A.N.R. project Iso-Galois

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:
1. From basic facts to linear isomonodromy, Charlotte Hardouin and Viktoria Heu
2. On local classification of q-difference equations: towards discrete isomonodromy, Jacques Sauloy
3. Non-linear isomonodromy, Guy Casale, Yohann Genzmer and Loïc Teyssier.
Below, we give a short description of each lecture.

All courses will be given at the Institute of Mathematics of Toulouse in the University Paul Sabatier (France).

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.univ-toulouse.fr
charlotte.hardouin@[REMOVE]math.univ-toulouse.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 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.
• Maruyama’s semi-continuity.
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.
[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 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).

# 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.
[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