Updated on 2024/08/22

写真a

 
TAKAKURA Tatsuru
 
Organization
Faculty of Science and Engineering Professor
Other responsible organization
Mathematics Course of Graduate School of Science and Engineering, Master's Program
Mathematics Course of Graduate School of Science and Engineering, Doctoral Program
Contact information
The inquiry by e-mail is 《here
External link

Degree

  • 博士(数理科学) ( 東京大学 )

Education

  • 1994.3
     

    The University of Tokyo   Graduate School, Division of Mathematical Sciences   doctor course   completed

Research History

  • 2023.4 - Now

    Chuo University High School at Bunkyo   Principal

  • 2014.4 - Now

    中央大学理工学部教授

  • 2007.4 - 2014.3

    中央大学理工学部准教授

  • 2001.4 - 2007.3

    中央大学理工学部助教授

  • 1997.4 - 2001.3

    中央大学理工学部専任講師

  • 1994.4 - 1997.3

    福岡大学理学部助手

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Professional Memberships

  • 日本数学会

Research Interests

  • Topology, Geometry

Research Areas

  • Natural Science / Geometry  / 幾何学

Papers

  • On Volume Functions of Special Flow Polytopes Associated to the Root System of Type A Reviewed

    Takayuki Negishi, Yuki Sugiyama, Tatsuru Takakura

    The Electronic Journal of Combinatorics   27 ( 4 )   4.56   2020.12

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  • On vector partition functions with negative weights Reviewed

    Tatsuru Takakura

    RIMS Kokyuroku Bessatsu   B39   183 - -195   2013

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  • Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group Reviewed

    Taro Suzuki, Tatsuru Takakura

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   61 ( 3 )   921 - 969   2009.7

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:MATH SOC JAPAN  

    We consider asymptotic behavior of the dimension of the invariant subspace in a tensor product of several irreducible representations of a compact Lie group G. It is equivalent to studying the symplectic volume of the symplectic quotient for a direct product of several coadjoint orbits of G. We obtain two formulas for the asymptotic dimension. The first formula takes the form of a finite sum over tuples of elements in the Weyl group of G. Each term is given as a multiple integral of a certain polynomial function. The second formula is expressed as an infinite series over dominant weights of G. This could be regarded as an analogue of Witten's volume formula in 2-dimensional gauge theory. Each term includes data such as special values of the characters of the irreducible representations of G associated to the dominant weights.

    DOI: 10.2969/jmsj/06130921

    Web of Science

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  • On asymptotic partition functions for root systems Reviewed

    Tatsuru Takakura

    TORIC TOPOLOGY   460   339 - 348   2008

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:AMER MATHEMATICAL SOC  

    We study the asymptotic partition functions associated to root systems with weights from the viewpoint of the theory of GKZ hypergeometric functions. We illustrate some explicit formulas of them for root systems of rank two.

    Web of Science

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  • Symplectic volumes of certain symplectic quotients associated with the special unitary group of degree three Reviewed

    Taro Suzuki, Tatsuru Takakura

    Tokyo Journal of Mathematics   31 ( 1 )   1 - 26   2008

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    Language:English   Publishing type:Research paper (scientific journal)  

    We consider the symplectic quotient for a direct product of several integral coadjoint orbits of SU(3) and investigate its symplectic volume. According to a fundamental theorem for symplectic quotients, it is equivalent to studying the dimension of the trivial part in a tensor product of several irreducible representations for SU(3), and its asymptotic behavior. We assume that either all of coadjoint orbits are flag manifolds of SU(3), or all are complex projective planes. As main results, we obtain an explicit formula for the symplectic volume in each case. © 2008 International Academic Printing Co. Ltd. All rights reserved.

    DOI: 10.3836/tjm/1219844821

    Scopus

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  • 書評 Ana Canas da Silva: Lecture on Symplectic Geometry (Springer Lecture Notes in Math., 1764) Reviewed

    数学   55 ( 3 )   332 - 335   2003.7

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    Language:Japanese   Publishing type:Research paper (other academic)   Publisher:日本数学会  

    CiNii Books

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  • A note on the symplectic volume of the moduli space of spatial polygons Reviewed

    T.Takakura

    Advanced Studies in Pure Mathematics 34, Minimal Surfaces, Geometric Analysis and Symplectic Geometry   34   255 - 259   2002.4

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  • Hamiltonian actions and equivariant indices Reviewed

    T.Takakura

    K monographs in Mathematics 7, Current Trends in Transformation Groups   7   217 - -229   2002.4

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  • Intersection theory on symplectic quotients of products of spheres, Reviewed

    TAKAKURA T.

    International Journal of Mathematics,   12 ( 1 )   97 - 111   2001.1

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  • Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections Reviewed

    Tatsuru Takakura

    Inventiones Mathematicae   123 ( 3 )   431 - 452   1996

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer New York  

    We construct a family of polarizations of the moduli space of flat SU(n)-connections on a closed 2-manifold of genus g(≧ 2). These are generalizations of various polarizations known until now. That is, our family of polarizations includes Weitsman's real polarizations in the case of n = 2 [17], as well as the Kähler polarizations which are well known since [2] and [18]. Our construction is based on an original formulation of degeneration of Riemann surfaces. The relation between our polarizations and the complex structures of the moduli spaces of parabolic bundles are also studied.

