Séminaire de topologie, géométrie et algèbre (archives)

Le principe fondamental de la topologie algébrique est d'associer des objets algébriques à des objets de nature géométrique ou topologique. Dans cet exposé, nous commencerons par rappeler rapidement la complexité croissante de ces objets algébriques : des nombres, des groupes, des algèbres associatives, puis différents types d'algèbres de plus en plus complexes. Les opérades sont un outil pour mieux comprendre ces algèbres, et mieux les étudier. Par exemple, la construction bar des opérades et la dualité de Koszul permettent de retrouver les complexes d'homologie (Hochschild, Chevalley-Eilenberg, etc) et des constructions similaires dans des contextes plus modernes. Ceux-ci font souvent intervenir des infini-catégories ou des algèbres à homotopie près. Le but final de l'exposé est de présenter la définition des infini-opérades et de leurs algèbres, et de définir des constructions bar associées.

Ceci est un travail en commun avec Ieke Moerdijk.

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.

We say that a link L in S^3 is negative amphichiral if there exists an orientation-reversing diffeomorphism of S3 that sends every component of L to itself with the opposite orientation. If such a map can be chosen to be an involution, then the link is said to be strongly negative amphichiral. Kawauchi proved that every strongly negative amphichiral link is rationally slice, i.e. it bounds a disjoint collection of disks in a rational homology 4-ball. In this talk, we prove that every negative amphichiral link is rationally slice, extending the aforementioned work of Kawauchi. Our proof relies on a careful analysis of the JSJ decomposition of the link complement of negative amphichiral links. This is joint work in progress with Jaewon Lee (KAIST, Daejeon) and Oğuz Şavk (CNRS, Nantes).

By a theorem of Kontsevich and Lambrechts-Volić, the little disks operad is formal in every dimension, meaning that its rational homotopy type is faithfully represented by its cohomology. This theorem has had significant consequences in both high and low-dimensional topology (e.g., Vassiliev invariants, diffeomorphisms of the disk) as well as in deformation theory (e.g., quantization deformation of Poisson manifolds). In my talk, I will review these results and explain a recent result on the equivariant formality of the little disks operad.

This is joint work with Pedro Boavida de Brito and Joana Cirici.

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.

More than twenty years ago, Etnyre and Ghrist established a connection between Reeb fields in contact geometry and a class of stationary solutions to the 3D Euler equations for ideal fluids. In this talk, we present a new framework that allows assigning contact/symplectic invariants to large sets of time-dependent solutions to the Euler equations on any three-manifold with an arbitrary fixed Riemannian metric, thus broadening the scope of contact topological methods in hydrodynamics. We use it to prove a general non-mixing result for the infinite-dimensional dynamical system defined by the equation and to construct new conserved quantities obtained from spectral invariants in embedded contact homology. This is joint work with Francisco Torres de Lizaur.

Le patchwork combinatoire est une méthode de construction de variétés algébriques réelles qui a été découverte par Oleg Viro dans les années 80, dont on est loin encore de comprendre toutes les propriétés. Dans un travail en commun avec Diego Matessi, nous nous intéressons aux patchworks dans les polytopes réflexifs, ce qui produit des hypersurfaces de Calabi-Yau. Par les travaux de Batyrev des années 90, ces hypersurfaces possèdent un miroir. Nous observons alors qu’un patchwork correspond à un diviseur dans le miroir et nous montrons certaines propriétés du patchwork en termes de la variété miroir. Par exemple, le patchwork produit une variété algébrique réelle connexe ssi le diviseur est non trivial.

A famous theorem of Laudenbach and Poénaru says that any diffeomorphism of the boundary of a 4-dimensional 1-handlebody extends to a diffeomorphism of the whole handlebody. I will present a new proof of this result using Heegaard splittings and a generalization to 4-dimensional compression bodies. I will also explain why this is essential in the theory of trisections.

Joint work with Trenton Schirmer.

Taut foliations have been a classical object of study in 3-manifolds theory. Recently, new interest in them has come from the investigation of the so-called "L-space conjecture", that predicts that manifolds containing a co-orientable taut foliation can be characterised in terms of their Heegaard Floer homology and their fundamental group. A possible approach to the study of this conjecture is analysing Dehn surgeries on knots and links. Most of the techniques employed for constructing taut foliations on Dehn surgeries usually make use of some property of the exterior of the link, for example its fiberedness. It is therefore interesting to address this study from a different perspective, using other types of properties of knots and links. In this talk I will present a result about the existence of taut foliations on all non-trivial surgeries on knots with a special diagram.