Foliations, contact structures, and symplectic structures on 3,4, and 5 dimensional manifolds

• Project/Area Number: 17H02845
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Chuo University
• Principal Investigator: Mitsumatsu Yoshihiko 中央大学, 理工学部, 教授 (70190725)
• Co-Investigator(Kenkyū-buntansha): 直江 央寛 中央大学, 理工学部, 助教 (10823255) ; 高倉 樹 中央大学, 理工学部, 教授 (30268974) ; 三好 重明 中央大学, 理工学部, 教授 (60166212)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥9,490,000 (Direct Cost: ¥7,300,000、Indirect Cost: ¥2,190,000); Fiscal Year 2020: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2019: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2018: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2017: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
• Keywords: 葉層構造 / 接触構造 / シンプレクティック構造 / Anosov 流 / カスプ特異点 / レフシェッツ・ファイブレーション / ポアソン構造 / ミルナー・ファイバー / Lefschetz fibration / シンプレクティック葉層構造 / Engel 構造 / 3次元Anosov流 / symplectic構造 / Milnor fiber / 複素構造 / Milnor fibre / 尖点特異点 / turbulization / 強擬凸性

Outline of Final Research Achievements

Based on the construction of Lefschetz fibrations and the existence of Anosov systems, we explained the structures of certain closed symplectic 4-manifolds and the regular Poisson structures on the 5-sphere. It also yields that the foliation on the 5-sphere constructed by Lawson is understood as a foliated Lefschetz fibration over the Reeb foliation on the 3-sphere.

Academic Significance and Societal Importance of the Research Achievements

次元横断的に関連する幾何構造を重要な例において有機的に研究することにより、特に重要な空間に対する幾何学的な理解を進化させた。多くの幾何構造、空間を取り扱った研究であるが、3次元のAnosov系という双曲的な力学系が背後で有機的に関連しており、個々の独立しうる理論が深く関連する地点を指示している。現代の数学理論の多くがK3曲面を舞台とするようになってきている状況で、K3曲面に新たな位相的理解を与えており、多くの数学の融合した発展に寄与すると思われる。