    DOI: 10.1007/s002220050035

    Scopus

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  • Representation spaces of genus zero Fuchsian groups and Hamiltonian torus action Reviewed

    Tatsuru Takakura

    J. of the Fac. of Sci., the Univ. of Tokyo, Sec. IA   39 ( 3 )   541 - 554   1992.12

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MISC

  • ICM98 部門別報告 トポロジー

    大槻知忠, 高倉樹

    数学   51 ( 1 )   84 - 87   1999.1

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    Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (other)   Publisher:日本数学会  

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  • 書評 D.McDuff、 D.Salamon:J-holomorphic Curves and Quantum Cohomology

    高倉樹

    数学   50 ( 1 )   104 - 105   1998.1

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    Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (other)   Publisher:日本数学会  

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  • 幾何学的量子化の理論概観

    高倉樹

    Surveys in Geometry「シンプレクティック幾何学」   1995.1

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Presentations

  • On Chern numbers of weight varieties

    高倉 樹

    中央大学幾何・トポロジー小研究集会 Geometry, Topology or Something  2024.7 

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    Event date: 2024.7    

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  • On non-abelian localization theorems

    高倉 樹

    Geometric Quantization and Related Topics  2023.3 

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  • Kostant function に対するLidskii の公式と微分方程式系について

    高倉 樹

    研究集会「接触構造、特異点、微分方程式及びその周辺」  2023.1 

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    Event date: 2023.1    

    Language:Japanese   Presentation type:Oral presentation (general)  

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  • On multiplicity varieties and weight varieties

    The 4th China-Japan Geometry Conference  2018.9 

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  • On multiplicity varieties

    研究集会「シンプレクティック幾何学とその周辺」, 熱海  2017.3 

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  • Multiplicities in equivariant indices and symplectic quotients 1,2

    Koriyama Geometry and Physics Days 2017, ``Geometric Quantization and related topics"  2017.2 

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  • On volume functions of special flow polytopes

    研究集会「接触構造、特異点、微分方程式及びその周辺」, 金沢大学サテライト・プラザ  2017.1 

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  • Special flow polytopes and associated toric varieties

    福岡大学幾何学研究会  2015.11 

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  • 同変指数とシンプレクティック商のトポロジーI, II

    非可換幾何若手勉強会2015  2015.3 

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  • Vector partition functions and the topology of multiple weight varieties

    ICM2014 Satellite conference 'Topology of torus actions and applications to geometry and combinatorics'  2014.8 

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  • An explicit formula for vector partition functions and applications

    第40回変換群論シンポジウム  2013.12 

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  • Vector partition functions and the topology of multiplicity varieties

    Knots, Manifolds and Group ActionsAdam Mickiewicz University, Slubice, Poland  2013.9 

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  • Vector partition functions and the topology of multiplicity varieties

    第60回トポロジーシンポジウム  2013.8 

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  • 分配多面体上の積分公式とその応用

    福岡大学幾何学研究会  2012.11 

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  • On multiplicity varieties

    第11回名古屋国際数学コンファレンス|rn|「Topology and Analysis on Foliations」,|rn|名古屋大学  2012.3 

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  • Intersection theory on double weight varieties

    第58回幾何学シンポジウム, 山口大学  2011.8 

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  • Vector partition functions with negative |rn|weights and some applications

    研究集会「変換群の幾何と組合せ論」,京都大学数理解析研究所  2011.6 

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  • On a vector partition function with negative weights

    第57回幾何学シンポジウム,神戸大学  2010.8 

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  • 負のウエイトをもつベクトル分配関数とその応用

    研究集会「接触構造、特異点、微分方程式及びその周辺」, 洞爺湖文化センター  2010.1 

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  • On a vector partition function with |rn|negative weights

    福岡微分幾何研究会,福岡大学セミナーハウス  2009.11 

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  • テンソル積表現における重複度と付随する|rn|シンプレクティック商の幾何

    第56回幾何学シンポジウム,佐賀大学  2009.8 

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  • 漸近的分配関数とGKZ方程式系

    福岡微分幾何研究会,福岡大学セミナーハウス  2008.1 

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  • テンソル積表現における重複度とシンプレクティック商のトポロジー

    シンポジウム「接触構造,特異点,微分方程式及びその周辺」,旭川  2007.1 

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  • Symplectic volumes of symplectic quotients for direct products of coadjoint orbits

    ICM MADRID 2006  2006.8 

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  • Symplectic volumes of certain symplectic quotients

    福岡微分幾何研究会,福岡大学セミナーハウス  2005.10 

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  • 余随伴軌道に関連するシンプレクティック商の体積について

    鈴木太郎, 高倉樹

    シンプレクティック幾何とその周辺,秋田大学  2004.11 

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  • 余随伴軌道に関連するシンプレクティック商の体積について

    鈴木太郎, 高倉樹

    日本数学会2004年度秋季総合分科会,日本数学会  2004.9 

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  • シンプレクティック商のトポロジー

    第30回 Encounter with Mathematics,中央大学  2004.3 

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  • Hamiltonian action and equivarinat index

    China-Japan Joint Workshop on Mathematical Physics, Beijin  2002.11 

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  • Hamiltonian actions and equivariant indices

