Annales de la Faculté des Sciences de Toulouse

Arpan Dutta

Generating sequences and semigroups of valuations on 2-dimensional normal local rings

Ismael Bailleul

Remi Catellier

Non-explosion criteria for rough differential equations driven by unbounded vector fields

Russell Lyons

Yuval Peres

Xin Sun

Tianyi Zheng

Occupation measure of random walks and wired spanning forests in balls of Cayley graphs

Nicholas McCleerey

Jian Xiao

Polar transform and local positivity for curves

Michel Bonnefont

Djalil Chafaï

Ronan Herry

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

Indranil Biswas

Kingshook Biswas

A Torelli type theorem for exp-algebraic curves

Ken’ichi Ohshika

Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces

Guy Casale

Lucia Di Vizio

Jean-Pierre Ramis

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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
GUY CASALE, LUCIA DI VIZIO AND JEAN-PIERRE RAMIS
Volume à la mémoire de Hiroshi Umemura: “Équations de Painlevé et théories
de Galois différentielles”
Tome XXIX, no 5 (2020), p. i–v.
https://doi.org/10.5802/afst.1654
© Université Paul Sabatier, Toulouse, 2020.
L’accès aux articles de la revue « Annales de la faculté des sciences de Toulouse
Mathématiques » (http://afst.centre-mersenne.org/) implique l’accord avec les
conditions générales d’utilisation (http://afst.centre-mersenne.org/legal/). Les
articles sont publiés sous la license CC-BY 4.0.
Hiroshi Umemura, Kyoto 2014
Préface
This special volume of the “Annales de la Faculté des Sciences de
Toulouse” contains several contributions in memory of Hiroshi Umemura.
Umemura died on March 8th, 2019, at the age of 74, in Nagoya, where
he graduated and spent the greatest part of his life, as a professor at Nagoya
University. He started working in algebraic geometry. More precisely, he
studied the subgroups of Cremona groups, subject that made him acquainted
with Paul Painlevé. He rapidly became a pillar of the Japanese school on
Painlevé equations. In 1996, he published his first article on Galois theory
for non-linear differential equations, which has been a turning point for the
domain. His last article, which closes this volume, is in its direct continuity
and shows how one can weaken the definition of Galois group of a func-
tional equation and hence make quantum groups naturally appear in the
framework.
Hiroshi was friendly and very enthusiastic when sharing knowledge or
ideas (mathematical or not). For this reason, he was deeply appreciated by
all colleagues. The contributions to this volume confirm that his vision of
mathematics will be a source of inspiration and of open problems for next
generations of mathematicians.
The articles published below touch several aspects of Umemura long and
rich activity.
Souvenirs and overall presentation of Umemura work.The first
two texts present the academic career of Umemura, his visits to France and
his main mathematical achievements.
J.-P. Ramis has written an historical survey on the relations of Umemura
with the French mathematical community. It can also be read as an in-
troduction to the difficult work of Umemura on differential Galois theory,
starting from the seminal work of E. Vessiot and presenting open questions
and possible expected applications.
In the second article, K. Okamoto and Y. Ohyama give an account of
Umemura mathematical career from his first work on Fano 3-fold to his
definition of his Galois infinitesimal group Inf-gal(L/K) of a differential field
extension.
– i –
Noriko et Hiroshi Umemura avec Monique Ramis, Carcassonne 2015
Hiroshi Umemura, Université de Nagoya 2013
– ii –
Umemura polynomials.K. Okamoto and Y. Ohyama introduce the
subsequent article written by Umemura in 1996, for the proceedings of
the conference Theory of nonlinear special functions: the Painlevé transcen-
dents, organised at Montréal. This work was never published, differently
from the paper Special polynomials associated with the Painlevé equations II
by Noumi, Okada, Okamoto and Umemura, that is its natural prolonga-
tion. Hiroshi was interested in rational solutions of Painlevé equations. After
an introduction to Yablonskii–Vorobiev polynomials, he defined a family of
polynomials related to Painlevé equations and gave the first step toward a
conjectural combinatorial description of those polynomials, that nowadays
we call Umemura polynomials.
In the next paper, M. Noumi generalizes the constructions of Umemura
polynomials, extending both presentations given in Umemura’s article. The
first construction is given in term of Bäcklund transformations of the 6th
Painlevé equation, while the second one is completely combinatorial. Noumi
