Geometric structures and foliations
Structures géométriques et feuilletages

18 – 22 October 2021

Guy Casale (Université de Rennes 1)
Frank Loray (Université de Rennes 1)
Jorge Vitorio Pereira (IMPA, Rio de Janeiro)
Erwan Rousseau (Université de Brest)

The notion of transversely projective foliations arose in the late ’70s in the real setting, and later in the ’90s for the singular holomorphic setting. One of the main interests is the link with integrability of the associated differential system. A particular case of this phenomenon appears in the work of Michael Singer relating Liouvillian integrability of codimension one foliations to the existence of an affine structure on the leaf space of the foliation. Several authors investigated other particular instances of this phenomenon. By the turn of the century, a conceptual breakthrough emerged from the work of Bernard Malgrange on non-linear differential Galois Theory. Guy Casale specialized Malgrange’s theory to codimension one foliations, shedding light on all previous investigations of the subject. More recently, Frank Loray, Jorge Pereira, and Frédéric Touzet established a structure theorem for codimension one transversely projective foliations, giving a rather concrete description of such objects, which proved to be a useful technical tool as exemplifies a recent preprint by Federico Lo Bianco, Jorge Pereira, Erwan Rousseau, and Frédéric Touzet on the structure of codimension one foliations invariant by rational endomorphisms. To summarize, the study of transversely homogeneous foliations of codimension one is mature. At the same time, the study of the transverse homogeneous structures for higher codimension foliations is still rather incipient. We believe that the community should strive to remedy this situation. The goal of the workshop is to revisit the definition of geometric structure in the sense of Élie Cartan and discuss its recent developments aiming at analogs for higher codimension of the results for codimension one foliations.

Guy Casale (Université de Rennes 1)    Malgrange pseudo-group of a foliation
Bertrand Deroin (Université de Paris)    Foliated projective structures
Sorin Dumitrescu (Université Côte d’Azur)    Holomorphic Foliations with Transverse Cartan geometries
Subhojoy Gupta (Indian Institute of Sciences) & Mahan Mj (Tata Institut, Mumbai)      Meromorphic projective structures and their monodromy
Jun-Muk Hwang (Institute for Basic Science, Daejeon)   Grauert’s formal principle and Cartan geometry