Research Products

• [Int'l Joint Research] 南ブルターニュ大学数学研究所(フランス)
• [Int'l Joint Research] ブリュッセル自由大学(ベルギー)
• [Int'l Joint Research] ベルリン自由大学数学研究所(ドイツ)
• [Int'l Joint Research] プリンストン高等研究所/American Institute of Mathematics/Stanford 大学(米国)
• [Int'l Joint Research] ベルリン自由大学(ドイツ)
• [Int'l Joint Research] ブリュッセル自由大学(ベルギー)
• [Int'l Joint Research] IMPA/PUC-RIO/UF-Fluminense(ブラジル)
• [Int'l Joint Research] 南ブルターニュ大学(フランス)
• [Int'l Joint Research] Universit\'e de Nantes/Universtit\'e de Sud-Bretagne(フランス)
• [Int'l Joint Research] Freie Universit\"at Berlin(ドイツ)
• [Int'l Joint Research] American Institute of Mathematics/Northwestern University/Stanford University(米国)
• [Int'l Joint Research] Université de Lyon 1/Université de Sud-Bretagne(フランス)
• [Int'l Joint Research] Freie Universität Berlin(ドイツ)
• [Int'l Joint Research] Université Libre de Bruxelles(ベルギー)
• [Journal Article] Corks with Large Shadow-Complexity and Exotic Four-Manifolds (2021)
  • Author(s): Naoe Hironobu
  • Journal Title: Experimental Mathematics Volume: 30 Issue: 2 Pages: 157-171
  • DOI: 10.1080/10586458.2018.1514332
  • Peer Reviewed
• [Journal Article] Shadows of acyclic 4-manifolds with sphere boundary (2020)
  • Author(s): Yuya Koda, Hironobu Naoe
  • Journal Title: Algebraic & Geometric Topology Volume: 20 Issue: 7 Pages: 3707-3731
  • DOI: 10.2140/agt.2020.20.3707
  • Peer Reviewed
• [Journal Article] On Volume Functions of Special Flow Polytopes Associated to the Root System of Type A (2020)
  • Author(s): Takayuki Negishi, Yuki Sugiyama and Tatsuru Takakura
  • Journal Title: The Electronic Journal of Combinatorics Volume: 27 Issue: 4
  • DOI: 10.37236/9062
  • Peer Reviewed
• [Journal Article] Leafwise symplectic structures on Lawson’s foliation (2018)
  • Author(s): Mitsumatsu Yoshihiko
  • Journal Title: Journal of Symplectic Geometry Volume: 16 Issue: 3 Pages: 817-838
  • DOI: 10.4310/jsg.2018.v16.n3.a6
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Leafwise Symplectic Structures on Lawson's Foliation (2018)
  • Author(s): Yoshihiko MITSUMATSU
  • Journal Title: Journal of Symplectic Geometry Volume: 印刷中
  • Peer Reviewed / Open Access
• [Presentation] カスプ特異点および単純楕円特異点の Milnor fiber 上の Lefschetz fibratio (2022)
  • Author(s): 三松佳彦
  • Organizer: 接触構造、特異点、微分方程式及びその周辺
• [Presentation] Presentation of the fundamental groups of complements of shadows (2022)
  • Author(s): 直江央寛
  • Organizer: Branched Coverings, Degenerations, and Related Topics 2022
  • Invited
• [Presentation] Lefschetz fibration on the Milnor fibers of simple elliptic and cusp singularities (2021)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: AIM Workshop on Conformal symplectic structures, contact topology, and foliations
  • Int'l Joint Research / Invited
• [Presentation] Lefschetz fibrations on the Milnor fibers of cusp singularities and applications (2021)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: IAS Princeton Workshop on the h-principle and beyond
  • Int'l Joint Research / Invited
• [Presentation] 正則 Poisson 構造=symplectic 葉層の存在と構成について (2021)
  • Author(s): 三松佳彦
  • Organizer: Poisson幾何とその周辺 21
  • Invited
• [Presentation] Positive flow-spines and contact 3-manifolds (2021)
  • Author(s): 石井一平、石川昌治、古宇田悠哉、直江央寛
  • Organizer: 日本数学会 2021 年度秋季総合分科会
• [Presentation] Rectification of Lefschetz-like singularities (2020)
  • Author(s): 三松 佳彦
  • Organizer: トポロジープロジェクト研究集会 「接触構造、特異点、微分方程式及びその周辺」(静岡)
  • Invited
• [Presentation] フロースパインと接触構造 II (2020)
  • Author(s): 直江 央寛
  • Organizer: トポロジープロジェクト研究集会 「接触構造、特異点、微分方程式及びその周辺」(静岡)
  • Invited
• [Presentation] The symmetry of Reeb components with complex leaves (2019)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: PUC-Rio(Brazil) Geometry Seminar
  • Invited
• [Presentation] Minicourse ``Lefschetz fibrations on Milnor fibres of cusp and simple elliptic singularities'' (2019)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: IMPA(Brazil) Thematic Semester on Symplectic Geometry and Poisson Geometry
  • Invited
• [Presentation] Lefschetz fibrations of divides and shadows (2019)
  • Author(s): Hironobu Naoe
  • Organizer: 東京大学 Hyper plane arrangements and Japanese Australian workshop on Real and Complex Singularities
  • Int'l Joint Research
• [Presentation] L'\'equation de Navier-Stokes/Euler sur la vari\'et\'e riemannienne compacte (2018)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: S\'eminaire de G\'eom\'etrie, Universit\'e de Nantes
  • Invited
• [Presentation] Composant de Reeb (un feuilletage de codimension 1 sur le tore solide de dim 3) aux feuilles complexes et son symmetrie (2018)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: S\'eminaire de Topologie, Universit\'e de Nantes
  • Invited
• [Presentation] Feuilletages aux feuilles symplectiques sur la sph\`ere de dimension 5 (I) (2018)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: S\'eminaire de G\'eom\'etrie, Universit\'e de Nantes
  • Invited
• [Presentation] Feuilletages aux feuilles symplectiques sur la sph\`ere de dimension 5 (II) (2018)
  • Author(s): Yoshihiko Mitsumatsu
  • Organizer: S\'eminaire de Topologie, Universit\'e de Nantes
  • Invited
• [Presentation] On multiplicity varieties and weight varieties (2018)
  • Author(s): Tatsuru Takakura
  • Organizer: The 4th China-Japan Geometry Conference
  • Int'l Joint Research / Invited
• [Presentation] Topological Flexibility of Symplectic Convexity and leafwise symplectic foliations (2018)
  • Author(s): Yoshihiko MITSUMATSU
  • Organizer: Topology Seminar FUB
  • Invited
• [Presentation] Symplectic structure on Lawson's foliation (2017)
  • Author(s): Yoshihiko MITSUMATSU
  • Organizer: Workshop on topological aspects of symplectic foliations
  • Int'l Joint Research / Invited
• [Presentation] Symplectic foliations of codimension 1 and topological flexibilty of convexity of complex surfaces' (2017)
  • Author(s): Yoshihiko MITSUMATSU
  • Organizer: RIMS Joint Usage Research Topology of pseudoconvex domains and analysis of reproducing kernels
  • Int'l Joint Research / Invited
• [Book] 幾何学百科II 幾何解析 (2018)
  • Author(s): 酒井隆、小林治、芥川和雄、西川青季、小林亮一 著（小島定吉、三松佳彦 編）
  • Total Pages: 436
  • Publisher: 朝倉書店
• [Book] Advanced Studies in Pure Mathematics 72 (2017)
  • Author(s): Edited by Taro Asuke, Shigenori Matsumoto, and Yoshihiko Mitsumatsu
  • Total Pages: 458
  • Publisher: The Mathematical Society of Japan
  • ISBN: 9784864970402
• [Remarks] Engel structures April 17 to April 21, 2017
  • URL: https://aimath.org/pastworkshops/engelstr.html
• [Remarks] Topological aspects of symplectic foliations
  • URL: http://math.univ-lyon1.fr/homes-www/niederkruger/symp-foliations/
• [Funded Workshop] Mini-Workshop on symplectic foliations (2018)
• [Funded Workshop] BGamma School III (2018)