Topological study of foliations and contact structures

2013 Fiscal Year Final Research Report

• Project/Area Number: 22340015
• 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): MIYOSHI Shigeaki 中央大学, 理工学部, 教授 (60166212) ; TAKAKURA Tatsuru 中央大学, 理工学部, 准教授 (30268974)
• Co-Investigator(Renkei-kenkyūsha): MATSUMOTO Shigenori 日本大学, 理工学部, 教授 (80060143) ; TSUBOI Takashi 東京大学, 数理科学研究科, 教授 (40114566) ; KIMURA Yoshifumi 名古屋大学, 多元数理科学研究科, 教授 (70169944) ; MORIYOSHI Hitoshi 名古屋大学, 多元数理科学研究科, 教授 (00239708) ; OHTA Hiroshi 名古屋大学, 多元数理科学研究科, 教授 (50223839)
• Project Period (FY): 2010-10-20 – 2014-03-31
• Keywords: 葉層構造 / 接触構造 / シンプレクティック構造 / Anosov 流・葉層 / 強擬凸性 / 葉向コホモロジー / ミルナー・ファイブレーション / 非圧縮流体

Research Abstract

Foliations and contact structures are studied, with a focus on those structures on 3, 4, and 5 dimensional spaces. Especially the construction of important examples and their mutual relationships are investigated. Interactions of many mathematical theories such as Milnor fibrations associated with singularities, fluid mechanics, symplectic geometry which is a refinement of classical mechanics, and several complex variables, re flected on those structures and objects are studied.
This study is also understood as looking at the tightness of those structures which is interpreted as the result of such mathematical theories are reflected on the geodesic flows of surfaces, which gives rise to a special class of flows called `Anosov flows'.

