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Study of antipodal sets in symmetric spaces with its extension and application

Research Project

Project/Area Number 24540064
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

TASAKI Hiroyuki  筑波大学, 数理物質系, 准教授 (30179684)

Co-Investigator(Renkei-kenkyūsha) TANAKA Makiko  東京理科大学, 理工学部, 教授 (20255623)
IKAWA Osamu  京都工芸繊維大学, 工芸科学研究科, 教授 (60249745)
AKAI Takashi  首都大学東京, 理工学研究科, 准教授 (30381445)
IRIYEH Hiroshi  東京電機大学, 未来科学部, 准教授 (30385489)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
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)
Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords対称空間 / 対蹠集合 / 実形の交叉 / 複素旗多様体 / 有向実Grassmann多様体 / 対称三対
Outline of Final Research Achievements

We showed a necessary and sufficient condition for the intersection of two real forms in a Hermitian symmetric space of compact type to be discrete and proved that the discrete intersection is an orbit of a Weyl group in a joint work with Makiko Tanaka and Osamu Ikawa. We partly extended these results to two real forms in a complex flag manifold in a joint work with Hiroshi Iriyeh and Takashi Sakai.
We associated antipodal sets in real oriented Grassmann manifolds with certain combinatorial objects and classified antipodal sets in the case where the rank is less than 5.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (65 results)

All 2015 2014 2013 2012 Other

All Journal Article (8 results) (of which Peer Reviewed: 8 results,  Acknowledgement Compliant: 4 results) Presentation (54 results) (of which Invited: 41 results) Remarks (3 results)

  • [Journal Article] The intersection of two real forms in Hermitian symmetric spaces of compact type II2015

    • Author(s)
      M. S. Tanaka and H. Tasaki
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 67 Issue: 1 Pages: 275-291

    • DOI

      10.2969/jmsj/06710275

    • NAID

      130004927759

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The fixed point set of a holomorphic isometry and the intersection of two real forms in the complex Grassmann manifold2014

    • Author(s)
      O. Ikawa, M. S. Tanaka and H. Tasaki
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 106 Pages: 319-327

    • DOI

      10.1007/978-4-431-55215-4_28

    • ISBN
      9784431552147, 9784431552154
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Lagrangian intersection theory and Hamiltonian volume minimizing problem2014

    • Author(s)
      H. Iriyeh, T. Sakai and H. Tasaki
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 106 Pages: 391-399

    • DOI

      10.1007/978-4-431-55215-4_35

    • ISBN
      9784431552147, 9784431552154
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Sequences of maximal antipodal sets of oriented real Grassmann manifolds2014

    • Author(s)
      H. Tasaki
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 106 Pages: 515-524

    • DOI

      10.1007/978-4-431-55215-4_46

    • ISBN
      9784431552147, 9784431552154
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Antipodal sets of symmetric $R$-spaces2013

    • Author(s)
      M. S. Tanaka and H. Tasaki
    • Journal Title

      Osaka J. Math.

      Volume: 50 Pages: 161-169

    • NAID

      120007136598

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Antipodal sets in oriented real Grassmann manifolds2013

    • Author(s)
      H. Tasaki
    • Journal Title

      Internat. J. Math.

      Volume: 24 Issue: 08 Pages: 1350061-1350061

    • DOI

      10.1142/s0129167x13500614

    • NAID

      120007136819

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type2013

    • Author(s)
      H. Iriyeh, T. Sakai and H. Tasaki
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 65 Issue: 4 Pages: 1135-1151

    • DOI

      10.2969/jmsj/06541135

    • NAID

      130003384667

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] The intersection of two real forms in Hermitian symmetric spaces of compact type2012

    • Author(s)
      M. S. Tanaka and H. Tasaki
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 64 Issue: 4 Pages: 1297-1332

    • DOI

      10.2969/jmsj/06441297

    • NAID

      10031177268

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] The intersection of two real flag manifolds in a complex flag manifold2015

    • Author(s)
      酒井高司
    • Organizer
      研究集会「ラグランジュ幾何と幾何学流」
    • Place of Presentation
      蔵王KKR白銀荘(山形県山形市)
    • Year and Date
      2015-03-26 – 2015-03-28
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Antipodal sets of symmetric R-spaces and their application2015

