Cédric De Groote (Max Planck Institute, Leipzig) will speak in the geometry seminar on Tuesday 26 November 2019 at 10am in the Salle de Profs. Cédric’s title is Orderability up to conjugation of certain open contact manifolds and his abstract is below.

Eliashberg and Polterovich introduced in 2000 a notion of orderability for the group of contact isotopies of a contact manifold, which provides insights into the geometry of that group. Later, this same notion “up to conjugation” was used by Borman, Eliashberg and Murphy in their proof of the flexibility of overtwisted contact manifolds of all dimensions. I will review some of the history of that problem, and then present a new result on the orderability up to conjugation of certain contact annuli. This involves restating the problem as a contact non-squeezing result, which is then shown using a version of contact homology.

Marine Fontaine (University of Antwerp) will speak in the geometry seminar on Tuesday 10 December 2019 at 11am in the Salle de Profs. Marine’s title is Braids in the N-body problem by cabling a body in a central configuration and her abstract is below.

We prove the existence of periodic solutions of the N = n+1-body problem in an even dimensional Euclidean space: We start with n bodies whose reduced motion is close to a central configuration and we replace one of them by a pair of bodies rotating uniformly around their center of mass. When the motion takes place in the standard Euclidean plane these solutions are a special type of braid solutions. We use a variational formulation and the result is obtained by performing a Lyapunov-Schmidt reduction and the use of the Lyusternik-Schnirelmann category. This work is in collaboration with Carlos García-Azpeitia.

Mélanie Theillière (Université Claude Bernard Lyon 1) will speak in the geometry seminar on Tuesday 19 November 2019 at 11am in the Salle de Profs. Melanie’s title is Convex Integration without Integration and her abstract is below.

The Convex Integration Theory was developped by Gromov in 70s. This theory allows to solve differential constraints seen as subsets of the jet space and called Differential Relations. In the case of a relation of order 1, it allows to build a solution F from a section (x,f(x),L(x)) of the bundle J^1(M,W) -> M whose image lies inside the differential relation using an iteration of suitable integrations called “Convex Integrations”. Recently this theory led to explicit constructions of C^1-isometric embeddings. In this talk, we will propose an alternative formula to the Convex Integrations called Corrugation Process and we will introduce a kind of differential relation that we call Kuiper relation. For these relations, the formula is greatly simplified. As an application of this result, we will give an idea of the construction of a new immersion of RP^2 and we will state a Nash-Kuiper C^1-isometric embedding theorem in the case of totally real applications.

Dmytro Yeroshkin (ULB) will speak in the geometry seminar on Tuesday 1 October 2019 at 11am in the Salle de Profs. Dmytro’s title is Manifolds with Density and Measure and his abstract is below.

Riemannian manifolds with measure, and more general metric spaces with measure have been studied under different guises for decades. In this talk I’ll present a new geometric approach to these spaces that introduces a torsion free affine connection, as opposed to a measure, as the fundamental object of study. This approach allows us to connect the notion of a manifold with measure and the Bakry-Emery notion of manifold with density. This allows us to extend the work on comparison results to the new case of negative synthetic dimension, as well as reveals a number of new structures for future study. This is joint work with Will Wylie and Lee Kennard.