Internal symmetry of soliton equations and its applications
Project/Area Number 
13640168

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kyoto University 
Principal Investigator 
SHIOTA Takehiro Graduate School of Science, Associate professor, 大学院・理学研究科, 助教授 (20243008)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)

Keywords  CalogeroMoser system / KP hierarchy / Krichever theory / Matrix Integrals / soliton方程式 / random permutation / 行列積分 / vicious random walk 
Research Abstract 
An algebrogeometric interpretation and construction (a la Krichever theory) of a general form of string equation, and some "genus zero" property it possesses in the situation which is closer to the bispectral problem, are shown. CalogeroMoser type KP solutions, related to bispectral problem and Krichever theory, is studied. Results include a proof, which does not assume bispectrality etc., of the fact that the spectral curve of any rank one ordinary differential operator with rational coefficients is unicursal. Various topics related to matrix integrals and combinatorics has been studied jointly with Adler and van Moerbeke. A formula counting the number of some vicious walks is given. BKP solutions of hypergeometric type is observed to be related to the sum of products of Schur Qfunctions over all the strict partitions λ = (λ_1 > λ_2 > ・・・) with λ_1 【less than or equal】 h, studied by Tracy and Widom in the context of combinatorics problems (result of Sasha Orlov) . Elementary approach to the Schottky problem  Based on the Krichever theory and yet without using complex analysis on principally polarized abelian varieties, we worked out details of characterization of Jacobian varieties in terms of finitely many differential equations in the KP hierarchy.

Report
(3 results)
Research Products
(3 results)