Face enumeration of convex polytopes and cell complexes

• Project/Area Number: 25400043
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University (2014-2015); Yamaguchi University (2013)
• Principal Investigator: Murai Satoshi 大阪大学, 情報科学研究科, 准教授 (90570804)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 凸多面体 / cd指数 / 三角形分割 / スタンレー・ライスナー環 / f列 / 次数付ベッチ数 / cd index / f-列

Outline of Final Research Achievements

In this research project, we study the face numbers of polytopes and cell complexes. The followings are main results of this project.
(1) We find new method to study the cd-index of a convex polytope by using commutative algebra, and by applying this new method, we find new upper bounds of the cd-index.
(2) We find new applications of polyhedral Morse inequality and graded Betti numbers to the study of face numbers of triangulated manifolds.

Research Products

• [Int'l Joint Research] Osnabrueck University(ドイツ)
• [Int'l Joint Research] University of Washington(米国)
• [Journal Article] Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein* simplicial complexes (2016)
  • Author(s): Naoyuki Matsuoka, Satoshi Murai
  • Journal Title: Journal of Algebra Volume: 455 Pages: 14-31
  • DOI: 10.1016/j.jalgebra.2016.02.005
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Invariance of Pontrjagin classes for Bott manifolds (2015)
  • Author(s): Suyoung Choi, Mikiya Masuda, Satoshi Murai
  • Journal Title: Algebraic and Geometric Topology Volume: 15 Issue: 2 Pages: 965-986
  • DOI: 10.2140/agt.2015.15.965
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Tight combinatorial manifolds and graded Betti numbers (2015)
  • Author(s): Satoshi Murai
  • Journal Title: Collectanea Mathematica Volume: 印刷中 Issue: 3 Pages: 367-386
  • DOI: 10.1007/s13348-015-0137-z
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Squarefree P-modules and the cd-index (2014)
  • Author(s): Satoshi Murai, Kohji Yanagawa
  • Journal Title: Advances in Mathematics Volume: 265 Pages: 241-279
  • DOI: 10.1016/j.aim.2014.07.037
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Ring isomorphisms of cohomologies of certain toric manifolds (2015)
  • Author(s): Satoshi Murai
  • Organizer: 第37回可換環論シンポジウム
  • Place of Presentation: 倉敷シーサイドホテル（岡山）
  • Year and Date: 2015-11-22
• [Presentation] 多面体・多様体のstackedな三角形分割 (2015)
  • Author(s): 村井 聡
  • Organizer: Algebraic Topology from Combinatorial Viewpoint
  • Place of Presentation: 松山市立子規記念博物館（愛媛）
  • Year and Date: 2015-10-31
• [Presentation] Average of polyhedral Morse inequalities and graded Betti numbers (2015)
  • Author(s): Satoshi Murai
  • Organizer: Combinatorial and Experimental Methods in Commutative Algebra and Related Fields
  • Place of Presentation: Osnabrueck (ドイツ)
  • Year and Date: 2015-10-09
  • Int'l Joint Research / Invited
• [Presentation] 多様体の三角形分割の組合せ論と可換代数 (2015)
  • Author(s): 村井 聡
  • Organizer: 第60回代数学シンポジウム
  • Place of Presentation: 静岡大学（静岡）
  • Year and Date: 2015-08-31
  • Invited
• [Presentation] d彩色可能なd次元単体的凸多面体の面の個数の下限について (2015)
  • Author(s): 村井 聡
  • Organizer: デザイン、符号、グラフおよびその周辺
  • Place of Presentation: 京都大学（京都）
  • Year and Date: 2015-07-10
• [Presentation] New inequalities for face numbers of d-colorable simplicial d-polytopes (2015)
  • Author(s): Satoshi Murai
  • Organizer: Toric Topology 2015 in Osaka
  • Place of Presentation: 大阪市立大学（大阪）
  • Year and Date: 2015-06-16
• [Presentation] 球面の直積の三角形分割のtight性について (2015)
  • Author(s): 村井 聡
  • Organizer: 日本数学会2015年度年会
  • Place of Presentation: 明治大学 (東京)
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] A Lefschetz property for barycentric subdivisions (2015)
  • Author(s): Satoshi Murai
  • Organizer: Workshop on Lefschetz properties in Goettingen
  • Place of Presentation: Goettingen (ドイツ)
  • Year and Date: 2015-03-07 – 2015-03-14
• [Presentation] Balanced generalized lower bound inequality for simplicial polytopes (2015)
  • Author(s): Satoshi Murai
  • Organizer: Geometric and Algebraic Combinatorics
  • Place of Presentation: Oberwolfach (ドイツ)
  • Year and Date: 2015-02-01 – 2015-02-07
• [Presentation] Tight triangulations and graded Betti numbers (2014)
  • Author(s): Satoshi Murai
  • Organizer: Meeting on Combinatorial Commutative Algebra
  • Place of Presentation: Levico Terme (イタリア)
  • Year and Date: 2014-09-08 – 2014-09-12
• [Presentation] 多面体的複体のflag f-列について (2014)
  • Author(s): 村井 聡, 柳川 浩二
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学
• [Presentation] cd-index for CW-posets (2013)
  • Author(s): Satoshi Murai
  • Organizer: Combinatorial Methods in Topology and Algebra
  • Place of Presentation: Cortona (イタリア）
• [Presentation] Squarefree P-modules and its applications (2013)
  • Author(s): Satoshi Murai
  • Organizer: Commutative Algebra and its interaction to Algebraic Geometry and Combinatorics
  • Place of Presentation: Hanoi (ベトナム)
  • Invited