Application of Abelian function theory to Integrable system

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05187
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Sasebo National College of Technology
• Principal Investigator: Matsutani Shigeki 佐世保工業高等専門学校, 一般科目, 教授 (30758090)
• Research Collaborator: Komeda Jiryo ; Previato Emma
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 代数曲線 / アーベル関数 / σ関数 / 可積分系 / 産業数理

Outline of Final Research Achievements

We have the following results: 1) The determination of the Riemann constant for every algebraic curve which is given by the Weierstrass normal form even with non-symmetric numerical semigroup as Weierstrass non-gap at its infinity, 2) Jacobi inversion formulae for a general Weierstrass normal form, 3) Clarification of the relation between covering of the curve and correspondence of the hyperelliptic quasi-periodic solution to periodic solution of the Toda lattice equation, 4) Formulation of the behavior of sigma function for a degeneration of trigonal cyclic curve, 5) Relation of the MKdV hierarchy of the isometric deformation of curve in a plane and the Faber polynomial, and 6) Applications of mathematics to other fields (Quantum walk and coloring, Graph zeta function and conductivity of carbon, path space of the Lie group and robotics, algebraic description of screw dislocation)

Free Research Field

可積分系

Academic Significance and Societal Importance of the Research Achievements

本研究は、数学以外の分野への応用を見据えて、アーベル関数を楕円関数と同レベルの具体性を持つように再構築し、その再構築の結果と可積分系を通して他分野へアーベル関数論を適用することを目指すものである。また、適用に向け、幅広い数学の異分野への応用も含め、数学が社会に役立つ事を示すことは重要である。上記成果は、このような方向性に沿った結果である。