Research Products

• [Journal Article] Helicity in differentrial topology and incompressible fluids on foliated 3-manifolds (2013)
  • Author(s): Yohihiko MITSUMATSU
  • Journal Title: Procedia IUTAM Volume: vol.7 Pages: 167-174
  • DOI: 10.1016/j.piutam.2013.03.020
  • Peer Reviewed
• [Journal Article] Homeomorphism groups of commutator width one (2013)
  • Author(s): Takashi TSUBOI
  • Journal Title: Proceedings Amer. Math. Soc Volume: 141 Pages: 1839-1847
  • DOI: 10.1090/S0002-9939-2012-11595-3
  • Peer Reviewed
• [Journal Article] On vector partition functions with negative weights (2013)
  • Author(s): Tatsuru TAKAKURA
  • Journal Title: RIMS Kokyuroku Bessatsu B39 Pages: 183-195
  • Peer Reviewed
• [Journal Article] Displacement of polydisks and Lagrangian Floer theory (2013)
  • Author(s): Kenji FUKAYA, Yong-Geun OH, Hiroshi OHTA, Kaoru ONO
  • Journal Title: J. Symplectic Geom.11 Volume: no.2 Pages: 231-268
  • Peer Reviewed
• [Journal Article] The space of (contact) Anosov flows on 3-manifolds (2013)
  • Author(s): Shigenori MATSUMOTO
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 20(3) Pages: 445-460
  • Peer Reviewed
• [Journal Article] Flows of flowable Reeb homeomorphisms (2012)
  • Author(s): MATSUMOTO Shigenori
  • Journal Title: Ann. Inst. Fourier (Grenoble) Volume: 62(3) Pages: 887-897
  • Peer Reviewed
• [Journal Article] Normally contracting Lie group actions (2012)
  • Author(s): Takashi INABA, Shigenori MATSUMOTO, Yoshihiko MITSUMATSU
  • Journal Title: Topology Appl Volume: 159(5) Pages: 1334-1338
  • DOI: 10.1016/j.topol.2011.12.012ns
  • Peer Reviewed
• [Journal Article] On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds (2012)
  • Author(s): Takashi TSUBOI
  • Journal Title: Commentarii Mathematici Helvetici Volume: 87 Pages: 141-185
  • DOI: 10.4171/CMH/251
  • Peer Reviewed
• [Journal Article] Energy spectra of stably stratified turbulence (2012)
  • Author(s): Yoshifumi KIMURA, J R Herring
  • Journal Title: Journal of Fluid Mech Volume: 698 Pages: 19-50
  • Peer Reviewed
• [Presentation] A geometric introduction to incompressible fluid motions on manifolds (2014)
  • Author(s): Yoshihiko MITSUMATSU
  • Organizer: Roever Lectures
  • Place of Presentation: Washington University in St.Luois, U.S.A
  • Year and Date: 2014-04-19
  • Invited
• [Presentation] Vector partition functions and the topology of multiplicity varieties (2013)
  • Author(s): Tatsuru TAKAKURA
  • Organizer: 国際研究集会「Knots, Manifolds and Group Actions」
  • Place of Presentation: Adam Mickiewicz University, Slubice, Poland
  • Year and Date: 2013-09-14
  • Invited
• [Presentation] Leafwise symplectic structures on codimension 1 foliations on S5 (2013)
  • Author(s): Y MITSUMATSU
  • Organizer: 国際会議 Geometry and Foliations 2013
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2013-09-10
  • Invited
• [Presentation] Vector partition functions and the topology of multiplicity varieties (2013)
  • Author(s): 高倉 樹
  • Organizer: 第60回トポロジーシンポジウム
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2013-08-08
  • Invited
• [Presentation] Incompressible fluids on foliated manifolds (2013)
  • Author(s): Y MITSUMATSU
  • Organizer: 国際会議 The 6th Pacific RIM Conference on Mathematics 2013, Session : Topological Problems in Fluid Dynamics
  • Year and Date: 2013-07-01
  • Invited
• [Presentation] Secondary cyclic cocycle for Connes (2013)
  • Author(s): MORIYOSHI Hitoshi
  • Organizer: Chern character on a Cantor set, China-Japan Conference on Noncommutative Geometry and K-Theory
  • Place of Presentation: Jilin University, Changchun, China
  • Year and Date: 2013-06-14
  • Invited
• [Presentation] Milnor's open-book and construction of leafwise symplectic foliations (2013)
  • Author(s): Y MITSUMATSU
  • Organizer: 国際会議 Branched Coverings,Degenerations, and Related Topics 2013
  • Place of Presentation: 首都大学東京
  • Year and Date: 2013-03-08
  • Invited
• [Presentation] Lawson's foliation and a regular Poisson structure on the 5-sphere (2012)
  • Author(s): Y MITSUMATSU
  • Organizer: Poisson 2012 : Poisson Geometry in Math.and Physics
  • Place of Presentation: Utrecht University(the Netherland)
  • Year and Date: 2012-07-31
  • Invited
• [Presentation] Helicity in differential topology (2012)
  • Author(s): Yoshihiko MITSUMATSU
  • Organizer: Topological Fluid Dynamics (IUTAM Symposium)
  • Place of Presentation: Newton Institute Cambridge, England
  • Year and Date: 2012-07-25
  • Invited
• [Presentation] Homeomorphism groups of commutator width one (2012)
  • Author(s): Takashi TSUBOI
  • Organizer: Geometry in Dynamics - Satellite Thematic Session, 6th European Congress of Mathematics
  • Place of Presentation: Krakow
  • Year and Date: 2012-07-01
• [Presentation] Leafwise symplectic structures on codimension one foliations on the 5-sphere (2012)
  • Author(s): Y MITSUMATSU
  • Organizer: Foliations 2012
  • Place of Presentation: University of Lodz (Poland)
  • Year and Date: 2012-06-29
  • Invited
• [Presentation] End-periodic symplectic structure on cubic Fermat surface and Lawson's foliation on the 5-sphere (2012)
  • Author(s): Y MITSUMATSU
  • Organizer: Max Planck Institute for Mathematics
  • Place of Presentation: Bonn, Germany
  • Year and Date: 2012-06-11
  • Invited
• [Presentation] Modifications de feuilletages de dimension 2 sur les varietes de dimension 4 (2012)
  • Author(s): Y MITSUMATSU
  • Organizer: Journee Topologie et Geometrie, Laboratoire de Mathematiques de Bretagne Atlantique
  • Place of Presentation: Quimper, France
  • Year and Date: 2012-05-25
  • Invited
• [Presentation] Modifications of 2-dim foliations on 4-manifolds (2011)
  • Author(s): Y MITSUMATSU
  • Organizer: Seminaire EDP, Universite de Toulouse 3
  • Place of Presentation: Toulouose, France
  • Year and Date: 2011-04-01
  • Invited
• [Presentation] Leafwise symplectic structures on Lawson's foliations (2011)
  • Author(s): Y MITSUMATSU
  • Organizer: Seminaire topologie differentielle, Universite de Bretagne Sud
  • Place of Presentation: Vannes, France
  • Year and Date: 2011-03-29
  • Invited
• [Presentation] Modifications of 2-dimensional foliations on 4-manifolds and tautness (2010)
  • Author(s): Y MITSUMATSU
  • Organizer: The 17th Brazilian Topology Meeting
  • Place of Presentation: PUC-Rio, Rio de Janeiro, Brazil
  • Year and Date: 2010-08-03
  • Invited
• [Presentation] Turbulization of 2-dimensional foliations on 4-manifolds and tautness (2010)
  • Author(s): Y MITSUMATSU
  • Organizer: Conference on Geometry and Topology of Foliations, CRM Univ. Autonoma de Barcelona
  • Place of Presentation: Barcelona, Spain
  • Year and Date: 2010-07-13
  • Invited
• [Presentation] A construction of a typical foliation on a 3-manifold (2010)
  • Author(s): Shigeaki MIYOSHI
  • Organizer: Conference on Geometry and Topology of Foliations, CRM Univ. Autonoma de Barcelona
  • Place of Presentation: Barcelona, Spain
  • Year and Date: 2010-07-12
  • Invited