• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Triangulations of polytopes and manifolds with nice coloring structure

Research Project

Project/Area Number 16K05102
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionWaseda University (2018-2019)
Osaka University (2016-2017)

Principal Investigator

Murai Satoshi  早稲田大学, 教育・総合科学学術院, 教授 (90570804)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords単体的複体 / 彩色 / スタンレー・ライスナー環 / 単体分割 / 凸多面体 / レフシェッツ性 / 単項式イデアル / f-列 / local h-vector / 三角形分割 / 頂点彩色
Outline of Final Research Achievements

In this research, we study combinatorial structures of triangulated manifolds having a nice coloring structure, called balanced. Here are main achievement.

[1] We give an affirmative answer to the conjecture of Novik and Klee on lower bounds of the face numbers of balanced triangulated manifolds. [2] We show that any (1,1,1)-balanced 3-polytope has the Lefschetz property with respect to a colored linear system of parameters, while (2,1)-balanced 3-polytopes do not always have this property. [3] We solve a problem posed by Izmestiev, Klee and Novik, on pentagon moves for balanced triangulated surfaces.

Academic Significance and Societal Importance of the Research Achievements

凸多面体や多様体の単体分割の組合せ構造の研究は組合せ論の分野における重要な研究テーマの一つである。一方、四色定理に代表される彩色の研究も数学における重要な研究テーマである。本研究では、これら二つのテーマの両方に関連する研究である、良い彩色構造をもつ単体分割の代数的・組合せ論的な構造に関する研究を行った。
今回の研究により、面の個数の下限・スタンレー・ライスナー環のレフシェッツ性・単体分割を変形によって構成する手法、などに関して未解決であった問題を解決することに成功し、単体分割の代数構造・組合せ構造の解明に貢献した。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (31 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (9 results) Journal Article (9 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 9 results,  Open Access: 7 results) Presentation (9 results) (of which Int'l Joint Research: 3 results,  Invited: 4 results) Remarks (3 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Hobart and William Smith Colleges/Cleveland State University/University of Kentucky(米国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Universidad Autonoma de Zacatecas(メキシコ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Universitaet Osnabrueck(ドイツ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] シアトル大学(米国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Uniersitaet Osnabrueck(ドイツ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Hebrew University(イスラエル)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Universidad de Barcelona(スペイン)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Universitaet Osnabrueck(ドイツ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Washington(米国)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Betti tables of monomial ideals fixed by permutations of the variables2020

    • Author(s)
      Satoshi Murai
    • Journal Title

      Transaction of the American Mathematical Society

      Volume: 論文受理

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Numbers of Edges of 5-Polytopes with a Given Number of Vertices2019

    • Author(s)
      Kusunoki Takuya、Murai Satoshi
    • Journal Title

      Annals of Combinatorics

      Volume: 23 Issue: 1 Pages: 89-101

    • DOI

      10.1007/s00026-019-00417-y

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Exceptional Balanced Triangulations on Surfaces2019

    • Author(s)
      Klee Steven、Murai Satoshi、Suzuki Yusuke
    • Journal Title

      Graphs and Combinatorics

      Volume: 印刷中 Issue: 6 Pages: 1361-1373

    • DOI

      10.1007/s00373-018-2001-x

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Lefschetz properties for complete intersection ideals generated by products of linear forms2018

    • Author(s)
      Juhnke-Kubitzke Martina、Miro-Roig Rosa M.、Murai Satoshi、Wachi Akihito
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 146 Issue: 8 Pages: 3249-3256

    • DOI

      10.1090/proc/14009

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Local h-Vectors of Quasi-Geometric and Barycentric Subdivisions2018

    • Author(s)
      Juhnke-Kubitzke Martina、Murai Satoshi、Sieg Richard
    • Journal Title

      Discrete & Computational Geometry

      Volume: 61 Issue: 2 Pages: 364-379

    • DOI

      10.1007/s00454-018-9986-z

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A generalized lower bound theorem for balanced manifolds2018

    • Author(s)
      Juhnke-Kubitzke Martina、Murai Satoshi、Novik Isabella、Sawaske Connor
    • Journal Title

      Mathematische Zeitschrift

      Volume: 印刷中 Issue: 3-4 Pages: 921-942

    • DOI

      10.1007/s00209-017-1981-1

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Lefschetz properties of balanced 3-polytopes2018

    • Author(s)
      David Cook II, Martina Juhnke-Kubitzke, Satoshi Murai, Eran Nevo
    • Journal Title

      Rocky Mountain Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Face numbers and the fundamental group2017

    • Author(s)
      Murai Satoshi、Novik Isabella
    • Journal Title

      Israel Journal of Mathematics

      Volume: 222 Issue: 1 Pages: 297-315

    • DOI

      10.1007/s11856-017-1591-y

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Balanced subdivisions and flips on surfaces2017

    • Author(s)
      Murai Satoshi、Suzuki Yusuke
    • Journal Title

      Proceedings of the American Mathematical society

      Volume: 146 Issue: 3 Pages: 939-951

    • DOI

      10.1090/proc/13775

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] 凸多面体の面の個数に関するあれこれ2020

    • Author(s)
      村井 聡
    • Organizer
      第3回数理新人セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] h-vectors and the number of generators of fundamental groups2018

    • Author(s)
      Satoshi Murai
    • Organizer
      New Trends in Syzygies
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] 基本群の生成元の個数と単体分割の辺の数2018

    • Author(s)
      村井 聡
    • Organizer
      変換群論における幾何・代数・組み合わせ論
    • Related Report
      2018 Research-status Report
  • [Presentation] 凸多面体の面の数え上げ論の近況2017

    • Author(s)
      村井 聡
    • Organizer
      2017年日本数学会年会
    • Place of Presentation
      首都大学東京(東京都八王子市)
    • Year and Date
      2017-03-24
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 多様体の単体分割の持つ組合せ論的・代数的対称性2017

    • Author(s)
      村井 聡
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Face numbers and the fundamental group2017

    • Author(s)
      Satoshi Murai
    • Organizer
      2017 CMS Winter Meeting
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] A duality of Buchsbaum rings and triangulated manifolds2016

    • Author(s)
      Satoshi Murai
    • Organizer
      Asymptotic Phenomena in Local Algebra and Singularity Theory
    • Place of Presentation
      Oberwolfach (ドイツ)
    • Year and Date
      2016-12-11
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Face numbers of balanced simplicial polytopes and complexes2016

    • Author(s)
      村井 聡
    • Organizer
      第38回可換環論シンポジウム
    • Place of Presentation
      IPC生産国際交流センター(神奈川県三浦郡)
    • Year and Date
      2016-11-18
    • Related Report
      2016 Research-status Report
  • [Presentation] 多様体の単体分割の組合せ論と代数2016

    • Author(s)
      村井 聡
    • Organizer
      第63回トポロジーシンポジウム
    • Place of Presentation
      神戸大学(兵庫県神戸市)
    • Year and Date
      2016-07-05
    • Related Report
      2016 Research-status Report
    • Invited
  • [Remarks] 研究代表者webページ

    • Related Report
      2018 Research-status Report
  • [Remarks] 研究代表者webページ

    • URL

      http://www.f.waseda.jp/s-murai/

    • Related Report
      2017 Research-status Report
  • [Remarks] 研究代表者webページ

    • URL

      http://www.math.sci.osaka-u.ac.jp/~s-murai/

    • Related Report
      2016 Research-status Report
  • [Funded Workshop] Combinatorics on Convex Polytopes2018

    • Related Report
      2018 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2022-02-16  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi