Limiting procedure and behavior of solutions for integrable dynamical systems

• Project/Area Number: 19740084
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Tokyo University of Marine Science and Technology
• Principal Investigator: TAKENAWA Tomoyuki Tokyo University of Marine Science and Technology, 海洋工学部, 准教授 (70361805)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,020,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥720,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 可積分系 / パンルヴェ方程式 / トロピカル幾何学 / 離散系 / 力学系 / ハミルトン系 / モノドロミー / 楕円曲面 / 超離散化 / トロピカル幾何 / テータ関数 / 実代数幾何 / セルオートマトン / 双有理写像

Research Abstract

I studied how property of solutions reflects through limiting procedure from discrete dynamical systems to continuous or ultra-discrete systems. Especially, on ultra-discrete systems, by using tropical algebraic geometry, I found tropical spectral curve for ultra-discrete periodic Toda lattice, which is essentially equivalent to the integrable box and ball system. On discrete systems, I clarified relationship between some dynamical systems and higher dimensional rational variety. Moreover, using rational elliptic surfaces, I classified integrable autonomous systems.

Research Products

• [Journal Article] リーの理論と可積分性-解析学におけるガロアの影響 (2011)
  • Author(s): 竹縄知之
  • Journal Title: 現代思想 39-5 Pages: 150-163
• [Journal Article] リーの理論と可積分性---解析学におけるガロアの影響 (2011)
  • Author(s): 竹縄知之
  • Journal Title: 現代思想 Volume: 39-5 Pages: 150-163
• [Journal Article] 超離散系に付随するトロピカルスペクトル曲線とテータ関数 (2009)
  • Author(s): 井上玲, 岩尾慎介, 竹縄知之
  • Journal Title: 数理解析研究所講究録 1650 Pages: 41-58
• [Journal Article] Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic (2009)
  • Author(s): R. Inoue, T. Takenawa
  • Journal Title: RIMS Kokyuroku Bessatsu B 13 Pages: 175-190
  • Peer Reviewed
• [Journal Article] Tropical representation of Weyl groups associated with certain rational varieties (2009)
  • Author(s): R. Inoue, T. Takenawa
  • Journal Title: Advances in Mathematics 221 Pages: 93-954
  • Peer Reviewed
• [Journal Article] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2009)
  • Author(s): R. Inoue, T. Takenawa
  • Journal Title: Communications in Mathematical Physics 289 Pages: 995-1021
  • Peer Reviewed
• [Journal Article] Tropical representation of Weyl groups associated with certain rational varieties (2009)
  • Author(s): T. Tsuda, T. Takenawa
  • Journal Title: Advances in Mathematics 221 Pages: 936-954
  • Peer Reviewed
• [Journal Article] Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic (2009)
  • Author(s): R. Inoue, T. Takenawa
  • Journal Title: RIMS Kokyuroku Bessatsu B13 Pages: 175-190
  • Peer Reviewed
• [Journal Article] Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic (2009)
  • Author(s): 井上玲, 竹縄知之
  • Journal Title: RIMS Kokyuroku Bessatsu B13 Pages: 175-190
  • Peer Reviewed
• [Journal Article] Tropical Spectral Curves and Integrable Cellular Automata (2008)
  • Author(s): R. Inoue, T. Takenawa
  • Journal Title: International Mathematics Research Notices 2008 rnn019 Pages: 1-27
  • Peer Reviewed
• [Journal Article] Tropical Spectral Curves and Integrable Cellular Automata (2008)
  • Author(s): R. Inoue, T. Takenawa
  • Journal Title: International Mathematics Research Notices 2008rnn019 Pages: 1-27
  • Peer Reviewed
• [Journal Article] A Class of Integrable and Nonintegrable Mappings and their Dynamics (2007)
  • Author(s): T. Tsuda, B. Grammaticos, A. Ramani, T. Takenawa
  • Journal Title: Letters in Mathematical Physics 82 Pages: 39-49
  • Peer Reviewed