    T.Takakura

    Foliations and Geometry 2001, Rio de Janeiro/FoliationsandGeometry2001,RiodeJaneiro  2001.8 

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  • シンプレクティック商のコホモロジー交叉積とフェアリンデの公式

    高倉樹

    Symplectic Topology,城崎  2001.3 

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  • Intersection theory on certain symplectic qunotients

    高倉樹

    接触構造・特異点・微分方程式,福岡  2001.1 

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  • Intersection theory on certain symplectic quotients and Verlinde formula,

    高倉樹

    共形幾何学とシンプレクティック幾何学,福岡  2000.12 

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  • あるシンプルティック商のコホモロジー交換積

    高倉樹

    日本数学会秋季総合分科会,京都  2000.9 

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  • Intersection pairings of certain symplectic quotients

    高倉樹

    共形幾何学・シンプレクティック幾何学と関連する話題(福岡)  1999.12 

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  • Hamiltonian group actions and related topics

    高倉樹

    第2回日韓変換群論シンポジウム(岡山)  1999.8 

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  • Equivariant characteristic numbers and Hamiltonian bordism

    高倉樹

    概複素構造の幾何学について、新潟  1998.11 

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  • Intermediate Polarization of the moduli space of flat connections;

    高倉樹

    複素幾何・シンプレクティック幾何学と関連する話題、福岡  1998.11 

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  • Equivariant characteristic numbers and Hamiltonian bordism

    高倉樹

    変換群論シンポジウム、山形  1998.10 

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  • Hamiltonian group action and equivariant index

    高倉樹

    4th Int. Workshop on complex structures and vector fields,Bulgaria  1998.9 

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  • Hamiltonian group action, equivariant index and cobordism

    高倉樹

    日本数学会1998年度秋季総合分科会トポロジー分科会特別講演アブストラクト,日本数学会トポロジー分科会  1998.9 

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  • Hamiltonian group action and multiplicity formula

    高倉樹

    福岡大学微分幾何学研究集会,福岡  1998.1 

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  • シンプレクティック商のトポロジーと幾何

    高倉樹

    研究集会「接触幾何とシンプレクティック幾何」,北見  1998.1 

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  • An application of the multiplicity formula

    高倉樹

    Warwick Symposium on Symplectic Geometry, Workshop on Moment Maps and Quantization, Warwick.  1997.12 

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  • Intermediate polurization of the moduli space of flat connections

    T.Takakura

    IAS/Park City Mathematics Institute 1997 Summer Session on Symplectic Geometry & Topology, Utah./Institute for Advanced Study  1997.7 

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  • A symplectic proof of Verlinde factorization

    高倉 樹

    福岡大学微分幾何学研究集会,福岡  1997.3 

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  • Surveys on the topology of the moduli space of flat connections

    高倉 樹

    京都大学数理解析研究所短期共同研究「基本群の表現空間の幾何」,京都(招待講演)  1996.6 

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  • リーアン面の退化とヤコビ多様体の偏極およびデータ関数

    高倉樹

    日本数学会1995年度秋季総合分科会幾何学分科会講演アブストラクト,日本数学会幾何学分科会  1995.9 

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  • 幾何学的量子化の理論概観

    高倉樹

    研究集会資料Surveys in Geometry 1995年1月30日-2月2日シンプレクティック幾何学,日本数学会幾何学分科会  1995.1 

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  • 平坦接続のモジュライ空間の中間偏極

    高倉樹

    日本数学会1994年度秋季総合分科会幾何学分科会講演アブストラクト,日本数学会幾何学分科会  1994.9 

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  • Intermediate polarizations of the moduli space of flat connections

    高倉樹

    第41回幾何学シンポジウム講演要旨,日本数学会幾何学分科会  1994.7 

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  • シンプレクティック・トーリック多様体の幾何学的量子化

    高倉樹

    日本数学会1993年度秋季総合分科会幾何学分科会講演アブストラクト,日本数学会幾何学分科会  1993.9 

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  • Geometric quantizations of symplectic toric manifolds

    Tatsuru Takakura

    First MSJ International Reserch Institute on GEOMETRY AND GLOBAL ANALYSIS, Lecture Notes Volume 2/THE MATHEMATICAL SOCIETY OF JAPAN  1993.7 

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  • あるFuchs群の表現空間とHamiltonian torus action

    高倉樹

    第39回幾何学シンポジウム講演要旨,日本数学会幾何学分科会  1992.7 

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  • あるFuchs群のSU2-表現空間とSymplectic幾何

    高倉樹

    日本数学会1992年度年会幾何学分科会講演アブストラクト,日本数学会幾何学分科会  1992.4 

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  • Representation Spaces of Seftert Homology Spheres

    高倉樹

    第37回トポロジーシンポジウム講演集,日本数学会トポロジー分科会  1989.7 

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Research Projects

  • 力学的微分トポロジーによる葉層・接触・シンプレクティック構造の研究

    Grant number:21H00985  2021.4 - 2026.3

    日本学術振興会  科学研究費助成事業  基盤研究(B)  中央大学

    三松 佳彦, 直江 央寛, 高倉 樹, 太田 啓史, 三好 重明, 粕谷 直彦

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    Grant amount: \10270000 ( Direct Cost: \7900000 、 Indirect Cost: \2370000 )

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  • リー群の表現の分解に関連するシンプレクティック商の研究

    2019.4 - 2023.3

    基盤研究(C)(一般) 

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    Grant type:Competitive

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  • 分岐問題に関連するシンプレクティック商の大域的構造の研究

    2019.4 - 2022.3

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    Grant type:Competitive

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

    Grant number:17H02845  2017.4 - 2021.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Chuo University

    Mitsumatsu Yoshihiko

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    Grant amount: \9490000 ( Direct Cost: \7300000 、 Indirect Cost: \2190000 )

    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.

