A survey of Isomonodromic Deformations
A MasterClass in Toulouse, June 29th to July 8th 2015
PDF version
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. LinsNeto, 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 masterclass, 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 nonlinear setting (isoholonomic deformations) in the framework of the Galois pseudogroup of a foliation and the Mattei unfoldings theory. We are going to study such isoGaloisian deformations, with a special focus on the first nontrivial 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 qdifference equations: towards discrete isomonodromy, Jacques Sauloy

Nonlinear isomonodromy, Guy Casale, Yohann Genzmer and Loïc Teyssier.
Below, we give a short description of each lecture.
Local organizers: Yohann GENZMER and Charlotte HARDOUIN
National organizers: Guy CASALE, Viktoria HEU, Loïc TEYSSIER
This event benefits from the support of the
CIMI.
Attendance and registration fees: 10€
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:
yohann.genzmer@[REMOVE]math.univtoulouse.fr
charlotte.hardouin@[REMOVE]math.univtoulouse.fr
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 masterclass almost selfcontained. We will introduce some classical results related to these objects, such as

the RiemannHilbert correspondence between vector bundles with connection and monodromy representations.

the BirkhoffGrothedieck theorem of classification of vector bundles on Riemann sphere.

Maruyama’s semicontinuity.
In a second step, we will describe the integrability equations for Fuchs systems over the projective line, the socalled Schlesinger equations, and study their properties. In particular, we will focus on isomonodromic deformations of linear systems such as
(dY)/(dz) = ⎛⎝(A_{1})/(z) + (A_{2})/(z − t) + (A_{3})/(z − 1)⎞⎠Y
and their link with the Painlevé VI equation. An overview of recent properties of the latter equation will be given.
[1] Claude Sabbah: Isomonodromic deformations and Frobenius manifolds, Springer, 2007
[2] 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
[3] Viktoria Heu: Stability of rank 2 vector bundles along isomonodromic deformations, Mathematische Annalen June 2009, Volume 344, Issue 2, pp 463490
2 On local classification of qdifference equations: towards discrete isomonodromy.
This course is given by Jacques Sauloy (Institut de Mathématiques de Toulouse).
$q$Lax Pairs. In their 1995 paper “A $q$Analog of the Sixth Painlevé Equation”, Jimbo and Sakai showed how to obtain a Lax Pair form for $q{P}_{VI}$. Let $q\in {C}^{*},\leftq\right<1$. They consider a family of fuchsian systems $Y\left(qx\right)={A}_{t}\left(x\right)Y\left(x\right)$, where ${A}_{t}\left(x\right):={A}_{0}\left(t\right)+{A}_{1}\left(t\right)x+{A}_{2}\left(t\right){x}^{2}$ 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 $t\leftarrow qt$, i.e. it is “$q$constant”. ${P}_{t}\left(x\right)={P}_{qt}\left(x\right)$. Jimbo and Sakai show that their $q$isomonodromy condition ${P}_{t}={P}_{qt}$ for the family of systems $Y\left(qx\right)={A}_{t}\left(x\right)Y\left(x\right)$ is equivalent to the existence of a “Lax Pair”, i.e. another family ${B}_{t}\left(x\right)$ such that, with obvious notations $A\left(x,qt\right)B\left(x,t\right)=B\left(qx,t\right)A\left(x,t\right)$. Then they derive $q{P}_{VI}$ 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 $q$Painlevé equations”, and many more authors, extended this to families of irregular systems. Murata's list is the basis for our case studies.
3 Nonlinear 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 nonlinear 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 nonlinear differential equations [1,2,3], and isoGaloisian deformations.

Monodromy of MatteiMarin for local singularities of differential equations in the complex plane [4,5].

Schlesinger equations for deformations of general foliations.
[1] G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, Ann. Inst. Fourier (Grenoble), vol. 56, #3: pp. 735–779, 2006
[2] 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
[3] B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, vol. 23, #2: pp. 219–226, 2002
[4] 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
[5] 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