Any (at most countable) family of pairwise transverse (even singular, with saddle or prong singularities) foliations on the plane R² admits a compactification (as the disc D²) by a circle at infinity so that every ray in a leaf tends to a point on the circle, and this compactification is unique up to two natural requirements. Thus every leaf corresponds to a pair of points on the circle. With Th. Barthelmé and K. Mann, we consider the reverse problem and we give a complete answer to the two following questions:

Q1: (realization) Under which hypotheses two sets L⁺, L⁻ of pairs of points on the circle are precisely the pairs of endpoints of leaves of two transverse foliations (we give the answer for singular, and also for nonsingular foliations, and we prove that the foliations are uniquely determined).

More important is the second question:
Q2: (completion) Under which hypotheses two sets l⁺,l⁻ or pair of points on the circle correspond the pairs of endpoints of a dense subset of leaves of two transverse foliations (singular or nonsingular, and uniqueness). The uniqueness implies that any group action on the circle preserving l⁺,l⁻ extends in an action on the disc preserving the corresponding foliations. This allows us to prove that if an action G -> Homeo_+(S¹) of a group on the circle is induced by an Anosov-like action G -> Homeo_+(D²), then this action is unique and completely determined by the action on the circle. With Th. Marty, we consider the case of 1 (singular or not) foliation and we give a complete answer to the following questions

Q3: (realization) Under which hypotheses a set L of pairs of points on the circle is precisely the set of pairs of endpoints of leaves of a foliation,

Q4: (completion) Under which hypotheses a set l of pairs of points on the circle corresponds to the pairs of endpoints of a dense subset of leaves of a foliation and we prove again the uniqueness.[-]

An element g of an abstract group G is a distortion element if there exists a finite family S in G such that g belongs to the subgroup generated by S and the wordlength of gn (w.r.t. S) grows sublinearly in n. In this talk, we will be interested in the distortion elements of the group of Cr orientation-preserving diffeomorphisms of the closed interval, for different values of r. In particular, we will present some natural obstructions to distortion (such that the presence of hyperbolic fixed points in C1 regularity and the positivity of the so-called asymptotic distortion in C2 regularity (and higher)), and we will wonder whether they are the only ones.[-]

In the late '90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by both positive and negative contact structures. Additionally, if the foliation is taut then its contact approximations are tight. In this talk, I will present a converse result on constructing taut foliations from suitable pairs of contact structures. While taut foliations are rather rigid objects, this viewpoint reveals some degree of flexibility and offers a new perspective on the L-space conjecture. A key ingredient is a generalization of a result of Burago and Ivanov on the construction of branching foliations tangent to continuous plane fields, which might be of independent interest.[-]

Studying the (closure of the) (semi-)conjugacy class of a given group action on a 1-manifold is interesting from many points of view. Depending on the manifold and/or the differentiability involved, one is faced with problems concerning small denominators, growth of groups / orbits, distortion elements, bounded cohomology, group orderability, etc. In this minicourse we will explore several general results on this topic such as the $C^1$ smoothing via (semi-)conjugacies of small group actions and obstructions in class $C^2$ and higher. We will also explore some of the ideas involved in the proof of the connectedness of the space of $\mathbb{Z}^d$ actions by diffeomorphisms of $C^{1+ac}$ regularity (obtained in collaboration with H. Eynard-Bontemps).[-]

Taut co-orientable foliations are associated with non-trivial elements of Heegard-Floer homology, hence, if a 3-manifold admits a taut, co-oriented foliation, it is not an L-space (Kronheimer-Mrowka-Ozsváth-Szabó). Conjecturally (Boyer-Gordon-Watson, Juhász), the converse is also true for irreducible manifolds. Thus far, the evidence from Dehn surgery on knots in S3 is consistent with this conjecture. We consider the L-space Knot Conjecture: if a knot has no reducible or L-space surgeries, then it is persistently foliar, meaning that for each boundary slope there is a taut, co-oriented foliation meeting the boundary of the knot complement in curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a taut, co-oriented foliation in every manifold obtained by Dehn surgery on that knot. I will describe an approach, applicable in a variety of settings, to constructing families of foliations realizing all boundary slopes. Recalling the work of Ghiggini, Hedden, Ni, Ozsváth-Szabó (and more recently, Juhász and Baldwin-Sivek) revealed that Dehn surgery on a knot in S3 can yield an L-space only if the knot is fibered and strongly quasipositive, we note that this approach seems to apply more easily when the knot is far from being fibered. As applications of this approach, we find that among the alternating and Montesinos knots, all those without reducible or L-space surgeries are persistently foliar. In addition, we find that any connected sum of alternating knots, Montesinos knots, or fibered knots is persistently foliar. Furthermore, any composite knot with a persistently foliar summand is easily shown to be persistently foliar. This work is joint with Charles Delman.[-]