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  • 分岐問題に関わるシンプレクティック商の研究

    2016.4 - 2019.3

    文部科学省  科学研究費補助金(日本学術振興会・文部科学省)-基盤研究(C) 

    高倉 樹

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  • Development of the index theorem on foliated manifolds

    Grant number:25400085  2013.4 - 2016.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Nagoya University

    Moriyoshi Hitoshi, NATSUME TOSHIKAZU, MAEDA YOSHIAKI, MITSUMATSU YOSHIHIKO, ONO KAORU, MIYAZAKI NAOYA, TAKAKURA TATSURU, TATE TETSUYA

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    Grant amount: \4940000 ( Direct Cost: \3800000 、 Indirect Cost: \1140000 )

    First, we extended the index theorem to fractal sets such as the Cantor set and the Sierpinski gasket. Second, by exploiting the framework of Noncommutative Geometry we generalized the Atiyah-Patodi-Singer index theorem to a Galois covering of compact manifold with boundary, which gives a formula for the pairing between K-group and cyclic cohomology. Third, we clarified the relation of the Dixmier-Douady class and the Godbillon-Vey class, which respectively appears as a characteristic class for Gerbe and foliated circle bundles. It turned out that they are connected via the Cheeger-Chern-Simons invariant. As a byproduct we succeeded to describe the universal central extension of circle diffeomorphism group in terms of the Calabi invariant.

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  • Topological study of foliations and contact structures

    Grant number:22340015  2010.10 - 2014.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Chuo University

    MITSUMATSU Yoshihiko, MIYOSHI Shigeaki, TAKAKURA Tatsuru, MATSUMOTO Shigenori, TSUBOI Takashi, KIMURA Yoshifumi, MORIYOSHI Hitoshi, OHTA Hiroshi

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    Grant amount: \8710000 ( Direct Cost: \6700000 、 Indirect Cost: \2010000 )

    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'.

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  • Studies on Floer thoery, theory of holomorphic curves and symplectic structures, contact structures

    Grant number:21244002  2009.4 - 2014.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (A) 

    ONO Kaoru, IZUMIYA Shyuichi, JINZENJI Masao, MATSUSHITA Daisuke, ISHIKAWA Goo, YAMAGUCHI Keizo, TAKAKURA Tatsuru

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    Grant amount: \41340000 ( Direct Cost: \31800000 、 Indirect Cost: \9540000 )

    Symplectic structure is a geometric structure, which appeared in the understanding of Hamilton's equation of motion. In recent years, there has been profound development in the geometric study of symplectic structures. In particular, combined with the mathematical study on mirror symmetry, symplectic geometry
    attracts attentions from many researchers. The investigator has been working on Floer theory, which plays a significant role in symplectic geometry, and its applications. In this research project, we studied Floer theory for Lagrangian torus fibers in toric manifold in a concrete way and obtained various interesting results.

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  • A research on Thurston's inequality for foliations and contact topology

    Grant number:23540106  2011 - 2013

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Chuo University

    MIYOSHI Shigeaki, MITSUMATSU Yoshihiko, TAKAKURA Tatsuru

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    Grant amount: \2990000 ( Direct Cost: \2300000 、 Indirect Cost: \690000 )

    As a foliation is an integrable tangent plane field, in the case of 1-dimensional line fields, we treated a completely integrable vector field, which satisfies an extra integrability condition. We consider a placement problem of a closed leaf (i.e., a periodic trajectory) of the 1-dimensional foliation tangent to a completely integrable vector field on an open 3-manifold. Such a foliation is given by the inverse images of a submersion to the plane. We gave a necessary and sufficient condition for the realization for any given link, and in the case of a knot, we described the condition by the words of classical invariants.
    A 2-dimensional foliation transverse to a completely integrable vector field satisfies Thurston's inequality and it gives a natural class of such foliations. Thus, we can consider such a class of foliations is a natural class on an open 3-manifold.

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  • シンプレクティック空間の大域的構造の研究

    2008.3 - 2009.3

    中央大学  中央大学在外研究費 

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    Grant type:Competitive

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  • Research on foliations, contact structures and Euler class

    Grant number:18540095  2006 - 2007

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Chuo University

    MIYOSHI Shigeaki, MITSUMATSU Yoshihiko, TAKAKURA Tatsuru

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    Grant amount: \2340000 ( Direct Cost: \2100000 、 Indirect Cost: \240000 )