• [Journal Article] A Class of Integrable and Nonintegrable (2007)
  • Author(s): T. Tsuda, B. Grammaticos, A. Ramani, T. Takenawa
  • Journal Title: Mappings and their Dynamics Letters in Mathematical Physics 82 Pages: 39-49
  • Peer Reviewed
• [Journal Article] A Class of Integrable and Nonintegrable Mappings and their Dynamics (2007)
  • Author(s): T. Tsuda, B.Grammaticos, A. Ramani, T. Takenawa
  • Journal Title: Letters in Mathematical Physics 82 Pages: 39-49
  • Peer Reviewed
• [Journal Article] A tropical analogue of Fays trisecant identity and the ultra-discrete periodic Toda lattice
  • Author(s): R. Inoue and T. Takenawa
  • Journal Title: Communications in Mathematical Physics (to appear)
  • Peer Reviewed
• [Journal Article] Tropical representation of Weyl groups associated with certain rational varieties
  • Author(s): T. Tsuda and T. Takenawa
  • Journal Title: Advances in Mathematics (to appear)
  • Peer Reviewed
• [Presentation] Tropical Spectral Curves and Integrable Cellular Automata (2010)
  • Author(s): 竹縄知之
  • Organizer: コロラド州立大学セミナー
  • Place of Presentation: コロラド州立大学(Fort Collins(アメリカ))
  • Year and Date: 2010-12-20
• [Presentation] Tropical spectral curve and integrable cellular automata (2010)
  • Author(s): T.Takenawa
  • Organizer: Seminar at Colorado State University
  • Place of Presentation: コロラド州立大学(Fort Collins(アメリカ))
  • Year and Date: 2010-12-20
• [Presentation] A classification of two dimensional integrable mappings and rational elliptic surfaces (2010)
  • Author(s): 竹縄知之
  • Organizer: the 7th International Conference on Differential Equations and Dynamical Systems
  • Place of Presentation: 南フロリダ大学(Tampa(アメリカ))
  • Year and Date: 2010-12-17
• [Presentation] A classification of two dimensional integrable mappings and rational elliptic surfaces (2010)
  • Author(s): T.Takenawa
  • Organizer: The 7th International Conference on Differential Equations and Dynamical Systems
  • Place of Presentation: 南フロリダ大学(Tampa(アメリカ))
  • Year and Date: 2010-12-17
• [Presentation] A classification of two dimensional integrable mappings and rational elliptic surfaces (2010)
  • Author(s): 竹縄知之
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: 東京大学玉原国際セミナーハウス(群馬県)
  • Year and Date: 2010-07-30
• [Presentation] 有理楕円曲面と2次元可積分写像の分類について (2010)
  • Author(s): 竹縄知之
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: 東京大学玉原国際セミナーハウス(群馬県)
  • Year and Date: 2010-07-30
• [Presentation] Toropical geometry and integrable ultra-discrete dynamical systems (2010)
  • Author(s): 竹縄知之
  • Organizer: The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: ドレスデン工科大学(Dresden(ドイツ))
  • Year and Date: 2010-05-26
• [Presentation] 実イデアルと多項式最適化に対する半正定値計画 (2010)
  • Author(s): 竹縄知之
  • Organizer: 研究集会「トロピカル幾何と超離散系の新展開」
  • Place of Presentation: 京都大学工学部(京都府)
  • Year and Date: 2010-03-11
• [Presentation] 実イデアルと多項式最適化に対する半正定値計画 (2010)
  • Author(s): 竹縄知之
  • Organizer: 研究集会「トロピカル幾何と超離散系の新展開」
  • Place of Presentation: 京都
  • Year and Date: 2010-03-11
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2009)
  • Author(s): 竹縄知之
  • Organizer: Mathematical physics seminar
  • Place of Presentation: ラフバラ大学(Loughborough,(イギリス))
  • Year and Date: 2009-04-23
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2009)
  • Author(s): 竹縄知之
  • Organizer: Mathematical physisc seminar
  • Place of Presentation: Loughborough, (イギリス)
  • Year and Date: 2009-04-23
• [Presentation] Tropical geometry and integrale cellular automata II-Bilinear form and tropical Fay's identity (2009)
  • Author(s): 竹縄知之
