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

• Project/Area Number: 24540064
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: University 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: ¥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.

Research Products

• [Journal Article] The intersection of two real forms in Hermitian symmetric spaces of compact type II (2015)
  • 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
  • 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 manifold (2014)
  • 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
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Lagrangian intersection theory and Hamiltonian volume minimizing problem (2014)
  • 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
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Sequences of maximal antipodal sets of oriented real Grassmann manifolds (2014)
  • 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
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Antipodal sets of symmetric $R$-spaces (2013)
  • Author(s): M. S. Tanaka and H. Tasaki
  • Journal Title: Osaka J. Math. Volume: 50 Pages: 161-169
  • NAID: 120007136598
  • Peer Reviewed
• [Journal Article] Antipodal sets in oriented real Grassmann manifolds (2013)
  • Author(s): H. Tasaki
  • Journal Title: Internat. J. Math. Volume: 24 Issue: 08 Pages: 1350061-1350061
  • DOI: 10.1142/s0129167x13500614
  • NAID: 120007136819
  • Peer Reviewed
• [Journal Article] Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type (2013)
  • 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
  • Peer Reviewed
• [Journal Article] The intersection of two real forms in Hermitian symmetric spaces of compact type (2012)
  • 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
  • Peer Reviewed
• [Presentation] The intersection of two real flag manifolds in a complex flag manifold (2015)
  • 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
  • Invited
• [Presentation] Antipodal sets of symmetric R-spaces and their application (2015)
  • 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
  • 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
• [Presentation] Antipodal sets in oriented real Grassmann manifolds (2015)
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
• [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
• [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
  • 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
  • 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
  • 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
  • Invited
• [Presentation] The fixed point set of a holomorphic isometry and the intersection of two real forms in the complex Grassmann manifold (2014)
  • 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
  • Invited
• [Presentation] Sequences of maximal antipodal sets of oriented real Grassmann manifolds (2014)
  • 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
  • Invited
• [Presentation] Lagrangian intersection theory and Hamiltonian volume minimizing problem (2014)
  • 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
  • Invited
• [Presentation] Symmetric triads and their applications (2014)
  • 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
  • Invited
• [Presentation] Antipodal structure of the intersection of real forms and its applications (2014)
  • 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
  • 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
• [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
  • 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
  • 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
• [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
• [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
  • 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
  • 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
  • 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
  • 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
  • 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
• [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
  • 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
• [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
  • 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
  • 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
  • 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
  • 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
  • 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)
• [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)
• [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)
  • 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)
  • 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)
  • 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)
• [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)
• [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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
• [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)
  • 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)
  • 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)
  • 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)
  • Invited
• [Remarks] 田崎博之のホームページ
  • URL: http://www.math.tsukuba.ac.jp/~tasaki/
• [Remarks] 田崎博之のホームページ
  • URL: http://www.math.tsukuba.ac.jp/~tasaki/
• [Remarks] 田崎博之のホームページ
  • URL: http://www.math.tsukuba.ac.jp/~tasaki/