    W. Thurston showed that a foliation on a 3-manifold which has no Reeb component enjoys the property that the Euler class of the tangent bundle satisfies an inequality, Thurston's inequality. On the other hand, the Reeb foliation on the three sphere satisfies Thurston's inequality and a previous research followed by this research showed that there is a class of foliations each of which has Reeb components and satisfi es Thurston's inequality.
    In the research in 2006, for a class of foliations which are called spinnable foliations, we obtained a sufficient condition for the foliation satisfying Thurston's inequality. Moreover, we revealed an aspect where Thurston's inequality does not hold. They are described by properties of the monodromy diffeomorphisms which determine the spinnable foliations
    In view of the research with respect to the convergence of contact structures to foliations, we studied a finer inequality, the relative version of Thurston's inequality, which deepens the research until 2006. In fact, for spinnable foliations we showed that the relative version implies the absolute version. The same statement for contact structures was known however, it does not hold in general for foliations. Also in 2007, we found the method to construct a foliation which satisfies Thurston's inequality with Euler class of infinite order. Until then, all foliations which satisfies Thurston's inequality have trivial Euler class. Indeed, we can find'a spinnable foliation whose Euler class is of infinite order by the research in 2006. Then we can perform Dehn surgery along the Reeb component and with certain condition on the original spinnable foliation we can conclude that with finitely many exceptions the resultant satisfies Thurston's inequality with Euler class of infinite order by virtue of D. Gabai's sutured manifold theory.

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  • シンプレクティック空間の不変量とその表現論的構造の研究

    2003.4 - 2005.3

    文部科学省  科学研究費補助金(基盤研究C2) 

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  • A study on foliations, contactstrucures, and symplectic styructures on 3 and 4 dimensional manifolds

    Grant number:16540080  2004 - 2005

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Chuo University

    MITSUMATSU Yoshihiko, MIYOSHI Shigeaki, TAKAKURA Tatsuru, MATSUMOTO Shigenori, TSUBOI Takashi, ONO Kaoru

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    Grant amount: \3700000 ( Direct Cost: \3700000 )

    The head Mitsumatsu and an investigator Miyoshi colaborated with others to study the euler class of tangent bundles to foliations and (so called Thurston-Winkelnkemper's) contact structures which are associate with spinnable structures, as a typical class of convergences of contact structures to foliations. Especially they studied the (non-)vanishing of the euler class and the violation of Thurston's inequality and Bennequin's inequlity, from the topological view point of monodromies. As a consequence, a certain class of mapping classes of a surface with boundary can be presented neither as a product of only right-handed Dehn twists nor as that of only right-handed ones. This result was presented in several symposiums including the annual meeting of MSJ in March 2006 as a special invited talk by Miyoshi. The paper is under submission.
    The investigator Ono studied the symplectec homology from Floer theory as well as from Seiberg-Witten theory. Including the solution to the Flux conjecture as well as the detemination of the symplectic filling of the link of simple singularities, his contributions to this area are profound.
    The investigator Tsuboi studied the relationship between foliation theory and that of contact structures from the view point of the group of contact diffeomorphisms. The investigator Matsumoto stepped further to the foliation theory and studied the ends of Lie foliations.
    The head investigator also studied the incompressible fluid dynamics in the framework of the geometry of volume preserving diffeomorphisms and infinite dimensional Hamiltonian systems. He proved that looking from the point of view of such global differential geometry is still valid even to viscous fluides with dissipations. This study is presented in many symposiums, especially as a special project talk in the annual meeting of MSJ in September 2005.

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  • シンプレティック空間の不変量とその代数的構造の研究

    2002.4 - 2003.3

    文部科学省  科学研究費補助金(若手研究B) 

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    Grant type:Competitive

    Grant amount: \1000000

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  • Study on contac structures and foliations on 3 and 4 dimensional manifolds

    Grant number:13440026  2001 - 2003

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Chuo University

    MITSUMATSU Yoshihiko, MIYOSHI Shigeaki, TSUBOI Takashi, SATO Hajime, TAKAKURA Tatsuru, ONO Kaoru

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    Grant amount: \8600000 ( Direct Cost: \8600000 )

    Based on the notion of asymptotic linking, the head investigator proposed the framework where the study on contact structures and that on foliations would be unified, and began the research. Op the side of foliations, it turned out that exotic classes and the 1st foliated cohomology are strongly related to this framework. On the other side, the torsion invariant of contact structures has a deep relation with it. For algebraic Anosov foliations, we also established the computation of its 1st foliated cohomology and found its relation to local orbit rigidity.
    A research group including Miyoshi and Mitsumatsu investigated Thurston's inequality for foliations on the boundary of compact Stein surfaces and established the absolute version of the inequality for certain cases. The relative ersion and more general case are left for the future as important subject. Tsuboi and Mitsumatsu worked on the perfectness of groups of diffeomorphisms preservein certain geometric structures. Especially Tsuboi provednthe perfectness for contact diffeomorphisms and analytic diffeomorphisms of certain manifolds. Tsuboi also classified regular bi-contact structures on Seifert fibered spaces.
    Another group including Ono and Ohta, mainly working on contact/symlectic topology, characterized the symplectic diffeo-types of the filling of the link of simple and hyper-elliptic singularities. Also they got started the construction of obstruction theory for Lagrangian Floer homology theory.

