An algebraic topological approach to Schubert calculus

• Project/Area Number: 20840041
• Research Category: Grant-in-Aid for Young Scientists (Start-up)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Fukuoka University
• Principal Investigator: KAJI Shizuo Fukuoka University, 理学部, 助教 (00509656)
• Project Period (FY): 2008 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,276,000 (Direct Cost: ¥2,520,000、Indirect Cost: ¥756,000); Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2008: ¥1,716,000 (Direct Cost: ¥1,320,000、Indirect Cost: ¥396,000)
• Keywords: 位相幾何学 / ホモトピー論 / リー群 / 群作用をもつ空間 / の位相不変量 / 旗多様体 / 幾何学 / トポロジー / コホモロジー / シューベルトカルキュラス / 等質空間 / グラスマン多様体

Research Abstract

The symmetry of spaces transformed continuously is mathematically described through actions of Lie groups, which can be caught by invariants of the Lie groups and the spaces acted by them. In this research project, we investigate Schubert calculus, the study of the geometry of a certain class of spaces called flag varieties, which appear as the quotient of Lie groups. We took an algebraic topological approach to it and calculated some invariants of flag varieties explicitly.

Research Products

• [Journal Article] Homotopy nilpotency in $p$-regular loop spaces (2010)
  • Author(s): Shizuo Kaji, Daisuke Kishimoto
  • Journal Title: Mathematische Zeitschrift vol.264,no.1
  • Peer Reviewed
• [Journal Article] Homotopy nilpotency in $p$-regular loop space (2010)
  • Author(s): Shizuo KAJI, Daisuke Kishirnoto
  • Journal Title: Mathematische Zeitschrift 26-1-1
  • Peer Reviewed
• [Presentation] SCHUBERT CALCULUS, SEEN FROM TORUS EQUIVARIANT TOPOLOGY (2010)
  • Author(s): Shizuo KAJI
  • Organizer: KAIST Toric Topology Workshop 2010
  • Place of Presentation: Korea Advanced Institute of Science and Technology
  • Year and Date: 2010-02-25
• [Presentation] Schubert Calculus, seen from trus equivariant topology (2010)
  • Author(s): 鍛治静雄
  • Organizer: KAIST Toric Topology Workshop 2010
  • Place of Presentation: Korea Advanced Institute of Science and Technology
  • Year and Date: 2010-02-05
• [Presentation] G2型旗多様体の同変コホモロジー (2010)
  • Author(s): 鍛冶静雄
  • Organizer: 福岡ホモトピー論セミナー
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2010-01-09
• [Presentation] Torus equivariant cohomology of flag varieties (2009)
  • Author(s): Shizuo KAJI
  • Organizer: The Third East Asia Conference on Algebraic Topology
  • Place of Presentation: Vietnam National University, Hanoi
  • Year and Date: 2009-12-17
• [Presentation] Torus equivariant cohomology of flag varieties (2009)
  • Author(s): 鍛治静雄
  • Organizer: The Third East Asia Conference on Algebraic Topology
  • Place of Presentation: Vietnam National University
  • Year and Date: 2009-12-17
• [Presentation] リー群のトポロジーから見るシューベルトカリキュラス (2009)
  • Author(s): 鍛冶静雄
  • Organizer: 第56回トポロジーシンポジウム
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-08-10
• [Presentation] リー群のトポロジーから見るシューベルトカリキュラス (2009)
  • Author(s): 鍛治静雄
  • Organizer: 第56回トポロジーシンポジウム
  • Place of Presentation: 化海道大学
  • Year and Date: 2009-08-10
• [Presentation] Divided difference operator and equivariant cohomology (2009)
  • Author(s): 鍛冶静雄
  • Organizer: トポロジー小研究集会
  • Place of Presentation: 信州大学
  • Year and Date: 2009-03-18
• [Presentation] Divided difference operator and equivariant cohomology (2009)
  • Author(s): 鍛治静雄
  • Organizer: 信州トポロジー小研究集会
  • Place of Presentation: 信州大学
  • Year and Date: 2009-03-18
• [Presentation] Bott-Samelson cycle in Schubert calculus (2008)
  • Author(s): 鍛治静雄
  • Organizer: Very informal seminar on Schubert calculus
  • Place of Presentation: 岡山大学
  • Year and Date: 2008-12-25
• [Presentation] An algebraic topological approach toward concrete Schubert calculus (2008)
  • Author(s): Shizuo KAJI
  • Organizer: The 2nd East Asia Conference on Algebraic Topology
  • Place of Presentation: National University of Singapore
  • Year and Date: 2008-12-19
• [Presentation] An algebraic topological approach toward concrete Schubert calculus (2008)
  • Author(s): 鍛治静雄
  • Organizer: The 2nd East Asia Conference on Algebraic Topology
  • Place of Presentation: National Univ. of Singapore
  • Year and Date: 2008-12-19
• [Presentation] Computers work exceptionally on Lie groups (2008)
  • Author(s): 鍛治静雄
  • Organizer: 第一回GCOE「計算機を用いた数学研究」セミナー
  • Place of Presentation: 京都大学
  • Year and Date: 2008-10-24
• [Presentation] Mod 2 cohomology of some low rank 2-local finite groups (2008)
  • Author(s): Shizuo KAJI
  • Organizer: Shizuo KAJI, International Conference on Algebraic Topology
  • Place of Presentation: Korea University
  • Year and Date: 2008-10-05
• [Remarks]
  • URL: http://www.math.kyoto-u.ac.jp/~kaji
• [Remarks] 研究会などの開催第3回Lusternik-Schnirelmannカテゴリー研究集会「L-SカテゴリとRobot Motion Planning」,2008年10月29日～10月31日,唐津市・共同オーガナイザー