conjectures that the polynomials obtained actually coincide. In the case of
Umemura polynomials, this conjecture is a theorem by Taneda, which, fol-
lowing Umemura, confirm that the Painlevé transcendents are special func-
tions.
Discrete Painlevé equations.Y. Ohyama, J. Sauloy and J.-P. Ramis
explore some q-analogs of the notion of isomonodromic deformations. This
is a first step toward the discrete Riemann-Hilbert correspondence. On one
side of the Riemann–Hilbert correspondence, we have the spaces of initial
conditions, classified by Sakai. On the other side, we must have the q-version
of the character varieties. To defined it, the authors restrict their interest to
a family of linear difference systems rich enough to obtain the celebrated
Jimbo-Sakai Lax pair, that produces the q-PV I equation. Then, using a new
tool called Mano decomposition, they describe the algebraic structure of the
character variety.
The Hamiltonian presentation of Painlevé equations is one of their
essential properties. The associated Newton polygon gives the weights used
to prove the irreducibility of Painlevé transcendents. The article written
by T. Mase, A. Nakamura et H. Sakai gives a discrete version of this
Hamiltonian. In a case by case computation, they obtain Hamiltonians for
the discrete Painlevé equations. In contrast with the differential case, the
discrete Hamiltonians not only are not rational but they contain logarithmic
and dilogarithmic terms.
– iii –
Hiroshi Umemura avec Daniel Bertrand, 2004
Hiroshi Umemura avec Pierre Cartier, Université de Nagoya 2013
– iv –
“Unlikely intersections” in the functional setting.The reader
may compare the form given to PVI and the following Proposition 1 in D.
Bertrand’s article and the Schwarzian equation and the Proposition 4.10 in
D. Blàzquez Sanz, G. Casale, J. Freitag and J. Nagloo paper. They study
the differential equation satisfied by the uniformization of triangle groups.
They show the strong minimality of the set of solutions of this equation and
explore some consequences of this property. Strong minimality is a notion
from Model Theory generalising the celebrated (J) condition of Umemura.
From a Galoisian perspective, an equation with this property should have
a Galois group whose Lie algebra has no algebraic sub Lie-algebra. To give
a proof of the strong minimality of Schwarzian equations computing their
Umemura groups Inf-gal is a non-trivial open problem. . . The authors used
other technics from model theory.
The article written by D. Bertrand is a survey on Manin map and Manin
kernel. Different maps are introduced and, following Manin, Coleman, Chai
and André, D. Bertrand shows the equality of the different kernels. In a first
part, he recalls the presentation of PVI using Manin map and show how
properties of its kernel give informations on solutions of the sixth Painlevé
equation.
Quantum groups as Galois groups.The volume ends with Umemura’s
last article in collaboration with A. Masuoka et K. Saito. They develop a
new point of view on the construction of Picard–Vessiot extensions starting
from the Galois theory devised by Umemura in his seminal article Differ-
ential Galois theory of infinite dimension. The functional equations studied
are given by an action of a non-cocommutative bialgebra. The weakening
of the construction gives rise to quantum groups. The foundations of this
new theory are detailed on three examples and for equations with constant
coefficients.
Hiroshi Umemura’s interests for Painlevé equations and differential Galois
theory have brought him to France very often. Hence this volume is published
in France to honor his longstanding mathematical collaboration and the
profound friendship that he has left behind.
Guy Casale, Lucia Di Vizio, Jean-Pierre Ramis
– v –


Volume à la mémoire de Hiroshi Umemura: “Équations de Painlevé et théories de Galois différentielles”

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Generating sequences and semigroups of valuations on 2-dimensional normal local rings

Non-explosion criteria for rough differential equations driven by unbounded vector fields

Occupation measure of random walks and wired spanning forests in balls of Cayley graphs

Polar transform and local positivity for curves

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

A Torelli type theorem for exp-algebraic curves

Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces

Volume à la mémoire de Hiroshi Umemura: “Équations de Painlevé et théories de Galois différentielles”

Toward quantization of Galois theory

Extension operator for the MIT Bag Model

1,744

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