    • Author(s)
      田中真紀子
    • Organizer
      Oberseminar Differentialgeometrie
    • Place of Presentation
      University of Augsburg(ドイツ連邦共和国)
    • Year and Date
      2015-03-26
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の二つの実旗多様体の交叉2015

    • Author(s)
      奥田隆幸
    • Organizer
      日本数学会2015年度年会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Annual Research Report
  • [Presentation] Antipodal sets in oriented real Grassmann manifolds2015

    • Author(s)
      田崎博之
    • Organizer
      Hakata Workshop 2015
    • Place of Presentation
      Reference Eki Higashi Building at Hakata(福岡県福岡市)
    • Year and Date
      2015-02-15
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の二つの実旗多様体の交叉2014

    • Author(s)
      田崎博之
    • Organizer
      広島大学大学院理学研究科数学専攻談話会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-12-09
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の二つの実形の交叉2014

    • Author(s)
      田崎博之
    • Organizer
      部分多様体論・湯沢2014
    • Place of Presentation
      湯沢グランドホテル(新潟県南魚沼郡)
    • Year and Date
      2014-11-20 – 2014-11-22
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の二つの実形の交叉2014

    • Author(s)
      酒井高司
    • Organizer
      福岡大学微分幾何研究会
    • Place of Presentation
      福岡大学
    • Year and Date
      2014-10-31 – 2014-11-03
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 対称空間の対蹠集合2014

    • Author(s)
      田崎博之
    • Organizer
      広島幾何学研究集会2014
    • Place of Presentation
      広島大学
    • Year and Date
      2014-10-08 – 2014-10-10
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の四元数旗多様体の交叉の構造2014

    • Author(s)
      酒井高司
    • Organizer
      日本数学会2014年度秋季総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Annual Research Report
  • [Presentation] 対蹠的部分集合の系列と評価2014

    • Author(s)
      田崎博之
    • Organizer
      日本数学会2014年度秋季総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Annual Research Report
  • [Presentation] 対称三対の基礎と応用2014

    • Author(s)
      井川治
    • Organizer
      日本数学会2014年度秋季総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の四元数旗多様体の交叉の構造2014

    • Author(s)
      酒井高司
    • Organizer
      研究集会「部分多様体幾何とリー群作用2014」
    • Place of Presentation
      東京理科大学森戸記念館(東京都新宿区)
    • Year and Date
      2014-09-05 – 2014-09-06
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の実形の交叉の対蹠性2014

    • Author(s)
      酒井高司
    • Organizer
      幾何学シンポジウム
    • Place of Presentation
      名城大学
    • Year and Date
      2014-08-23 – 2014-08-26
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 対称三対の実形の交叉への応用2014

    • Author(s)
      井川治
    • Organizer
      幾何学シンポジウム
    • Place of Presentation
      名城大学
    • Year and Date
      2014-08-23 – 2014-08-26
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The fixed point set of a holomorphic isometry and the intersection of two real forms in the complex Grassmann manifold2014

    • Author(s)
      田中真紀子
    • Organizer
      ICM 2014 Satellite Conference on Real and Complex Submanifolds
    • Place of Presentation
      NIMS, Daejeon, Korea
    • Year and Date
      2014-08-10 – 2014-08-12
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Sequences of maximal antipodal sets of oriented real Grassmann manifolds2014

    • Author(s)
      田崎博之
    • Organizer
      ICM 2014 Satellite Conference on Real and Complex Submanifolds
    • Place of Presentation
      NIMS, Daejeon, Korea
    • Year and Date
      2014-08-10 – 2014-08-12
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Lagrangian intersection theory and Hamiltonian volume minimizing problem2014

    • Author(s)
      酒井高司
    • Organizer
      ICM 2014 Satellite Conference on Real and Complex Submanifolds
    • Place of Presentation
      NIMS, Daejeon, Korea
    • Year and Date
      2014-08-10 – 2014-08-12
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Symmetric triads and their applications2014

    • Author(s)
      井川治
    • Organizer
      RIMS研究集会「群作用と部分多様体論の展開」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-06-25 – 2014-06-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Antipodal structure of the intersection of real forms and its applications2014

