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Topological toric theory and combinatorics

Research Project

Project/Area Number 17540092
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka City University

Principal Investigator

MASUDA Mikiya  Osaka City University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (00143371)

Co-Investigator(Kenkyū-buntansha) HASHIMOTO Yoshitake  Osaka City University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (20271182)
HIBI Takayuki  Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (80181113)
加藤 信  大阪市立大学, 大学院・理学研究科, 助教授 (10243354)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2006: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsToric manifold / fan / convex polytope / equivariant cohomology / Bott tower / GKM graph / topology / combinatorics / トーリック多様体論 / face ring / グラフ理論
Research Abstract

I have beein developing the theory of toric varieties in algebraic geometry from a topological point of view for these years. At the same time, a research from the same viewpoint has been independently developed in England and Russia, and now this emerging subject is called Toric Topology. In this research, I mainly studied the relation between topology and Combinatorics jointly with Taras Panov, Zhi Lu etc, and have made effort for developing this new subject. In particular, I organized an international conference on Toric Topology for a week from the end of May 2006 at Osaka City University. The organizers are Megumi Harad (Toronto), Yael Karshon(Toronto) and Taras Panov(Moscow) except me. This conference was supported mainly by 21COE program "Constitution of wide-angle mathematical basis focused on knots", and other resources are research grants of Shigenori-Matsumoto, Taras Panov, Yoshitake Hashimoto and this my research grant. Fortunately, we had 140 participants and half of them … More are foreigners. We realized that many people are interested in this emerging area.
As for my own research, I got the following outcome.
(1) Gullemin-Zara have studied the relation between the topology of manifolds with torus actions and graph theory. This is a very interesting study because it introduces ideas of topology to the study of graphs. Motivated by their research, we have introduced the notion of torus graph and showed that its equivariant cohomology agrees with the face ring of a simplicial poset which was already introduced in commutative algebra. The paper of this research has appeared in Adv. Math. 212 (2007), 458-483.
(2) It is known that equivariant cohomology contains a rich geometrical of a toric manifold and fits very well to the study of toric manifolds. In this research I have proved that equivariant cohomology completely distinguishes toric manifolds.
(3) A Bott tower is an iterated CP^1 bundle and provides a good family of toric manifolds. I worked with Taras Panov on the topological classification of those Bott towers (the paper is submitted). Also, I worked with Dong Youp Suh on the topological classification of higher Bott towers which are higher dimensional generalization of Bott towers. Less

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • Research Products

    (14 results)

All 2007 2006 2005

All Journal Article (14 results)

  • [Journal Article] Torus graphs and simplicial posets2007

    • Author(s)
      Mikiya Masda
    • Journal Title

      Advances in Mathematics 212

      Pages: 458-483

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Torus graphs and simplicial posets2007

    • Author(s)
      Mikiya Masuda
    • Journal Title

      Advances in Mathematics 212

      Pages: 458-483

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the cohomology of torus manifolds2006

    • Author(s)
      Mikiya Masuda
    • Journal Title

      Osaka Journal of Mathematics 43

      Pages: 711-746

    • NAID

      120004847007

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Cohomology of open torus manifolds2006

    • Author(s)
      Mikiya Masuda
    • Journal Title

      Tr. Mat. Inst. Steklova 252

      Pages: 158-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Cohomology of open torus manifolds2006

    • Author(s)
      Mikiya Masuda
    • Journal Title

      Tr. Mat. Inst. Steklov 252

      Pages: 158-166

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Equivariant cohomology determines (quasi) toric manifolds2006

    • Author(s)
      Mikiya Masuda
    • Journal Title

      数理解析研究所講究録 1517

      Pages: 10-13

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Elliptic genera, torus orbifolds and multi-fans2005

    • Author(s)
      Mikiya Masuda
    • Journal Title

      International Journal of Mathematics 16

      Pages: 957-998

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] New infinite series of Einstein metrics on sphere bundles from AdS black hole2005

    • Author(s)
      Yoshitake Hashimoto
    • Journal Title

      Communications in Mathematical Physics 257

      Pages: 273-285

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Elliptic genera, torus orbifolds and multi-fans2005

    • Author(s)
      Mikiya Masuda
    • Journal Title

      Intern. Journal of Math. 16

      Pages: 957-998

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] New infinite series of Einstein metrics on sphere bundles from AdS black holes2005

    • Author(s)
      Yoshitake Hashimto
    • Journal Title

      Comm. in Math. Physics 257

      Pages: 273-285

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] h-vectors of Gorenstein* simplicial posets2005

    • Author(s)
      Mikiya Masuda
    • Journal Title

      Advances in Mathematics 193巻2号

      Pages: 332-344

    • Related Report
      2005 Annual Research Report
  • [Journal Article] New infinite series of Einstein metrics on sphere bundles from AdS black holes2005

    • Author(s)
      Yoshitake Hashimoto
    • Journal Title

      Communications in Mathematical Physics 257巻

      Pages: 273-285

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Ideals of fiber type and polymatroids2005

    • Author(s)
      Takayuki Hibi
    • Journal Title

      Osaka Journal of Mathematics 42巻4号

      Pages: 807-829

    • NAID

      120004846969

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Elliptic genera, torus orbifolds and multi-fans2005

    • Author(s)
      Mikiya Masuda
    • Journal Title

      International Journal of Mathematics 16巻

      Pages: 957-998

    • Related Report
      2005 Annual Research Report

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Published: 2005-04-01   Modified: 2016-04-21  

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