)
      T.Sunada, H.Urakawa
    • Journal Title

      Contemporary Math. 284

      Pages: 287-294

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] New metallic carbon crystal2009

    • Author(s)
      M.Itoh, M.Kotani, H.Naito, T.Sunada, Y.Kawazoe, T.Adschiri
    • Journal Title

      Phys.Rev.Lett. 102

      Pages: 55703-55704

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spectral structure of the Laplacian on a covering graph2009

    • Author(s)
      Y.Higuchi, Y.Nomura
    • Journal Title

      European Journal of Combinatorics 30

      Pages: 570-585

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Topological Crystallography2011

    • Author(s)
      T.Sunada
    • Organizer
      International Conference on Mathematical Crystal lography
    • Place of Presentation
      University of the Philippines
    • Year and Date
      2011-11-02
    • Related Report
      2011 Annual Research Report
  • [Presentation] Topological Crystallography2011

    • Author(s)
      T. Sunada
    • Organizer
      International Conference on Mathematical Crystallography
    • Place of Presentation
      Manila
    • Related Report
      2011 Final Research Report
  • [Presentation] Quantum Walks2010

    • Author(s)
      T. Sunada
    • Organizer
      ICMA2010
    • Year and Date
      2010-12-09
    • Related Report
      2011 Final Research Report
  • [Presentation] Quantum Walks2010

    • Author(s)
      砂田利一
    • Organizer
      ICMA2010
    • Place of Presentation
      Chulalongkorn University
    • Year and Date
      2010-12-09
    • Related Report
      2010 Annual Research Report
  • [Book] 現代幾何学への道2010

    • Author(s)
      砂田利一
    • Publisher
      岩波書店
    • Related Report
      2011 Final Research Report 2010 Annual Research Report
  • [Book] 新版バナッハ・タルスキーのパラドックス2009

    • Author(s)
      砂田利一
    • Publisher
      岩波書店
    • Related Report
      2011 Final Research Report 2009 Annual Research Report

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Published: 2009-04-01   Modified: 2016-04-21  

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