Development of Representation Theory and Special Functions

• Project/Area Number: 15H03613
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: Ochiai Hiroyuki 九州大学, マス・フォア・インダストリ研究所, 教授 (90214163)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥14,430,000 (Direct Cost: ¥11,100,000、Indirect Cost: ¥3,330,000); Fiscal Year 2019: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2018: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2017: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2016: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2015: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
• Keywords: 表現論 / 特殊関数 / 超幾何関数 / 微分方程式 / 軌道分解 / リー群 / 分数階微分 / リー環 / optimal system / ドレスト光子 / 対称性 / 対称空間 / 特殊関数論 / 代数学 / 解析学 / 幾何学 / 統計学 / 特殊値 / 制限 / Witten ゼータ / 微分作用素 / 代数解析学 / 等質空間

Outline of Final Research Achievements

We focus the bridge between Representation theory treating symmetry and Theory of special functions. We describe the symmetry of fractional differential equations in terms of Lie algebra. This symmetry enables us to give an explicit expression of solutions of such equations.
We find the new holonomic system with three variables, and the restriction to a singular line of a solution turns out to be an integral representation of a solution of Dotsenko-Fateev equation arising in conformal field theory.

Academic Significance and Societal Importance of the Research Achievements

新しい関数を使うことで、今まで定性的にしか捉えられなかった現象(例えば天気が晴れ)を、定量的に記述すること(気圧が1018ヘクトパスカル)ができるようになる。我々の研究で取り扱った、３変数関数を１変数に制限したものや、分数階の微分方程式の対称性から生じた Mittag-Leffler関数などを使うことで、今までよりも詳細な記述が可能になった。例えば、関数が正負の符号を変える零点の位置は大切な情報だが、具体的な記述によって、計算機も援用することができ、それが可能になった。

