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Geometry of tropical variety and homogeneous spaces

Research Project

Project/Area Number 25610008
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionAoyama Gakuin University

Principal Investigator

Nishiyama Kyo  青山学院大学, 理工学部, 教授 (70183085)

Co-Investigator(Kenkyū-buntansha) Iwao Shinsuke  青山学院大学, 理工学部, 助教 (70634989)
Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords戸田格子 / Lax行列 / totally positive matrix / ヤコビ多様体 / 超離散化 / トロピカル幾何 / 量子コホモロジー / シューベルト多項式 / トロピカル曲線 / 離散可積分系 / グラスマン多様体 / totally positivity / 特異曲線 / 逆散乱法 / トロピカル幾何学 / 超離散ソリトン方程式 / 非負行列 / プリュッカー座標 / グレブナー扇 / 行程計画問題 / トーリック幾何学
Outline of Final Research Achievements

We studies the methods of (ultra-)discretization of Toda lattice through the Lax formalism. There are so many researches in this field, but our emphasis is on the totally nonnegative matrices and algebro-geometric tools, such as singular (nodal) rational curves and Jacobian varieties. Thus our main result is proving the Lax matrices of Toda lattice which is totally nonnegative give the totally positive part of the Jacobian variety, which is summarized into the paper: Geometric interpretation of the totally nonnegative part of the finite Toda lattice via a singular rational curve. The paper is now being prepared.
We also studied on the quantum cohomology on the flag varieties and Peterson isomorphism. In this direction, we obtained various explicit formulas of Schubert polynomials and/or Grothendieck polynomials, especially their determinantal expressions. However, the results are somewhat rambling and it seems it will take some more time to organize them in a nice way.

Report

(4 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (9 results)

All 2015 2013 Other

All Presentation (4 results) (of which Int'l Joint Research: 2 results,  Invited: 1 results) Book (2 results) Remarks (2 results) Funded Workshop (1 results)

  • [Presentation] Totally non-negativity, singular curves and Kostant-Toda hierarchy2015

    • Author(s)
      Shinsuke Iwao
    • Organizer
      Workshop "Topics on tropical geometry, integrable systems and positivity"
    • Place of Presentation
      青山学院大学理工学部
    • Year and Date
      2015-12-22
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [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
    • Related Report
      2015 Annual Research Report
  • [Presentation] Tropical singular curves and Classical integrable systems2015

    • Author(s)
      Shinsuke Iwao
    • Organizer
      Students' Workshop on Tropical and Non-Archimedean Geometry Universität Regensburg 2015
    • Place of Presentation
      レーゲンスブルグ大学 (ドイツ)
    • Year and Date
      2015-08-05
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Tropical geometry and discrete integrable systems2013

    • Author(s)
      Shinsuke Iwao
    • Organizer
      JSPS-DST Asian Academic Seminar 2013, Discrete Mathematics & its Applications
    • Place of Presentation
      Graduate School of Mathematical Sciences, the University of Tokyo
    • Related Report
      2013 Research-status Report
    • Invited
  • [Book] 代数群と軌道2015

    • Author(s)
      太田琢也・西山享
    • Total Pages
      448
    • Publisher
      数学書房
    • Related Report
      2015 Annual Research Report
  • [Book] 射影幾何学の考え方2013

    • Author(s)
      西山享
    • Total Pages
      225
    • Publisher
      共立出版
    • Related Report
      2013 Research-status Report
  • [Remarks] Workshop: Topics on tropical geometry...

    • URL

      https://sites.google.com/site/tropintpos/home

    • Related Report
      2015 Annual Research Report
  • [Remarks] トロピカル・セミナー

    • URL

      http://www.gem.aoyama.ac.jp/~kyo/tropical_seminar.html

    • Related Report
      2015 Annual Research Report
  • [Funded Workshop] Topics on tropical geometry, integrable systems and positivity2015

    • Place of Presentation
      青山学院大学理工学部
    • Year and Date
      2015-12-22
    • Related Report
      2015 Annual Research Report

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Published: 2014-07-25   Modified: 2019-07-29  

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