Study of algebraic solutions of the differential equations determined by isomonodromic deformations

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14506
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: University of Hyogo (2023); Kobe University (2019-2022)
• Principal Investigator: Komyo Arata 兵庫県立大学, 理学研究科, 准教授 (90760976)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: モノドロミー保存変形 / 接続のモジュライ理論 / ベクトル束のモジュライ空間 / 見かけの特異点 / 不確定特異点

Outline of Final Research Achievements

Algebraic solutions of the Garnier systems have been studied. The Garnier systems are non-linear differential equations determined by the isomonodromic deformations of some linear ODEs. First, generalization of Girand's algebraic solution was studied. Next algebraic solutions of irregular Garnier systems, which is corresponding to the isomonodromic deformations of some linear ODEs with irregular singularities. By the classification theorem due to Diarra--Loray, there's a list of irregular Garnier systems which have algebraic solutions. In this list, there was an irregular Garnier system whose algebraic solution was not found. To give that algebraic solution, the theory of apparent singularities was studied. As the result, that algebraic solution was found.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

パンルヴェ方程式は19世紀最後の年に発見された非線型常微分方程式である. この方程式は数理物理への応用が見つかって以降, 様々な分野の多くの研究者によって研究されてきた. パンルヴェ方程式の特殊解を求めるという問題はパンルヴェ方程式の研究では基本的であり, これまでに多くの数学者・数理物理学者によって取り組まれた. 本研究ではパンルヴェ方程式の仲間であるガルニエ系についての特殊解について研究してきた. ガルニエ系はパンルヴェ方程式に比べまだわかっていないことが多く, 本研究はガルニエ系の特殊解の研究に新たな進展をもたらしたとともに, 得た特殊解を用いたガルニエ系の研究の展開が期待される.