An application of global analysis of differential equations to representation theory

• Project/Area Number: 26800072
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Josai University
• Principal Investigator: Hiroe Kazuki 城西大学, 理学部, 助教 (50648300)
• Research Collaborator: OSHIMA Toshio 城西大学, 理学部数学科, 教授 (50011721) ; YAMAKAWA Daisuke 東京理科大学, 理学部第一部数学科, 講師 (20595847)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 不確定特異点 / ワイル群 / ミドル・コンボリューション / 箙の表現論 / Stokes現象 / 結び目 / スペクトル曲線 / ストークス現象 / 平面代数曲線の特異点 / 絡み目 / モノドロミー保存変形 / 微分方程式の不確定特異点 / 箙多様体 / Kac-Moody Lie環 / 代数曲線の特異点

Outline of Final Research Achievements

I gave an isomorphism between a moduli space of algebraic differential equations on the Riemann sphere and　quiver variety as symplectic manifolds when differential equations have at most one unramified　irregular singular point. When the differential equations have arbitrary number of unramified irregular singular　points, although it was known that the moduli space can not be isomorphic to any quiver varieties, I constructed　an open embedding into a quiver variety which is not surjective in general and clarified a　relation between middle convolution of differential equations and Wely group of the quiver, also determined a　necessary and sufficient conditon to the nonemptiness of the regular part of the moduli space.

Research Products

• [Journal Article] Linear differential equations on the Riemann sphere and representations of quivers (2017)
  • Author(s): Kazuki Hiroe
  • Journal Title: Duke Mathematical Journal Volume: 166 Issue: 5 Pages: 855-935
  • DOI: 10.1215/00127094-3769640
  • NAID: 130008004973
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Moduli spaces of meromorphic connections and quiver varieties (2014)
  • Author(s): Kazuki Hiroe, Daisuke Yamakawa
  • Journal Title: Advances in Mathematics Volume: 266 Pages: 120-151
  • DOI: 10.1016/j.aim.2014.05.024
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] On additive Deligne-Simpson problem (2017)
  • Author(s): Kazuki Hiroe
  • Organizer: Modern interactions between algebra, geometry and physics
  • Place of Presentation: Tohoku forum for creativity
  • Year and Date: 2017-04-21
  • Int'l Joint Research / Invited
• [Presentation] 線型常微分方程式のアクセサリーパラメーターを巡って (2017)
  • Author(s): 廣惠一希
  • Organizer: 日本数学会年会函数方程式分科会特別講演
  • Place of Presentation: 首都大学東京
  • Year and Date: 2017-03-24
  • Int'l Joint Research / Invited
• [Presentation] Invariants of differential equations and algebraic curves (2017)
  • Author(s): Kazuki Hiroe
  • Organizer: Branched covering, degeneration and related topics 2017
  • Place of Presentation: 東北学院大学
  • Year and Date: 2017-03-11
  • Int'l Joint Research / Invited
• [Presentation] On additive Deligne-Simpson problem (2017)
  • Author(s): Kazuki Hiroe
  • Organizer: Representation theory of quivers and finite dimensional algebras
  • Place of Presentation: MathematischesForschungstitut Oberwolfach
  • Year and Date: 2017-02-22
  • Int'l Joint Research
• [Presentation] On index of rigidity (2017)
  • Author(s): Kazuki Hiroe
  • Organizer: Hitchin Systems in Mathematics and Physics
  • Place of Presentation: Perimeter Institute
  • Year and Date: 2017-02-15
  • Int'l Joint Research / Invited
• [Presentation] 不確定スペクトル曲線の特異性について (2016)
  • Author(s): 廣惠一希
  • Organizer: 微分トポロジーセミナー
  • Place of Presentation: 京都大学
  • Year and Date: 2016-06-14
  • Int'l Joint Research / Invited
• [Presentation] On additive Deligne-Simpson problem (2016)
  • Author(s): Kazuki Hiroe
  • Organizer: Modern Interactions between Algebra, Geometry and Physics
  • Place of Presentation: Tohoku Forum of Creativity
  • Year and Date: 2016-04-21
  • Int'l Joint Research / Invited
• [Presentation] Stokes現象と絡み目について (2016)
  • Author(s): 廣惠一希
  • Organizer: 日本数学会２０１６年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-18
• [Presentation] Stokes現象と絡み目について (2015)
  • Author(s): 廣惠一希
  • Organizer: 2015年度表現論シンポジウム
  • Place of Presentation: 伊豆長岡
  • Year and Date: 2015-11-18
• [Presentation] Ramified irregular singularities of meromorphic connections and plane curve singularities (2015)
  • Author(s): Kazuki Hiroe
  • Organizer: Analytic, Algebraic and Geometric Aspects of Differential Equations
  • Place of Presentation: The Mathematical Research and Conference Center in Bedlewo
  • Year and Date: 2015-09-17
  • Int'l Joint Research
• [Presentation] Applications of quiver varieties to moduli spaces of connections on P1 (2015)
  • Author(s): Kazuki Hiroe
  • Organizer: On Quivers: Computational Aspects and Geometric Applications
  • Place of Presentation: 神戸大学
  • Year and Date: 2015-07-31
  • Int'l Joint Research / Invited
• [Presentation] Local Fourier transform and blow up (2015)
  • Author(s): 廣惠一希
  • Organizer: Kobe Seminar on Integrable Systems
  • Place of Presentation: 神戸大学
  • Year and Date: 2015-02-28
  • Invited
• [Presentation] 微分方程式とルート系 (2015)
  • Author(s): 廣惠一希
  • Organizer: HMA セミナー・冬の研究会 2015
  • Place of Presentation: 広島大学
  • Year and Date: 2015-01-09
  • Invited
• [Presentation] 局所Fourier変換とブローアップについて (2014)
  • Author(s): 廣惠一希
  • Organizer: フックス型方程式の幾何 workshop
  • Place of Presentation: 広島大学
  • Year and Date: 2014-12-05 – 2014-12-06
  • Invited
• [Presentation] Local Fourier transform and blowing up (2014)
  • Author(s): 廣惠一希
  • Organizer: 表現論セミナー
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-11-21
  • Invited
• [Presentation] 局所Fourie変換とブローアップについて (2014)
  • Author(s): 廣惠一希
  • Organizer: 微分方程式の展望
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-10-19
  • Invited
• [Presentation] 局所Fourie変換とブローアップ (2014)
  • Author(s): 廣惠一希
  • Organizer: 日本数学会 2014年度秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25
• [Remarks]
  • URL: http://researchmap.jp/kazukiadvb/
• [Remarks] research map
  • URL: http://researchmap.jp/kazukiadvb/