Riemann surfaces and low dimensional manifolds

• Project/Area Number: 26400095
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Gakushuin University
• Principal Investigator: Matsumoto Yukio 学習院大学, 理学部, 研究員 (20011637)
• Co-Investigator(Renkei-kenkyūsha): Ashikaga Tadashi 東北学院大学, 工学部, 教授 (90125203) ; Komori Yohei 早稲田大学, 教育総合科学学術院, 教授 (70264794) ; Ohmoto Toru 北海道大学, 理学研究科, 教授 (20264400)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: リーマン面 / ドリーニュ・マンフォードコンパクト化 / オービフォールド / 結晶群 / ４次元多様体 / タイヒミュラー空間 / モジュライ空間 / カーブ複体 / 写像類群 / コンパクト化

Outline of Final Research Achievements

It is known that the moduli space of Riemann surfaces admits a natural compactification called the Deligne-Mumford compactification (DM-compactification). The main result of the present research is that we explicitly constructed a "natural" atlas consisting of orbifold charts on the DM-compactification of moduli space. These charts are indexed by the simplices of Harvey's curve complex. As a byproduct of the result, we discovered that certain higher dimensional euclidean crystallographic groups are attached to those orbifold charts that are indexed by the simplices of the maximum dimension. The theoretical meaning of this attachment of crystallographic groups will be studied in the future.

Research Products

• [Presentation] Orbifold charts of the compactified moduli spaces and crystallographic groups (2018)
  • Author(s): 松本幸夫
  • Organizer: リーマン面・不連続群研究集会
  • Invited
• [Presentation] ４次元多様体とLefschetzファイバー空間 (2017)
  • Author(s): 松本幸夫
  • Organizer: 第６４回トポロジーシンポジューム
  • Invited
• [Presentation] Compactification of moduli spaces and crystallographic groups (2016)
  • Author(s): Yukio Matsumoto
  • Organizer: Workshop on Grothendieck-Teichmuller Theories
  • Place of Presentation: Nankai University, Tianjin, China
  • Year and Date: 2016-07-25
  • Invited
• [Presentation] リーマン面のモジュライ空間と結晶群 (2016)
  • Author(s): 松本幸夫
  • Organizer: 第８回早稲田大学 数学・応用数学談話会
  • Place of Presentation: 早稲田大学 西早稲田キャンパス６３号館
  • Year and Date: 2016-06-30
  • Invited
• [Presentation] Moduli spaces and crystallographic groups (2016)
  • Author(s): Yukio Matsumoto
  • Organizer: Two lectures at Fudan University
  • Place of Presentation: Fudan University, Shanghai, China
  • Year and Date: 2016-06-01
  • Invited
• [Presentation] Riemann surfaces and crystallographic groups (2016)
  • Author(s): Yukio Matsumoto
  • Organizer: International workshop on low-dimensional topology
  • Place of Presentation: Dalian University of Technology, Dalian, China
  • Year and Date: 2016-05-06
  • Invited
• [Presentation] Riemann surfaces and crystallographic groups (2016)
  • Author(s): Yukio Matsumoto
  • Organizer: Colloquim Talk
  • Place of Presentation: University of South Florida, USA
  • Year and Date: 2016-04-22
  • Invited
• [Presentation] モジュライ空間のコンパクト化と結晶群 (2016)
  • Author(s): 松本幸夫
  • Organizer: 2016年日本数学会年会、特別講演
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-16
  • Invited
• [Presentation] Riemann surfaces and Crystallographic Groups (2016)
  • Author(s): Yukio Matsumoto
  • Organizer: 2016 Winter TAPU Workshop on Knots and Related Topics
  • Place of Presentation: College of Natural Sciences, Kyungpook National University, Daegu, Korea
  • Year and Date: 2016-02-16
  • Int'l Joint Research / Invited
• [Presentation] Crystallographic groups arising from Teichmuller spaces (2015)
  • Author(s): Yukio Matsumoto
  • Organizer: Master-Class, Finsler geometry and applications to low-dimensional geometry and topology and moduli spaces
  • Place of Presentation: Palazzo delle Scienze, Cagliari-Sardinia, Italy
  • Year and Date: 2015-09-24
  • Int'l Joint Research / Invited
• [Presentation] モジュライ空間、カーブ複体、結晶群 (2015)
  • Author(s): 松本幸夫
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2015-08-27
  • Invited
• [Presentation] Curve complexes and the DM-compactification of moduli spaces of Riemann surfaces (2015)
  • Author(s): Yukio Matsumoto
  • Organizer: RIMS Seminar "Topology, Geometry and Algebra of low dimensional manifolds"
  • Place of Presentation: KKR Numazu Hamayu
  • Year and Date: 2015-05-29
  • Int'l Joint Research / Invited
• [Presentation] Curve complexes and the DM-compactification of moduli spaces (2015)
  • Author(s): 松本幸夫
  • Organizer: Branched coverings, Degenerations and Related Topics
  • Place of Presentation: 東北学院大学、多賀城キャンパス
  • Year and Date: 2015-02-24
  • Invited
• [Presentation] Relatively non-singular hermitian forms in low and higher dimensional knot theory (2015)
  • Author(s): Yukio Matsumoto （松本幸夫）
  • Organizer: the 10th East Asian School of Knots and Related Topics
  • Place of Presentation: 華東師範大学、上海
  • Year and Date: 2015-01-26
• [Presentation] Ivanovの「切り詰めタイヒミュラー空間」とモジュライ空間のコンパクト化について (2014)
  • Author(s): 松本幸夫
  • Organizer: 研究集会「多様体のトポロジーの展望」
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2014-11-29
  • Invited
• [Presentation] Teichmuller spaces as infinite polyhedra (2014)
  • Author(s): Yukio Matsumoto （松本幸夫）
  • Organizer: Summer School-Finsler geometry with applications
  • Place of Presentation: Univ. the Aegean, Greece
  • Year and Date: 2014-09-25
  • Invited
• [Presentation] Teichmuller spaces as infinite polyhedra (2014)
  • Author(s): Yukio Matsumoto （松本幸夫）
  • Organizer: FJV2014-Sapporo (The 2nd Franco-Japanese-Vietnames Sympo.)
  • Place of Presentation: 北海道大学、理学部
  • Year and Date: 2014-08-28
  • Invited
• [Presentation] Four dimensional manifolds and Riemann surfaces (2 lectures) (2014)
  • Author(s): Yukio Matsumoto （松本幸夫）
  • Organizer: Fudan University, Geometry seminar
  • Place of Presentation: 復旦大学、上海
  • Year and Date: 2014-05-13 – 2014-05-16
  • Invited