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  • シンプレクティック空間の不変量とその代数的構造の研究

    2001.4 - 2002.3

    文部科学省  科学研究費補助金(奨励研究A) 

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    Grant type:Competitive

    Grant amount: \1100000

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  • Construction of the topological toric theory

    Grant number:13640087  2001 - 2002

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Osaka City University

    MASUDA Mikiya, HASHIMOTO Yoshitake, HIBI Takayuki, TAKAKURA Tatsuru, FURUSAWA Masaaki, KAWAYUCHI Akio

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    Grant amount: \3300000 ( Direct Cost: \3300000 )

    We developed the theory of toric varieties from the topological viewpoint. In these several years I worked with Professor Akio Hattori and found that geometrical properies of a torus manifold can be described in terms of a combinatorial object called a multi-fan. In particular, we found a neat formula describing the elliptic genus of a torus manifold in terms of the multi-fan associated with the torus manifold, and obtained a vanishing theorem saying that the level N elliptic genus of a torus manifold vanishes if the 1st Chern class of the manifold is divisible by N. As a corollary of this vanishing theorem, we obtained a result that if the 1st Chern class of a complete toric variety M of complex dimension n is divisible by N, then N must be less than or equal to n+1, and in case N=n+l, M is isomorphic to the complex protective space. This is a toric version of the famous Kobayashi-Ochiai or Mori's theorem.
    I invited Taras Panov from Moscow State University for a month and studied the equivariant cohomology of a torus manifold M and the cohomology of its orbit space. As a result, it turned out that when the cohomology ring of M is generated in degree two, the equivariant cohomology of M is a Stanley-Reisner ring and the orbit space of M has the same form as a convex polytope from a cohomological point of view. We also studied the case where M has vanishing odd degree cohomology. It turns out that this case is obtained by blowing down the previous case. Interestingly, the equivariant cohomology of M in this case provides a generalization of the Stanley-Reisner ring. The ring like this was already introduced by Stanley about ten years ago but we may think of our results as giving a geometrical meaning of the ring. Along this line, I proved a conjecture by Stanley about the h-vector of a Gorenstein* simplicial poset. The proof is purely algebraic but the idea stems from topology and this shows a close connection between combinatorics, commutative algebra and topology.

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  • シンプレクティック空間における局所化と大域的構造の研究

    2000.4 - 2001.3

    文部科学省  科学研究費補助金(奨励研究A) 

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    Grant type:Competitive

    Grant amount: \1000000

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  • シンプレクティック空間における局所化と大域的構造の研究

    1999.4 - 2000.3

    文部科学省  科学研究費補助金(奨励研究A) 

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    Grant type:Competitive

    Grant amount: \1200000

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  • Topological research of the theory of toric varieties

    Grant number:11640091  1999 - 2000

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Osaka City University

    MASUDA Mikiya, HASHIMOTO Yoshitake, HIBI Takayuki, TAKAKURA Tatsuru

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    Grant amount: \3300000 ( Direct Cost: \3300000 )

    We developed the theory of toric varieties from topological viewpoint. The theory of toric varieties says that there is a one-to-one correspondence between "toric varieties" (an object in algebraic geometry) and "fans" (an object in combinatorics). In our project, we studied "torus manifolds" or "torus orbifolds" which are topological counterparts to toric varieties and a wider object than that of toric varieties, and constructed a correspondence from those extended objects to an extended combinatorial object called "multi-fans". One of the fundamental problems in our correspondence is to characterize geometrically obtained multi-fans, and we completely characterized the multi-fans obtained form torus orbifolds. Moreover, we described signatures and T_y-genera of torus manifolds in terms of multi-fans. There is another fundamental correspondence given by moment maps. We introduced a notion of multi-polytopes, which appear as images of moment maps, and generalized Ehrhart polynomials and Khovanskii-Pukhlikov formula for convex polytopes to multi-polytopes.

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  • シンプレクティック幾何学の展開

    1998.4 - 1999.3

    文部科学省  科学研究費補助金(基盤研究C1) 

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    Grant type:Competitive

    Grant amount: \1300000

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  • シンプレクティック空間に対する局所化現象の研究

    1998.4 - 1999.3

    文部科学省  科学研究費補助金(奨励研究A) 

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    Grant type:Competitive

    Grant amount: \1100000

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  • Study of contact structures and foliations on 3-manifolds

    Grant number:09640130  1998 - 1999

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Chuo University

    MITSUHATSU Yoshihiko, MIZUTANI Tadayoshi, TAKAKURA Tatsuru, MATSUYAMA Yoshio, KANOA Yutaka, ONO Kaoru

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    Grant amount: \3100000 ( Direct Cost: \3100000 )

    Mitsumatsu and Mizutani studied and constructed many examples of bi-contact structures with a research group of foliations. Especially, they constructed a bicontact structure on the 3-sphere which consists both of over-twisted ones. Still the realization problem of homotopy class of plane fields as such structures remains to be studied.
    Ono has established in a colaboration with Fukaya foundamental theory in applying the J-curves to symplectic topology, overcoming the notorious problem of negative multiples. Major consequences from this are the definition of Gromov-Witten invariants for general symplectic manifolds and the positive solution for a version of the Arnold conjecture for the same class. He and Kanda also worked on applying Seiberg-witten theory to contact topology, colaborating with Ohta, and got toplogical constraints on the symplectic filling 4-manifolds around simple singularities. This streem of works continues and is expected to make a further progress, especially in relation with the last subject of study below.
    Kanda studied contact structures in more toplogical way, and classified tight contact structures on 3-torus and showed nonexactness of Bennequin's inequality.
    Takakura and Mitsumatsu have been searching for the formalism to study contact topology by using Lagrangian/Legendrian torus, instead of looking at J-curves in the symplectization. This is based on the theory of geometric quantization, on which Takakura has been working. They found that most of major concepts in the theory of algebraic functions in one variable can be suitably translated and planted in this framework. However, studying contact topology through this remains as the next step of research to go.

