Speaker: Gilbert Hector (Institut C. Jordan, Université Lyon 1)

Abstract: I will give a concrete and intuitive introduction to the theory of foliations, illustrating every concept and result with adapted examples. The course will consist of the following parts (time permitting):

(A) Fundamentals:

- Definitions, examples and constructions.

- Description of the transverse structure. The holonomy pseudogroup and its associated Schreier continuum.

- Holonomy groups of individual leaves and transversal holonomy. Genericity of leaves with trivial holonomy (Epstein et al., Hector).

- Transverse invariant measures, construction by Folner sequences. The theorem of Plante.

(B) Codimension one foliations of simply connected manifolds:

- Null homotopic closed transversals versus non analytic holonomy. Haefliger’s non existence theorem of analytic foliations on spheres.

- Vanishing cycles and Novikov’s theorem for the existence of Reeb components on 3-manifolds with finite fundamental group.

- Foliations of the Euclidean spaces (Hector) and the spheres (Lawson).

(C) Miscellaneous:

- Generic properties of leaves (following Ghys, Cantwell-Conlon and Blanc).

- Structure of Riemannian foliations according to Molino.

- Aspects of the theoy of matchbox manifolds.