A survey of Isomonodromic Deformations
A Master-Class in Toulouse, June 29th to July 8th 2015
Already in the nineteenth century, P. Painlevé, studying the properties of solutions of holomorphic differential equations in the complex field, saw the necessity of creating a geometric theory of foliations. The development of this theory has been started by C. Ehresmann and G. Reeb in the 1940’s, motivated by a conjecture of H. Hopf. The development of the theory of holomorphic foliations accelerated in the 1970’s and took place mainly in France (D. Cerveau, J.-F. Mattei, R. Moussu et al
) and Brazil (C. Camacho, A. Lins-Neto, P. Sad et al
). It is still very active nowadays (with centers in France, Brazil and Spain) and has applications among others in the study of differential equations, billiards, classifications of manifolds, the theory of certain moduli spaces and in mathematical physics. With this master-class, we plan to introduce Phd students and young researcher to the theory of isomonodromic deformations via the recently introduced concept of parameterized differential Galois group, and its generalization to the non-linear setting (isoholonomic deformations) in the framework of the Galois pseudogroup of a foliation and the Mattei unfoldings theory. We are going to study such iso-Galoisian deformations, with a special focus on the first non-trivial examples of such deformations, i.e.
the Painlevé differential and difference equations.
There will be three courses:
From basic facts to linear isomonodromy, Charlotte Hardouin and Viktoria Heu
On local classification of q-difference equations: towards discrete isomonodromy, Jacques Sauloy
Non-linear isomonodromy, Guy Casale, Yohann Genzmer and Loïc Teyssier.
Below, we give a short description of each lecture.
Yohann GENZMER and Charlotte HARDOUIN
Guy CASALE, Viktoria HEU, Loïc TEYSSIER
This event benefits from the support of the CIMI
Attendance and registration fees:
Students can apply for financial support by sending a CV to the organizers (address below). Deadline for candidature is March 1st 2015
Registration and further questions:
List of participants:
- Nakamura Akane
- Amaury Bittman
- Guy Casale
- Juan Sebastian Diaz
- Thomas Dreyfus
- Anton Eloy
- Jesus David Escobar
- Lourenco Fernando
- Yohann Genzmer
- Arnaud Giran
- Charlotte Hardouin
- Viktoria Heu
- Martin Klimes
- Oleg Lisovyy
- Jinan Loubani
- Matsubura Saeie
- Jacques Sauloy
- Loïc Teyssier
- Jesús de la Vega
1 From basic facts to linear isomonodromy.
This course is given by Charlotte Hardouin (Institut de Mathématiques de Toulouse) and Viktoria Heu (Institut de Recherche en Mathématiques Avancées, Strasbourg).
The concepts of monodromy and isomonodromic deformations rely on some natural objects and definitions such as foliated manifold, vector bundles endowed with a connection and integrability. The first objective of this lecture is to revisit these notions in order to make the master-class almost self-contained. We will introduce some classical results related to these objects, such as
the Riemann-Hilbert correspondence between vector bundles with connection and monodromy representations.
the Birkhoff-Grothedieck theorem of classification of vector bundles on Riemann sphere.
In a second step, we will describe the integrability equations for Fuchs systems over the projective line, the so-called Schlesinger equations, and study their properties. In particular, we will focus on isomonodromic deformations of linear systems such as
(dY)/(dz) = ⎛⎝(A1)/(z) + (A2)/(z − t) + (A3)/(z − 1)⎞⎠Y
and their link with the Painlevé VI equation. An overview of recent properties of the latter equation will be given.
 Claude Sabbah: Isomonodromic deformations and Frobenius manifolds, Springer, 2007
 Frank Loray: Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. in Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies RIMS Kôkyûroku Bessatsu, B2 (2007) 129–136
 Viktoria Heu: Stability of rank 2 vector bundles along isomonodromic deformations, Mathematische Annalen June 2009, Volume 344, Issue 2, pp 463-490
2 On local classification of q-difference equations: towards discrete isomonodromy.
This course is given by Jacques Sauloy (Institut de Mathématiques de Toulouse).
One geometric model of the derivation of Painlevé equations from isomonodromy conditions is the following. Consider a space of monodromy data
, for instance the space
of representations of
up to conjugacy, with
. This can be considered as a fibre bundle
endowed with a flat connection which expresses the fact that “small movements of
do not move
”. Pulling back the fibre bundle and its connection through the Riemann-Hilbert map then yields (in adequate coordinates) the Painlevé equation
. Thus, viewing the Riemann-Hilbert correspondence as an analytic mapping between varieties is an essential ingredient.