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  • Research for manifolds with conformal structure

    Grant number:09440044  1997 - 1999

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  FUKUOKA UNIVERSITY

    SUYAMA Yoshihiko, KUROSE Takashi, AKUTAGAWA Kazuo, SHIOHAMA Katsuhiro, INOGUCHI Jun-ichi, YAMADA Kataro

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    Grant amount: \6600000 ( Direct Cost: \6600000 )

    1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere.
    2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the conformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold.
    3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form.
    4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction.

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  • シンプレクティック空間に対する局所化現象の研究

    1997.4 - 1998.3

    文部科学省  科学研究費補助金(奨励研究A) 

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    Grant type:Competitive

    Grant amount: \1200000

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  • ENCOUNTER with MATHEMATICS

    Grant number:10894007  1998    

    日本学術振興会  科学研究費助成事業  基盤研究(C)  中央大学

    三松 佳彦, 高倉 樹, 小野 薫, 佐藤 肇, 村松 壽延, 服部 晶夫

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    Grant amount: \1700000 ( Direct Cost: \1700000 )

    本研究の補助金の執行期間中には以下の3回のENCOUNTER with MATHEMATICSの本会議が開催された。何れも中央大学理工学部に於いて金曜日の14:30または15:00から続く土曜日の
    17:00頃までの一日半である。
    1. 第8回目‘TORIC幾何''98.6.12(金)・13(土)
    トーリック幾何への招待I、II、小田忠雄氏(東北大・理)・佐藤拓氏(東北大・理)
    トポロジーから見たトーリック多様体論I、II、枡田幹也氏(阪市大・理)、
    log scheme理論入門 諏訪紀幸氏(中大・理工)
    2. 第9回目‘実1次元力学系"98.10.23(金)・24日(土)
    実1次元力学系I、II、III坪井俊氏(東大・数理)、
    有界オイラー類と円周の同相群の共役 松元重則氏(日大・理工)、
    双曲結晶群の\S^1\のPL同相群への表現:初等的な構成 皆川宏之氏(北大・理)
    3. 第10回‘応用特異点論'99.2.5(金)・6日(土)
    応用特異点論概説、一階偏微分方程式への応用、微分幾何学への応用 泉屋秀一氏(北大・理)、
    応用特異点論の基礎I、II、石川剛郎氏(北大・理)、
    微分位相幾何学への応用 佐伯修氏(広島大・理)
    各回とも100〜150人程度の参加者を得、また、講演も専門的になりすぎずに、この集会の本来の意義を十分に達成したと考えられる。
    上記の本会議以外には、各回の準備、及び、将来に向けての準備会議を何回か行った。また、代表者はドイツObervolfach数学研究所で毎年開かれているトポロイジー研究集会に出席し、Wolfgang Luck,Elmar Vogtの新旧主催者と定期的に集会を開催することの意義や問題点について意見交換を行った。本年度の問題点として、開催テーマで順延になったものがあったため、主催側に幾何系の人間が多いこともあって、急遽幾何系のテーマが繰り上がり、結果として、開催したテーマの分野に偏りがでたことがあげられる。今後の課題である。

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  • シンプレクティック空間上のD-加群の研究

    Grant number:08740077  1996    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  福岡大学

    高倉 樹

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    Grant amount: \1000000 ( Direct Cost: \1000000 )

    本研究課題において具体的な目標としていたことのうち、1.2次元多様体上の平坦接続のモジュライ空間の滑層構造の解明とD-加群の構成、2.幾何学的量子化の一般化としてのコホモロジー理論における消滅定理、について得られた結果と、その過程で明らかになった今後の課題等を以下に記す。
    1.については、平坦接続のモジュライ空間の幾何学的量子化の次元に関するフェアリンデの分解公式の、シンプレクティック幾何学的な証明を与えることと並行させて研究を進めた。その結果、「ハミルトン的群用の下での幾何学的量子化に関する重複度公式」を応用することにより、上記分解公式が自然に得られることが判った。その際、モジュライ空間の滑層構造は、運動量写像の臨界値におけるシンプレクティック商の滑層構造という形で一般的に考察が可能であり、Kirwanによる部分的特異点解消が有効であることを示したMeinrenken-Sjamaarの結果が鍵となった。なお、本研究者が有限次元のトーラス作用を用いているのに対し、Meinrenken-Woodwardは無限次元のループ群作用を解析して同公式を証明した。両者の相関の解明は、未達成のまま残ったD-加群の構成等に関連して興味深いと考えられ、今後の課題として挙げておく。
    2.に関連して、シンプレクティック・トーリック多様体上に退化付き不変偏極の族を構成し、同伴する幾何学的量子化の推移に祭し、特殊なラグランジュ部分多様体上への局所化現象が起こることを示した。さらに、この不変偏極による幾何学的量子化を層係数のコホモロジーとして記述し、上記局所化に対応するコホモロジーの消滅を示すことができた。昨年度得られたシンプレクティック・トーラスに対する同様の結果を踏まえつつ、これらをより広いクラスのシンプレクティック多様体に拡張することは、引続き今後の課題としておく。

