Next Seminar
27 May 2026 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Toshio Oshima (University of Tokyo)
A classification of linear differential equations on the Riemann Sphere
Abstract :
Fuchsian differential equations on the Riemann sphere are classified by their Riemann
schemes, or spectral types.
A correspondence between spectral types and the roots of a star-shaped Kac-Moody
root system, due to Crawley-Boevey, clarifies the orbit of a spectral type under
transformations generated by middle convolutions and additions.
We consider irreducible equations admitting unramified irregular singularities and
study transformations generated by M\"obius transformations, additions, Laplace
transformations, confluences, and unfoldings. We show that complete representatives of
equivalence classes defined by these transformations are given by Fuchsian equations
of $E_8$-fundamental spectral type. The representatives have only three singular
points, $0$, $1$, and $\infty$, and the degrees of the minimal polynomials of their
local monodromies at $0$ and $1$ are $2$ and $3$, respectively.
We also define $(D_4, E_6, E_7)$-fundamental spectral types and characterize them
in an equivalence class.
Painlevé Seminar Topics
Recently, we have encountered many interesting links between differential
equations of Painlevé types and other fields of mathematics such
as algebraic geometry, non-abelian Hodge theory and topological recursion
and so on.
In this web seminar, we will organize research talks on Painlevé
equations and related topics on web (zoom), basically about once every
two or three weeks.
From September 2022, Regular time for seminar talk will be on Wednesday from 17:30--18:30 + discussion time
in JST(Japan) or 10:30--11:30 in UTC+1. We are looking forward to your participation in
the seminar. In order to get zoom access, please register through the form.
List of Talks
You can find the titles and abstracts of all seminars. ⇒ List of Talks
Registration
Online. Please register at the form, we will send you about web seminar address.
Organizers
Arata Komyo (Hyogo), Frank Loray (Rennes 1),
Ryo Ohkawa (Osaka Metropolitan), Masa-Hiko Saito (Kobe Gakuin), Takafumi Matsumoto (RIMS, Kyoto)
Scientific Committee (other than the Organizing Committee)
Supported by
French ANR-16-CE40-0008 project "Foliage" (Frank Loray)
JSPS Grant-in-aid (A) 22H00094 (PI: Masa-Hiko Saito)
Osaka Central Advanced Mathematical Institute:
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.