Schubert calculus from algebraic topological viewpoint

• Project/Area Number: 22740051
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Yamaguchi University
• Principal Investigator: KAJI Shizuo 山口大学, 理工学研究科, 講師 (00509656)
• Project Period (FY): 2010-10-20 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: トポロジー / リー群 / シューベルトカルキュラス / 旗多様体 / コホモロジー / 幾何学 / 位相幾何学 / 同変トポロジー / 同変コホモロジー / シューベルトカリキュラス

Research Abstract

Some problems in Schubert calculus are studied from algebraic topological viewpoint. A type-uniform treatment for the torus equivariant cohomology ring of the flag varieties including exceptional ones is given. A generalisation of Schubert calculus to a class of manifolds with good Lie group actions is also obtained.

Research Products

• [Journal Article] Anti-commutative Dual Complex Numbers and 2D Rigid Transformation (2014)
  • Author(s): G. Matsuda, S. Kaji, and H. Ochiai
  • Journal Title: Mathematical Progress in Expressive Image Synthesis I Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Weyl group symmetry on the GKM graph of a GKM manifold with an extended Lie group action (2014)
  • Author(s): S. Kaji
  • Journal Title: Osaka J. Math. Volume: 印刷中
  • NAID: 120005986372
  • Peer Reviewed
• [Journal Article] A Lie Theoretic Parameterization of Affine Transformation (2013)
  • Author(s): S.Kaji, S.Hirose, H. Ochiai, and K.Anjyo
  • Journal Title: MI Lecture Notes Volume: 50
• [Journal Article] Mathematical Analysis on Affine Maps for 2D Shape Interpolation (2012)
  • Author(s): S.Kaji, S.Hirose, S.Sakata, Y.Mizoguchi, and K.Anjyo
  • Journal Title: Proceedings of SCA2012 Volume: 1 Pages: 71-76
  • DOI: 10.2312/SCA/SCA12/071-076
  • Peer Reviewed
• [Journal Article] A mathematical problem for security analysis of hash functions and pseudorandom generators (2011)
  • Author(s): Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata
  • Journal Title: Proceedings of IWSEC 2011 Volume: 7038 Pages: 144-160
  • NAID: 120007100293
  • Peer Reviewed
• [Journal Article] Equivariant Schubert calculus of Coxeter groups (2011)
  • Author(s): Shizuo Kaji
  • Journal Title: Proc. Stekelov. Inst. Math Volume: vol.275 Pages: 239-250
  • NAID: 120006665731
  • Peer Reviewed
• [Journal Article] Equivariant Schubert calculus of Coxeter groups (2011)
  • Author(s): Shizuo KAJI
  • Journal Title: Proc.Stekelov.Inst.Math Volume: 275 Pages: 239-250
  • NAID: 120006665731
  • Peer Reviewed
• [Journal Article] A mathematical problem for security analysis of hash functions and pseudorandom generators (2011)
  • Author(s): Kojl Nulda Takuro Abe, Sbizuo Kaji, Toshiaki Mano, Yasuhide Numata
  • Journal Title: Lecture Note in Computer Sciences Volume: 7,038 Pages: 144-160
  • NAID: 120007100293
  • Peer Reviewed
• [Journal Article] Schubert calculus, seen from torus equivariant topology (2010)
  • Author(s): Shizuo Kaji
  • Journal Title: Trends in Mathematics -New Series Volume: Volume12, no.1 Pages: 71-90
  • NAID: 120007145420
  • Peer Reviewed
• [Journal Article] Schubert calculus, seen from torus equivariant topology (2010)
  • Author(s): S.Kaji
  • Journal Title: Trends in Mathematics-New Series Volume: 12-1 Pages: 71-90
  • NAID: 120007145420
  • Peer Reviewed
• [Journal Article] Weyl group symmetry on the GKM graph of a GKM manifold with an extended Lie group action
  • Author(s): Shizuo KAJI
  • Journal Title: Osaka Journal of Mathematics Volume: (印刷中)
  • NAID: 120005986372
  • Peer Reviewed
• [Journal Article] Anti-commutative Dual Complex Numbers and 2D Rigid Transformation
  • Author(s): G. Matsuda, S. Kaji, and H. Ochiai
  • Journal Title: Proceedings of MEIS2013 (Mathematical Progress in Expressive Image Synthesis 2013) Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Mathematical Analysis on Affine Maps for 2D Shape Interpolation
  • Author(s): S. Kaji, S. Hirose, S. Sakata, Y. Mizoguchi, and K. Anjyo
  • Journal Title: Proceedings of SCA2012 (ACM SIGGRAPH/Eurographics Symposium on Computer Animation 2012) Pages: 71-76
