Study of integrable systems and tropical curves

• Project/Area Number: 26800062
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Tokai University (2017); Aoyama Gakuin University (2014-2016)
• Principal Investigator: Iwao Shinsuke 東海大学, 理学部, 講師 (70634989)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: トロピカル幾何学 / 対称多項式 / Young盤の組み合わせ論 / 旗多様体の量子K理論 / 特異曲線 / 超離散可積分系 / 正値性 / トロピカル曲線 / 旗多様体の量子コホモロジー / 離散可積分系 / 量子コホモロジー / Totally positivity / トロピカル幾何 / Totally Positive matrix

Outline of Final Research Achievements

The tropical geometry is a kind of geometry where the usual multiplication and addition are replaced with the addition and maximum. Since it has been known that the tropical geometry admits good applications to the study of integrable system theory, the main aim of this research is to study their essential relations. The main results of this research are as follows: 1. The relation between the "quantum K-theory" of the flag variety and some special symmetric polynomials are clarified by using the algebraic method to the relativistic Toda equation. 2. The application of the tropical KP equation to various combinatoric problems of Young tableau is obtained.

Research Products

• [Journal Article] Peterson isomorphism in K-theory and relativistic Toda lattice (2018)
  • Author(s): Takeshi Ikeda, Shinsuke Iwao and Toshiaki Maeno
  • Journal Title: International Mathematics Research Notices Volume: - Issue: 19 Pages: 6421-6462
  • DOI: 10.1093/imrn/rny051
  • Peer Reviewed
• [Journal Article] The discrete Toda equation revisited: dual β-Grothendieck polynomials, ultradiscretization, and static solitons (2018)
  • Author(s): S.Iwao and H.Nagai
  • Journal Title: J. Phys. A: Math. Theor. Volume: 51 Issue: 13 Pages: 134002-134002
  • DOI: 10.1088/1751-8121/aaae30
  • Peer Reviewed
• [Journal Article] ボゾンフェルミオン対応の基礎と線形代数のみから双対GROTHENDIECK多項式の行列式表示を導く (2017)
  • Author(s): 岩尾慎介
  • Journal Title: RIMS講究録 Volume: 印刷中
  • Open Access / Acknowledgement Compliant
• [Journal Article] 非周期的有限戸田格子方程式に対する一般化ヤコビ多様体とtotally non-negativity (2015)
  • Author(s): 岩尾 慎介
  • Journal Title: 九州大学応用力学研究所研究集会報告(講究録) Volume: 26AO-S2
  • Peer Reviewed
• [Presentation] 旗多様体の量子K理論と相対論的戸田方程式 (2017)
  • Author(s): 岩尾慎介
  • Organizer: RIMS研究集会2017「可積分系の数理と応用」
• [Presentation] 相対論的戸田方程式のLax行列と超離散戸田方程式の負の解 (2017)
  • Author(s): 岩尾慎介
  • Organizer: 研究集会「数理科学の広がり：可積分系・数理科学」
• [Presentation] Perron root の摂動とトロピカル固有値問題 (2016)
  • Author(s): 岩尾慎介
  • Organizer: RIMS 研究集会「可積 分系数理の現状と展望」
  • Place of Presentation: 京都大学
• [Presentation] K-theoretic analog of the Peterson isomorphism and the relativistic Toda equa- tion (2016)
  • Author(s): Shinsuke Iwao
  • Organizer: Infinite Analysis 16 summer school
  • Place of Presentation: 名古屋大学
• [Presentation] Totally non-negativity, singular curves and Kostant-Toda hierarchy (2015)
  • Author(s): Shinsuke Iwao
  • Organizer: Workshop "Topics on tropical geometry, integrable systems and positivity"
  • Place of Presentation: 青山学院大学
  • Year and Date: 2015-12-22
• [Presentation] Kostant-Toda 階層, Totally nonnegative matrix と特異曲線 (2015)
  • Author(s): 岩尾慎介、西山享、小川竜
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] Tropical singular curves and Classical integrable systems (2015)
  • Author(s): Shinsuke Iwao
  • Organizer: Students' Workshop on Tropical and Non-Archimedean Geometry
  • Place of Presentation: ドイツ・レーゲンスブルグ大学
  • Year and Date: 2015-08-05
  • Int'l Joint Research
• [Presentation] Totally positive matrix と特異曲線上の因子 (2014)
  • Author(s): 岩尾 慎介
  • Organizer: 九州大学応用力学研究所 共同利用研究集会 「非線形波動研究の現状―課題と展望を探る―」
  • Place of Presentation: 九州大学応用力学研究所（筑紫キャンパス、福岡）
  • Year and Date: 2014-10-30 – 2014-11-01