The Riemann-Hilbert correspondence for fuchsian
-difference equations was introduced by Birkhoff in his celebrated 1913 paper “The generalized Riemann problem for linear differential equations and the allied problems for linear difference and
-difference equations”. It has been deeply studied in the last decades, then generalized to irregular equations by Ramis, Sauloy and Zhang in “Local Analytic Classification of
-Difference Equations” (2013). However, the geometry
of the right hand side of the
-R.-H. correspondence (
-Stokes data) is presently not well understood.
In their 1995 paper “A
-Analog of the Sixth Painlevé Equation”, Jimbo and Sakai showed how to obtain a Lax Pair form for
. They consider a family of fuchsian systems
is subjected to adequate conditions. In a bold step, they consider the family as “isomonodromic” if the “Birkhoff connection matrix” of the system does not change when
it is “
. Jimbo and Sakai show that their
for the family of systems
is equivalent to the existence of a “Lax Pair”, i.e.
such that, with obvious notations
. Then they derive
from this (in adequate coordinates). Subsequently, Sakai in his 2007 paper “Problem: Discrete Painlevé equations in their Lax forms”, Murata in his 2009 paper “Lax forms of the
-Painlevé equations”, and many more authors, extended this to families of irregular systems. Murata's list is the basis for our case studies.
I shall present what I think to be the necessary tools for the study of
-isoGalois deformations. This will be applied using work in progress with Yousuke Ohyama and Jean-Pierre Ramis (on “space of monodromy data”), as well as some work done with Galina Filipuk (on variation of the local Galois group). The examples will be taken from Jimbo-Sakai and Murata papers.
 Jimbo, M., Sakai, H.: A -analog of the sixth Painlevé equation. Lett. Math. Phys.38, 145¡V154 (1996)
 Murata, M.: Lax form of the -Painlevé equations, Journal of Physics A: Mathematical and Theoretical, 42, 11 (2009)
 Ramis, J.-P., Sauloy, J., Zhang C.: Local analytic classification of -difference equations, Astérisque 355 (2013)
 Ramis, J.-P., Sauloy: The -analogue of the wild fundamental group and the inverse problem of the Galois theory of -difference equations, to appear in Annales Scientifiques de l'École Normale Supérieure (2015)
 Sauloy, J.: Systèmes aux q -différences singuliers réguliers : classification, matrice de connexion et monodromie, Annales de l'institut Fourier, Volume: 50, Issue: 4, page 1021-1071 (2000)
3 Non-linear isomonodromy.
This course is given by Guy Casale (Institut de Recherche en Mathématiques de Rennes), Yohann Genzmer (Institut de Mathématiques de Toulouse) and Loïc Teyssier (Institut de Recherche en Mathématiques Avancées, Strasbourg).
In the context of linear differential equations, the concept of isomonodromy deformations appears to be now quite well defined. There are various definitions adapted to different context but they all shared the properties of being generically equivalent to a natural notion of integrability. Although, the integrability notion can be easily extended to deformations of non-linear differential equations, the isomonodromy counter part is more difficult to obtain and there is no definitive definition. The objective of the lecture is to give an idea of the état de l’art and to introduce some recent answers such as
Galois groupoid for non-linear differential equations [1,2,3], and iso-Galoisian deformations.
Monodromy of Mattei-Marin for local singularities of differential equations in the complex plane [4,5].
Schlesinger equations for deformations of general foliations.
 G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, Ann. Inst. Fourier (Grenoble), vol. 56, #3: pp. 735–779, 2006
 B. Malgrange, Le groupoïde de Galois d’un feuilletage, in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38 : pp. 465–501, 2001
 B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, vol. 23, #2: pp. 219–226, 2002
 D. Marín and J.-F. Mattei, Incompressibilité des feuilles de germes de feuilletages holomorphes singuliers, Ann. Sci. Éc. Norm. Supér. (4), vol. 41, #6 : pp. 855–903, 2008
 D. Marín and J.-F. Mattei, Monodromy and topological classification of germs of holomorphic foliations, Ann. Sci. Éc. Norm. Supér. (4), vol. 45, #3 : pp. 405–445, 2012