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  • 正曲率及びワイル共形曲率と多様体の空間

    Grant number:08640146  1996    

    日本学術振興会  科学研究費助成事業  基盤研究(C)  福岡大学

    陶山 芳彦, 荻 秀和, 秋田 利之, 高倉 樹, 黒瀬 俊, 吉田 守

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    Grant amount: \2100000 ( Direct Cost: \2100000 )

    1.どのような共形平坦な多様体が,定曲率空間の超曲面として実現されるかという問題を研究し,4次元以上の(ある種の)共形平坦な多様体に関して,それらの多様体から定曲率空間への共形的はめ込みの具体的構成法を発見した。更に,上の構成ではめ込み可能な多様体の共形類を決定するために,それらの超曲面から球面への展開写像の構成を行った。
    2.射影平坦で捩れをもたないアフィン接続が与えられた単連結多様体の射影展開写像について研究し,次ぎの結果を得た。3次元以上で接続に関して極を持つ多様体のリッチ曲率が対称で負定値ならば,その展開写像は単射であり,像は射影空間の凸集合となる。
    3.シンプレクティック・トーリック多様体上に,退化した不変偏極の族を構成し,同伴する幾何学的量子化の推移において,特殊なラグランジュ部分多様体上への極所化現象が起こることを示した。
    4.閉曲面上の平坦接続のモジュライ空間の幾何学的量子化(一般化されたテ-タ関数の空間)の次元に関するフェアリンデの分解公式の,シンプレクティック幾何的な証明について研究した。その方法は,ハミルトン的群作用の下での幾何学的量子化に関する重複度公式を応用するというものである。

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  • シンプレクティック多様体に対する層理論とコホモロジー理論

    Grant number:07740081  1995    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  福岡大学

    高倉 樹

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    Grant amount: \1000000 ( Direct Cost: \1000000 )

    本研究課題において具体的な目標としていたことのうち、1.テ-タ関数の理論の偏極シンプレクティックトーラスへの一般化、2.1の結果の、リーマン面およびそのヤコビ多様体への応用、3.偏極シンプレクティック多様体のコホモロジーに関する消滅定理に対して、ある程度の結果と新たな知見を得ることができた。
    1については、必ずしもケーラー的でない偏極に対しても、対応するテ-タ関数を定義できること、異なる偏極を用いて得られるテ-タ関数の空間の間には(ベクトル空間としての)同型対応が存在することが示された。実際、同型写像を2通りの方法で構成することができる。これらの間の関係を調べることは未達成であるが、同型のユニタリ性の問題等も含めて興味深い。なお、多少修正が必要だが、シンプレクティック・トーリック多様体に対しても同様の結果が得られると思われる。
    2に関しては、リーマン面および偏極の退化という点に対しては、2次元閉多様体上の平坦G主束の同型類の空間(ただしGは一般のコンパクト単純リー群)を含めて統一的に扱うことができることがわかり、さらにG=U(1)の場合にはリーマン面の退化との関連がかなり明確に記述できた。
    3に対しては、偏極シンプレクティック多様体の層係数コホモロジーについて小平消滅定理の拡張が成り立つことが判ったが、退化のないきれいな(中間)偏極を許容するものはかなり限られたものしかなく、応用上は、偏極に退化を許すあるいは滑層構造を持つシンプレクティック空間としての偏極まで範囲を広げて考察することが重要と思われ、今後の課題として挙げておく。
    なお、J.E.Andersenも上記1、3の結果の一部を独立に得ていることをコメントしておく。

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  • 多様体の曲率と大域的性質

    Grant number:06640184  1994    

    日本学術振興会  科学研究費助成事業  一般研究(C)  福岡大学

    陶山 芳彦, 高倉 樹, 黒瀬 俊

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    Grant amount: \1100000 ( Direct Cost: \1100000 )

    リーマン多様体の研究における1つの大きなテーマとして、“局所的性質から、その大域的構造がどの程度決定されるか?"という問題がある。ここでいう局所的性質とは、他様体の曲り方を表わす曲率を指し、大域的構造とは位相構造あるいは微分構造による分類の問題を指す。この研究では主として、次の2つのテーマに関して研究を行った。
    1.微分可能球面定理の研究
    位相的球面定理を精密化して、微分構造を決定する微分可能球面定理の研究を行い、重要な成果を上げた。
    定理:完備、単連結なリーマン多様体(M,g)の断面曲率Kが、0.654<K<1となるとき、Mは標準的球面と微分同相である。
    2.共形構造を持つ多様体の研究
    リーマン多様体(M,g)が共形構造を持つとは、ワイルの共形曲率テンソルが消えるときをいう。この研究では、その様な多様体を余次元1でユークリッド空間に共形的に埋め込む問題を考えた。もしその様な埋め込みが存在するならば、(M,g)はショトキ-多様体に限られる。逆に、“全てのショトキ-多様体は共形的埋め込みを持つか"という問題である。これについては一部成果を上げているが継続して研究中である。

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