Lagrangian submanifolds exhibit surprising rigidity in view of their small dimension (only half that of the ambient symplectic manifold). A famous manifestation of this rigidity is that some of them cannot be displaced from themselves by any Hamiltonian diffeomorphism or even by any diffeomorphism which preserves the symplectic form. In comparison, any submanifold of the same dimension which is not Lagrangian can be displaced from itself by a Hamiltonian diffeomorphism.

Moreover, that diffeomorphism can be chosen arbitrarily C⁰-small, so that one can wonder whether a Lagrangian L (which can be displaced from itself) admits a neighborhood of Hamiltonian (resp. symplectic) non-displacement. Such a neighborhood W is defined by the property that any Lagrangian obtained from L by a Hamiltonian (resp. symplectic) diffeomorphism, and included in W, must intersect L.

On the one hand, I will give conditions which ensure the existence of such neighborhoods for a large class of Lagrangians. On the other hand, I will construct a Lagrangian which does not admit any, in any symplectic manifold of dimension at least 6. Then, I will discuss several applications of our techniques to the topology of orbits of Lagrangians. This is based on a joint work with Marcelo Attalah, Jean-Philippe Chassé, and Egor Shelukhin.

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.

Spectral invariants are essential tools for studying symplectic and contact rigidity, such as non-squeezing problems and camel problems, as well as Hamiltonian dynamics. These invariants have been constructed through various approaches. In this talk, we focus on the Floer-theoretic construction of Legendrian spectral invariants in one-jet bundles.

Typically, Floer theory on contact manifolds, such as one-jet bundles, involves the use of symplectization. However, in this talk, I will introduce the concept of contact instantons, which enables the construction of Legendrian Floer cohomology and its spectral invariants without relying on symplectization.

Unsupervised learning aims to capture the underlying structure of potentially large and high-dimensional datasets. Traditionally, this involves using dimensionality reduction methods to project data onto lower-dimensional spaces or organizing points into meaningful clusters (clustering). Typically, this process involves aligning two graphs depicting the relationship between samples in the input high-dimensional space and their corresponding positions in the output low-dimensional space. In this talk we will present a new perspective on these approaches that is based on optimal transport and the Gromov-Wasserstein distance. Precisely, we will propose a new general framework, called distributional reduction, that recovers dimension reduction and clustering as special cases and allows us to address them jointly with a single optimization problem. We then empirically showcase the relevance of our approach on both image and genomics datasets.