Research Products

• [Int'l Joint Research] Anges 大学(フランス)
• [Int'l Joint Research] 国立モンゴル大学(モンゴル)
• [Int'l Joint Research] Angers(France)
• [Int'l Joint Research] National University of Mongolia(Mongolia)
• [Int'l Joint Research] アンジェ(フランス)
• [Journal Article] Zeta functions of periodic cubical lattices and cyclotomic-like polynomials (2020)
  • Author(s): Y. Hiraoka, H. Ochiai, T. Shirai
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 84 Pages: 93-121
  • DOI: 10.2969/aspm/08410093
  • Peer Reviewed / Open Access
• [Journal Article] On solutions of linear fractional differential equations and systems thereof (2019)
  • Author(s): Dorjgotov Khongorzul、Ochiai Hiroyuki、Zunderiya Uuganbayar
  • Journal Title: Fractional Calculus and Applied Analysis Volume: 22 Issue: 2 Pages: 479-494
  • DOI: 10.1515/fca-2019-0028
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Exact solutions to a class of time fractional evolution systems with variable coefficients (2018)
  • Author(s): Dorjgotov, Khongorzul; Ochiai, Hiroyuki; Zunderiya, Uuganbayar
  • Journal Title: J. Math. Phys. Volume: 59 Pages: 81504-81504
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Lie symmetry analysis of a class of time fractional nonlinear evolution systems (2018)
  • Author(s): Dorjgotov, Khongorzul; Ochiai, Hiroyuki; Zunderiya, Uuganbayar
  • Journal Title: Appl. Math. Comput. Volume: 329 Pages: 105-117
• [Journal Article] Lie symmetry analysis of a class of time fractional nonlinear evolution systems (2018)
  • Author(s): Khongorzul Dorjgotov, Hiroyuki Ochiai and Uuganbayar Zunderiya
  • Journal Title: Appl. Math. Comput. Volume: 329
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the Letac-Massam Conjecture on cones Q_{A_n} (2017)
  • Author(s): Piotr Graczyk, Hideyuki Ishi, Salha Mamane, and Hiroyuki Ochiai
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 93
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the Letac-Massam Conjecture on cones $Q_{A_n}$ (2017)
  • Author(s): Piotr Graczyk, Hideyuki Ishi, Salha Mamane, and Hiroyuki Ochiai
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 93(3) Pages: 16-21
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A concise parametrisation of affine transformation (2016)
  • Author(s): Shizuo Kaji and Hiroyuki Ochiai
  • Journal Title: SIAM J. Imaging Sci. Volume: 9(3) Issue: 3 Pages: 1355-1373
  • DOI: 10.1137/16m1056936
  • Peer Reviewed
• [Journal Article] リーマンの球面 (2016)
  • Author(s): 落合啓之
  • Journal Title: 現代思想 2016年3月臨時増刊号 総特集=リーマン リーマン予想のすべて Volume: 2016-3 Pages: 176-185
• [Presentation] 安楽椅子探偵による算術幾何平均 (2021)
  • Author(s): 落合啓之
  • Organizer: 九州代数的整数論2021
  • Invited
• [Presentation] Nilpotent orbits in complex simple Lie algebras and prehomogeneous vector spaces (2019)
  • Author(s): 落合啓之
  • Organizer: 研究集会 “Representation theory of reductive Lie groups and algebras”
  • Int'l Joint Research / Invited
• [Presentation] 超幾何関数のリー環対称性について (2018)
  • Author(s): 落合啓之
  • Organizer: 研究集会「微分方程式と表現論」
  • Invited
• [Presentation] Gamma factors of Selberg's zeta functions and absolute zeta functions (2018)
  • Author(s): 落合啓之
  • Organizer: 研究集会「多重三角関数とその一般化」神戸大学
  • Invited
• [Presentation] Zeros of Eulerian polynomials (2017)
  • Author(s): H. Ochiai
  • Organizer: 研究集会「多重ゼータ関数の諸相」名古屋大学
  • Int'l Joint Research / Invited
• [Presentation] Gamma factors of Selberg’s zeta functions and absolute zeta functions (2016)
  • Author(s): Hiroyuki Ochiai
  • Organizer: Zeta Functions in Okinawa 2016
  • Place of Presentation: Okinawa Convention Center
  • Year and Date: 2016-10-29
  • Invited
• [Presentation] An Introduction to Matrix Exponential for CG (2016)
  • Author(s): Ken Anjyo, Hiroyuki Ochiai
  • Organizer: Vic talk, Victoria University of Wellington
  • Place of Presentation: Victoria University of Wellington, NZ
  • Year and Date: 2016-02-16
  • Int'l Joint Research / Invited
• [Presentation] Covariant differential operators and Heckman-Opdam hypergeometric systems (2016)
  • Author(s): Hiroyuki Ochiai
  • Organizer: International Conference for Korean Mathematical Society 70th Anniversary
  • Place of Presentation: Seoul National University
  • Int'l Joint Research / Invited
• [Presentation] CG映像制作の数理 (2015)
  • Author(s): 落合啓之
  • Organizer: 日本数学会秋季総合分科会・応用数学分科会特別講演
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-14
  • Invited
• [Presentation] Covariant differential operators and Heckman-Opdam hypergeometric systems (2015)
  • Author(s): Hiroyuki Ochiai
  • Organizer: Workshop, Analytic Representation Theory of Lie Groups
  • Place of Presentation: 東京大学柏キャンパス IPMU
  • Year and Date: 2015-07-01
  • Int'l Joint Research / Invited
• [Book] Mathematical Basics of Motion and Deformation in Computer Graphics, Second Edition (2017)
  • Author(s): Ken Anjyo and Hiroyuki Ochiai
  • Total Pages: 95
  • Publisher: Morgan and Claypool
• [Remarks] 落合啓之のページ
  • URL: https://user.math.kyushu-u.ac.jp/index.php?ochiai
• [Remarks] 研究者情報
  • URL: https://hyoka.ofc.kyushu-u.ac.jp/search/details/K003560/index.html
• [Remarks]
  • URL: http://user.math.kyushu-u.ac.jp/index.php?ochiai
• [Remarks]
  • URL: http://user.math.kyushu-u.ac.jp/index.php?ochiai
• [Remarks] 落合啓之のページ
  • URL: http://user.math.kyushu-u.ac.jp/index.php?ochiai
• [Remarks] 研究者情報（落合啓之）
  • URL: http://hyoka.ofc.kyushu-u.ac.jp/search/details/K003560/index.html
• [Remarks] 落合啓之のページ
• [Funded Workshop] Symposium MEIS 2016: Mathematical Progress in Expressive Image Synthesis (2016)
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2016-11-11
• [Funded Workshop] International Conference Geometry, Representation Theory, and Differential Equations (2016)
  • Place of Presentation: 九大IMIオーディトリアム
  • Year and Date: 2016-02-16