  • Peer Reviewed
• [Presentation] An Application of Lie theory to Computer Graphics (2013)
  • Author(s): Shizuo KAJI
  • Organizer: Applied Topology
  • Place of Presentation: Banach center, Bedlewo (ポーランド)
  • Year and Date: 2013-07-25
• [Presentation] Cohomology of a GKM graph with symmetry (2012)
  • Author(s): Shizuo KAJI
  • Organizer: Toric Topology 2012 in Osaka
  • Place of Presentation: Osaka city university (大阪市)
  • Year and Date: 2012-11-18
• [Presentation] 数学がつなぐカタチ - 幾何学的な形状補間法 - (2012)
  • Author(s): Shizuo KAJI
  • Organizer: CEDEC2012
  • Place of Presentation: Pacifico Yokohama (横浜市)
  • Year and Date: 2012-08-20
• [Presentation] The equivariant cohomology of a manifold with a G-action (2012)
  • Author(s): Shizuo KAJI
  • Organizer: Topology of Mapping space and around
  • Place of Presentation: Okinawa Senin Kaikan (那覇市)
  • Year and Date: 2012-05-14
• [Presentation] Constructing a Schubert-like basis for GKM G-manifolds (2011)
  • Author(s): Shizuo KAJI
  • Organizer: Toric Topology 2011 in Osaka
  • Place of Presentation: 大阪市大(大阪)
  • Year and Date: 2011-11-28
• [Presentation] Schubert calculus for G-manifolds (2011)
  • Author(s): Shizuo KAJI
  • Organizer: International Conference Toric Topology and Automorphic Functions
  • Place of Presentation: Pacific National University, Khabarovsk(ロシア)
  • Year and Date: 2011-09-09
• [Presentation] Schubert calculus for G-manifolds (2011)
  • Author(s): Shizuo KAJI
  • Organizer: International Conference Toric Topology and Automorphic Function
  • Place of Presentation: Khabarovsk, Russia
  • Year and Date: 2011-09-09
• [Presentation] 旗多様体の同変コホモロジー (2010)
  • Author(s): 鍛冶静雄
  • Organizer: トポロジーの多様性
  • Place of Presentation: 豊岡市、城崎大会議館
  • Year and Date: 2010-12-02
• [Presentation] Equivariant Schubert calculus of Coxeter groups (2010)
  • Author(s): 鍛冶静雄
  • Organizer: International Conference Japan-Mexico on Topology and its Applications
  • Place of Presentation: コリマ大学(メキシコ)
  • Year and Date: 2010-09-29
• [Presentation] Equivariant Schubert calculus of Coxeter group I_2(m) (2010)
  • Author(s): Shizuo KAJI
  • Organizer: The International Conference "GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS" dedicated to the 120th anniversary of Boris Delone
  • Place of Presentation: Moscow(ロシア)
  • Year and Date: 2010-08-19
• [Presentation] Equivariant Schubert calculus of Coxeter group$I_2(m)$ (2010)
  • Author(s): 鍛冶静雄
  • Organizer: The International Conference GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS dedicated to the 120th anniversary of Boris Delone
  • Place of Presentation: モスクワ大学
  • Year and Date: 2010-08-19
• [Presentation] Equivariant Schubert calculus of Coxeter group (2010)
  • Author(s): Shizuo KAJI
  • Organizer: International Conference Japan-Mexico on Topology and its Applications
  • Place of Presentation: Colima( メキシコ)
• [Presentation] An Application of Lie theory to Computer Graphics
  • Author(s): S. Kaji
  • Organizer: Applied Topology
  • Place of Presentation: Banach center (Bedlewo, Poland)
• [Presentation] A topological algorithm for Blending Shapes
  • Author(s): S. Kaji
  • Organizer: Directed Algebraic Topology and Concurrency
  • Place of Presentation: Universite Lyon 1 (Lyon, France)
• [Presentation] Ordinary vs Double Schubert polynomials
  • Author(s): S. Kaji
  • Organizer: MSJ-SI2012 Schubert calculus
  • Place of Presentation: 大阪市立大学
• [Presentation] 数学がつなぐカタチ - 幾何学的な形状補間法 -
  • Author(s): 鍛冶静雄
  • Organizer: CEDEC2012
  • Place of Presentation: Pacifico Yokohama
  • Invited
• [Presentation] Cohomology of a GKM-graph with symmetry
  • Author(s): S. Kaji
  • Organizer: Toric Topology 2012 in Osaka
  • Place of Presentation: 大阪市立大学
• [Presentation] The equivariant cohomology of a manifold with a G-action
  • Author(s): S. Kaji
  • Organizer: Topology of Mapping space and around
  • Place of Presentation: 沖縄船員会館
• [Remarks]
  • URL: http://skaji.org/home/publications
• [Remarks]
  • URL: http://www.math.kyoto-u.ac.jp/~kaji/index.php