    • Author(s)
      酒井高司
    • Organizer
      幾何コロキウム
    • Place of Presentation
      東京大学
    • Year and Date
      2014-06-19
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素旗多様体内の四元数旗多様体の交叉の構造2014

    • Author(s)
      田崎博之
    • Organizer
      筑波大学微分幾何学火曜セミナー
    • Place of Presentation
      筑波大学
    • Year and Date
      2014-06-17
    • Related Report
      2014 Annual Research Report
  • [Presentation] 対蹠的部分集合の系列と評価2014

    • Author(s)
      田崎博之
    • Organizer
      離散数理セミナー
    • Place of Presentation
      山形大学
    • Year and Date
      2014-06-06
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 対称三対とその応用2014

    • Author(s)
      井川治
    • Organizer
      秋葉原微分幾何セミナー
    • Place of Presentation
      秋葉原ダイビル(東京都千代田区)
    • Year and Date
      2014-05-24
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 複素Grassmann多様体の正則等長変換の不動点集合と二つの実形の交叉2014

    • Author(s)
      田崎博之
    • Organizer
      筑波大学微分幾何学火曜セミナー
    • Place of Presentation
      筑波大学
    • Year and Date
      2014-05-13
    • Related Report
      2014 Annual Research Report
  • [Presentation] 有向実Grassmann多様体の対蹠集合の系列と評価2014

    • Author(s)
      田崎博之
    • Organizer
      筑波大学微分幾何学火曜セミナー
    • Place of Presentation
      筑波大学
    • Year and Date
      2014-04-15
    • Related Report
      2014 Annual Research Report
  • [Presentation] 有向実Grassmann多様体の対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      首都大学東京幾何学セミナー
    • Place of Presentation
      首都大学東京
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Geometry of symmetric $R$-spaces

    • Author(s)
      田中真紀子
    • Organizer
      多様体上の幾何と解析(小林昭七先生記念シンポジウム)
    • Place of Presentation
      東京大学大学院数理科学研究科
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 複素旗多様体内の実旗多様体の交叉の構造

    • Author(s)
      酒井高司
    • Organizer
      研究集会「部分多様体の微分幾何学の深化」
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      東北大学組合せ論セミナー
    • Place of Presentation
      東北大学情報科学研究科
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 二つの実形の交叉と対称三対

    • Author(s)
      井川治
    • Organizer
      研究集会「部分多様体幾何とリー群作用2013」
    • Place of Presentation
      東京理科大学森戸記念館
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      日本数学会2013年度秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Research-status Report
  • [Presentation] Fixed point sets of isometries and the intersection of real forms in a Hermitian symmetric space of compact type

    • Author(s)
      田中真紀子
    • Organizer
      The 17th International Workshop on Differential Geometry
    • Place of Presentation
      NIMS, Daejeon, Korea
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 極大対蹠的部分集合の系列

    • Author(s)
      田崎博之
    • Organizer
      部分多様体論・湯沢2013
    • Place of Presentation
      湯沢グランドホテル
    • Related Report
      2013 Research-status Report
  • [Presentation] 正則等長変換の不動点集合、実形の交叉と対称三対

    • Author(s)
      井川治
    • Organizer
      部分多様体論・湯沢2013
    • Place of Presentation
      湯沢グランドホテル
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の極大対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      離散数理セミナー
    • Place of Presentation
      山形大学理学部
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の対蹠集合の系列と評価

    • Author(s)
      田崎博之
    • Organizer
      組合せ数学セミナー
    • Place of Presentation
      東京大学駒場キャンパス
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Antipodal structure of the intersection of real forms and its applications

    • Author(s)
      酒井高司
    • Organizer
      Seminars
    • Place of Presentation
      University of California, Irvine, USA
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Fixed point sets of holomorphic isometries of Hermitian symmetric spaces

    • Author(s)
      田中真紀子
    • Organizer
      Oberseminar Differentialgeometrie
    • Place of Presentation
      University of Augsburg, Germany
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 複素旗多様体内の実旗多様体の交叉の構造

    • Author(s)
      田崎博之
    • Organizer
      筑波大学微分幾何学火曜セミナー
    • Place of Presentation
      筑波大学(茨城県)
    • Related Report
      2012 Research-status Report
  • [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉II