  • Organizer: Workshop "Geometric Aspects of Discrete and Ultra-discrete Integrable Systems"
  • Place of Presentation: グラスゴー大学(Glasgow(イギリス))
  • Year and Date: 2009-03-31
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2009)
  • Author(s): 竹縄知之
  • Organizer: Workshop "Geometric Aspects of Discrete and Ultra-discrete Integrable System"
  • Place of Presentation: グラスゴー
  • Year and Date: 2009-03-31
• [Presentation] 実イデアルと多項式最適化に対する半正定値計画 (2009)
  • Author(s): 竹縄知之
  • Organizer: 第1回日本数式処理学会理論分科会研究会
  • Place of Presentation: 東京大学工学部(東京都)
  • Year and Date: 2009-02-13
• [Presentation] Real ideal and the duality of semidefinite programming for polynomial optimization (2009)
  • Author(s): 竹縄知之
  • Organizer: 計算機代数セミナー
  • Place of Presentation: 東京
  • Year and Date: 2009-02-13
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: 大阪大学代数幾何・複素幾何セミナー
  • Place of Presentation: 大阪大学大学院理学研究科
  • Year and Date: 2008-10-17
• [Presentation] A tropical analogue of Fays trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: Workshop "Crystals and Tropical Combinatorics"
  • Place of Presentation: 京都
  • Year and Date: 2008-10-17
• [Presentation] A tropical analogue of Fays trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: 大阪大学代数幾何・複素幾何セミナー
  • Place of Presentation: 大阪
  • Year and Date: 2008-10-17
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: Workshop "Crystals and Tropical Combinatorics"
  • Place of Presentation: 関西セミナーハウス(京都府)
  • Year and Date: 2008-08-27
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: RIMS研究集会「可積分数理の拡がり」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-08-13
• [Presentation] A tropical analogue of Fays trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: RIMS研究集会「可積分数理の拡がり」
  • Place of Presentation: 京都
  • Year and Date: 2008-08-13
• [Presentation] A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice (2008)
  • Author(s): 竹縄知之
  • Organizer: 東京無限可積分系セミナー
  • Place of Presentation: 東京大学大学院数理科学研究科(東京都)
  • Year and Date: 2008-02-23
• [Presentation] A tropical analogue of Fay's trisecant identity and its application to the ultra-discrete periodic Toda equation (2008)
  • Author(s): 竹縄知之
  • Organizer: 函数方程式サマーセミナー
  • Place of Presentation: 目黒区
  • Year and Date: 2008-02-23
• [Presentation] Tropical Spectral Curves and Integrable Cellular Automata (2007)
  • Author(s): 竹縄知之
  • Organizer: 研究集会「トロピカル幾何学と関連分野」
  • Place of Presentation: 北海道大学大学院理学研究科(北海道)
  • Year and Date: 2007-09-11
• [Presentation] Tropical Spectral Curves and Integrable Cellular Automata (2007)
  • Author(s): 竹縄知之・井上玲
  • Organizer: 研究集会「トロピカル幾何学と関連分野」
  • Place of Presentation: 札幌市
  • Year and Date: 2007-09-11
• [Presentation] Tropical Spectral Curves and Integrable Cellular Automata (2007)
  • Author(s): 竹縄知之
  • Organizer: 研究集会「可積分数理の新潮流」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2007-08-20
• [Presentation] Tropical Spectral Curves and Integrable Cellular Automata (2007)
  • Author(s): 竹縄知之・井上玲
  • Organizer: 研究集会「可積分数理の新潮流」
  • Place of Presentation: 京都市
  • Year and Date: 2007-08-20
• [Presentation] Tropical Spectral Curves and Integrable Cellular Automata (2007)
  • Author(s): 竹縄知之
  • Organizer: 函数方程式サマーセミナー
  • Place of Presentation: ビレッジ安曇野(長野県)
  • Year and Date: 2007-08-06
• [Presentation] トロピカル超楕円曲線のAbel-Jacobi写像について (2007)
  • Author(s): 竹縄知之
  • Organizer: 函数方程式サマーセミナー
  • Place of Presentation: 安曇野市
  • Year and Date: 2007-08-06
• [Book] コア・テキスト微分積分 (2009)
  • Author(s): 竹縄知之
  • Total Pages: 222
  • Publisher: サイエンス社
• [Remarks] ホームページ等
  • URL: http://www2.kaiyodai.ac.jp/~takenawa/