Algebraic, geometric, and analytic studies on irregular singularity and their applicaitions

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K14222
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Chiba University (2019-2020); Josai University (2017-2018)
• Principal Investigator: Hiroe Kazuki 千葉大学, 大学院理学研究院, 准教授 (50648300)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 不確定特異点 / 有理型接続のモジュライ空間 / 箙多様体 / オイラー変換 / 特異点の合流理論

Outline of Final Research Achievements

I found a formula which connects Komatsu-Malgrange irregularity of differential equations and Milnor number of corresponding plane curve singularities. Moreover, I studied the local Laplace transform of differential equations and determined
isomorphism classes of knots arising from the analytic continuation of these Laplace kernels. By using this formula, I showed that
the Euler characteristics of spectral curves of differential equations coincide with Katz' indices of rigidity of corresponding differential equations.

Free Research Field

基礎解析学

Academic Significance and Societal Importance of the Research Achievements

不確定特異点の解析学は一般的に難しいが，この研究よって微分方程式に対応する代数曲線の不変量が微分方程式の不確定特異点の様子を非常によくとらえていることが明らかになった．これによって代数曲線における代数幾何学的な研究手法が微分方程式論へ応用できることが期待され，微分方程式論に新しい研究手法を与えることになる．