    • Author(s)
      田崎博之
    • Organizer
      筑波大学微分幾何学火曜セミナー
    • Place of Presentation
      筑波大学(茨城県)
    • Related Report
      2012 Research-status Report
  • [Presentation] 複素旗多様体の実形の交叉の対蹠性

    • Author(s)
      酒井高司
    • Organizer
      明治大学幾何学セミナー
    • Place of Presentation
      明治大学生田キャンパス(神奈川県)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] コンパクト対称空間の対蹠集合と部分多様体の交叉

    • Author(s)
      田中真紀子
    • Organizer
      研究集会「部分多様体の微分幾何学及び関連課題」
    • Place of Presentation
      佐賀大学理工学部(佐賀県)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉

    • Author(s)
      田中真紀子
    • Organizer
      研究集会「部分多様体幾何とリー群作用2012」
    • Place of Presentation
      東京理科大学森戸記念館(東京都)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉II

    • Author(s)
      田中真紀子、田崎博之
    • Organizer
      日本数学会2012年度秋季総合分科会
    • Place of Presentation
      九州大学伊都キャンパス(福岡県)
    • Related Report
      2012 Research-status Report
  • [Presentation] 複素旗多様体内の実旗多様体の交叉の構造

    • Author(s)
      酒井高司、入江博、田崎博之
    • Organizer
      日本数学会2012年度秋季総合分科会
    • Place of Presentation
      九州大学伊都キャンパス(福岡県)
    • Related Report
      2012 Research-status Report
  • [Presentation] The intersection of two real forms in Hermitian symmetric spaces of compact typ

    • Author(s)
      田中真紀子
    • Organizer
      Oberseminar Differentialgeometrie
    • Place of Presentation
      University of Augsburg(ドイツ)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 複素旗多様体内の実旗多様体の交叉の構造

    • Author(s)
      酒井高司
    • Organizer
      首都大学東京幾何学セミナー
    • Place of Presentation
      首都大学東京(東京都)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Antipodal structure of the intersection of real flag manifolds in a complex flag manifold

    • Author(s)
      酒井高司
    • Organizer
      The 16th International Workshop on Differential Geometry
    • Place of Presentation
      Kyungpook National University(韓国)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] The intersection of two real forms in Hermitian symmetric spaces of compact type

    • Author(s)
      田中真紀子
    • Organizer
      The 16th International Workshop on Differential Geometry
    • Place of Presentation
      Kyungpook National University(韓国)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      部分多様体論・湯沢2012
    • Place of Presentation
      湯沢グランドホテル(新潟県)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      筑波大学微分幾何学火曜セミナー
    • Place of Presentation
      筑波大学(茨城県)
    • Related Report
      2012 Research-status Report
  • [Presentation] Antipodal sets in compact Riemannian symmetric spaces

    • Author(s)
      田崎博之
    • Organizer
      The 6th OCAMI-KNUGRG Joint International Workshop on Submanifold Theory in Symmetric Spaces and Lie Theory in Finite and Infinite Dimensions
    • Place of Presentation
      大阪市立大学(大阪府)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces and its application

    • Author(s)
      入江博
    • Organizer
      Symplectic Geometry Seminar
    • Place of Presentation
      ETH Zurich(スイス)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] コンパクト対称空間の対蹠集合と全測地的部分多様体の交叉

    • Author(s)
      田中真紀子
    • Organizer
      日本数学会2012年度年会
    • Place of Presentation
      京都大学(京都府)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 有向実Grassmann多様体の対蹠集合

    • Author(s)
      田崎博之
    • Organizer
      首都大学東京幾何学セミナー
    • Place of Presentation
      首都大学東京(東京都)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Remarks] 田崎博之のホームページ

    • URL

      http://www.math.tsukuba.ac.jp/~tasaki/

    • Related Report
      2014 Annual Research Report
  • [Remarks] 田崎博之のホームページ

    • URL

      http://www.math.tsukuba.ac.jp/~tasaki/

    • Related Report
      2013 Research-status Report
  • [Remarks] 田崎博之のホームページ

    • URL

      http://www.math.tsukuba.ac.jp/~tasaki/

    • Related Report
      2012 Research-status Report

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Published: 2013-05-31   Modified: 2019-07-29  

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