Centre International de Mathématiques Pures et Appliquées



Ministère Enseignement Supérieur et Recherche, France UNESCO
Universitè de Montpellier UNS
Norwegian Ministry of Education and Research

Accueil > Écoles de recherche > Ecoles de recherche 2015 > Liste chronologique des écoles de recherche 2015 > Transformation Groups and Dynamical Systems

Transformation Groups and Dynamical Systems

English Version

CIMPA-PERU research school

Pontificia Universidad Católica del Perú, Lima, May 25-June 5, 2015

Report by Nuria Corral Perez & Francisco Ugarte Guerra

Report by Mercedes Siles & Jorge Mozo

Many problems in mathematics concern with classification once a notion of equivalence is given in a class of mathematical objects. Classically, the theory of invariants proposes to study the orbits of a group action over a geometric object and the properties which remain stable under this action.
The aim of this CIMPA school is to give an introduction to different techniques in this context related with the study of holomorphic foliations and differential equations from a local and global viewpoint. More precisely, the courses will cover the following subjects :
differential Galois theory, algebraic properties of groups of local diffeomorphisms, automophism group of a foliation on a projective variety, algebraic and dynamical properties of birational maps of the complex proyective plane.

Administrative and scientific coordinators

- Nuria Corral Perez (Universidad de Cantabria, España) nuria.corral@unican.es
- Francisco Ugarte Guerra (Pontificia Universidad Católica del Perú) fugarte@pucp.edu.pe

Scientific Committee

  • Jorge Mozo Fernández jmozo@maf.uva.es (Universidad de Valladolid, Spain)
  • Francisco Ugarte Guerra fugarte@pucp.edu.pe (Pontificia Universidad Católica del Perú)
  • Lorena López lorena@mat.ufmg.br (Universidade de Minas Gerais, Brazil)

Local Organizing Committee

  • Christiam Figueroa cfiguer@pucp.pe (Pontificia Universidad Católica del Perú)
  • Jaime Cuadros Valle jcuadros@pucp.edu.pe (Pontificia Universidad Católica del Perú)
  • Liliana Puchuri Medin lpuchuri@pucp.pe (Universidad Nacional de Ingeniería - Pontificia Universidad Católica del Perú)

Scientific Program

Course 1
Lecturer : Guy Casale (Univesité de Rennes I, France, guy.casale@univ-rennes1.fr )
Title : Pseudo-groupes algébriques de transformations et théories de Galois
Abstract : Ce cours est une introduction au pseudo-groupe de Galois d’une application f de M dans M, rationnelle dominante sur une variété algébrique.
Le plan du cours est le suivant (il pourra être raccourci, le point 3 peut disparaître, suivant les connaissances mathématiques des participants)

  1. Définitions
    • Pseudogroupes
    • Espaces de jets
    • Pseudo-groupes algébriques et pseudogroupe de Galois
    • premier exemples, pesudogroupes préservant une structure géométrique
    • Théorème de Chevalley pour les pseudogroupes algébriques
  2. Le cas ou M est de dimension 1 Nous donnerons la liste des applications f ayant un petit pseudogroupe de Galois.
  3. Quelques remarques lorsque M est de dimension deux
  4. Les équations aux différences. Les équations aux différences sont des exemples d’rationnelle dominante. Nous montrerons que les équations résolubles par quadratures discretes ont un pseudogroupe de Galois résoluble.

Course 2
Lecturer : José Manuel Aroca (Universidad de Valladolid, Spain, aroca@agt.uva.es )
Title : Grupoids, stacks, etc.
Abstract : As a complement of the Guy Casale course, I will present, in an elementary way, the Grothendieck viewpoint of the Galois theory.

Course 3
Lecturer : Percy Fernández (Pontificia Universidad Católica del Perú, pefernan@pucp.edu.pe )
Title : Transformation Groups of Holomorphic Foliations
Abstract : We study the automorphism group of a foliation on a projective. Along the course we study the structure of arbitrary codimension foliations on projective varieties invariant by an infinite linear algebraic group. We also study holomorphic foliation of the projective plane with rich, but finite, automorphism group does not have invariant algebraic curves.

Course 4
Lecturers : Lorena López (Universidade de Minas Gerais, Brazil, lorena@mat.ufmg.br ) and Javier Ribón (Universidade Federal Fluminense, Brazil, javier@mat.uff.br )
Title : Algebraic properties of groups of complex analytic local diffeomorphisms.
Abstract : A solvable group of complex analytic local diffeomorphisms in dimension one is metabelian. It is interesting to ask whether or not the derived length of a solvable group of local diffeomorphisms is bounded by a constant that depends only on the dimension. The answer is positive and it is possible to provide an assymptotically optimal upper bound. We can give the best possible upper bound for dimensions 2 and 3. The idea of the proof is replacing the initial solvable group with its algebraic closure and then studying well chosen actions of this group in finite dimensional vector spaces.

Course 5
Lecturer : Julie Déserti (Université Paris VII, France, deserti@math.jussieu.fr )
Title : On the birational maps of the complex projective plane.
Abstract : I will introduce the basic tools which are useful to study such maps by presenting some known properties of the Cremona group. The mentioned properties are algebraic properties : generators, relations, finite subgroups, subgroups of finite type, automorphisms of the Cremona group, Tits alternative... but also dynamical properties : classification of birational maps, centralizers of Cremona maps, dynamics of an Heisenberg subgroup and of course construction of automorphisms with positive entropy.

Most of the talks will be given in Spanish. Guy Casale and Julie Déserti will lecture in French or English, according to the students preference.

Deadline for registration

February 8, 2015

Application procedure only for applicants not from Peru.

Applicants from Peru must contact local organizer : Francisco Ugarte Guerra (Pontificia Universidad Católica del Perú) fugarte@pucp.edu.pe

Voir en ligne : Local web site