Schedule

Mélanie Bertelson (Université Libre de Bruxelles)

On certain quantitative aspects of locally conformal symplectic topology

Locally conformally symplectic structures appear naturally in contact geometry and present new and interesting quantitative aspects. Some of them will be described in the talk.

Alain Chenciner (Observatoire de Paris et Université Paris-Cité)

Radial attraction and angular twist

Some questions of an analytic nature about local analytic diffeomorphisms of the plane with a non-resonant elliptic fixed point and families of local diffeomorphisms unfolding them.

Sylvain Courte (Université Grenoble Alpes)

On the parametrised Whitehead torsion of families of nearby Lagrangian submanifolds

The space of all closed exact Lagrangian submanifolds in the cotangent bundle of a closed connected manifold is conjecturally contractible: this is a strong version of the nearby Lagrangian conjecture. Parametrised Whitehead torsion defines a map from this space to a certain h-cobordism space, which at the level of connected componentsboils down to the classical Whitehead torsion. In a recent work with Noah Porcelli, we use generating functions to recover Abouzaid-Kragh’s theorem that the Whitehead torsion of a nearby Lagrangian vanishes and give a generalization to the parametrized setting. This allows us to give constraints on the possible monodromy diffeomorphisms of a loop of closed exact Lagrangians: for instance in the case of the torus Tⁿ for large n, we obtain that this monodromy must be isotopic to the identity through homeomorphisms.

Mihai Damian (Université de Strasbourg)

Morse-type theories with differential graded coefficients and applications

In two joint papers with Barraud, Humilière and Oancea we developed Morse and Floer theory with coefficients in a differential graded module over the algebra of singular chains on a based loop space, following seminal ideas of some previous works of Barraud and Cornea. In this talk I will present these theories and some applications to the study of the existence of periodic Hamiltonian orbits in cotangent bundles.

Oliver Edtmair (ETH Zürich)

On the Topological Invariance of Helicity II (joint work with Sobhan Seyfaddini)

Helicity is an invariant of divergence free vector fields on three-manifolds. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic linking number, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows.

In these two lectures, we will present affirmative answers to both questions in the case of non-singular flows and will outline the main ideas behind the proofs. Our approach relies on recent advances in C⁰-symplectic topology, particularly new insights into the algebraic structure of the group of area-preserving homeomorphisms, which we will also review.

Hélène Eynard-Bontemps (Université Grenoble Alpes)

Distorted elements in groups of diffeomorphisms of one-manifolds (joint work with Emmanuel Militon, Nice)

In a group G, an element g is called “distorted” if there exists a finite family S in G which generates g and such that the word-length of gⁿ w.r.t. S grows sublinearly in n. This very general notion of geometric group theory is particularly interesting in the context of transformation groups, as it provides obstructions for some groups to act faithfully on some spaces. In this talk, I will focus on the groups of homeo/diffeomorphisms of the line (with compact support) and of the circle, and I will give a concrete dynamical description of the distorted elements in regularity C^∞. Interestingly, this requires ingredients which are specific to this regularity (among which a “local uniform perfection” result), and such a description remains unknown in finite regularity.

Albert Fathi (Ens de Lyon et Georgia Tech)

TBA

Anna Florio (Université Paris Dauphine-PSL)

Genericity of transverse homoclinic points for analytic convex billiards

A celebrated result by Zehnder in the '70s states that a generic analytic area-preserving map of the disk, having the origin as elliptic fixed point, exhibits a transverse homoclinic orbit in every neighborhood of the origin. In an ongoing project with Inmaculada Baldomà, Martin Leguil and Tere Seara, we adapt the strategy of Zehnder and use Aubry-Mather theory for twist maps in order to show that a generic analytic strongly convex billiard has, for every rational rotation number, a periodic orbit with a transverse homoclinic intersection.

Patrice Le Calvez (Sorbonne Université)

Area preserving surface homeomorphisms without horseshoe

Let S be an orientable closed surface and f an area preserving homeomorphism of S with no horseshoe (a particulary case is a homeomorphism with zero entropy). In a join work with Pierre-Antoine Guihéneuf, Alejandro Passeggi and Fabio Tal, we prove that there is a decomposition of S in three sets: a finite union of invariant open sets of positive genus where the map is transitive; a union of fixed point free open annuli; a separating set made of fixed points and homo/heteroclinic orbits. Such a result generalizes a well-known result for area preserving flows.

Ailsa Keating (University of Cambridge)

Mirror symmetry for K3 surfaces: a gentle introduction

The goal of this talk is to give an introduction to recent approaches to (homological) mirror symmetry for K3 surfaces, aimed at a  non-specialist geometry & topology audience. We will build intuition from the corresponding story in complex dimension one, and emphasise throughout the role of Lagrangian torus fibrations and their degenerations. Partly based on joint work with Paul Hacking.

Gabriel Rivière (Nantes Université)

Poincaré series and linking of closed geodesics

If we are given two closed geodesics on a compact Riemannian surface and if we suppose that they are homologically trivial, then we can define their linking number. When the surface has negative sectional curvature, I will explain how this number can be expressed as the value at zero of a certain zeta function involving the lengths of the geodesics arcs orthogonal to these two geodesics. This will combine tools from hyperbolic dynamical systems, spectral theory and the theory of de Rham currents. This is a joint work with Nguyen Viet Dang (Strasbourg).

Sobhan Seyfaddini (ETH Zürich)

On the Topological Invariance of Helicity I (joint work with Oliver Edtmair)

Helicity is an invariant of divergence free vector fields on three-manifolds. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic linking number, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows.

In these two lectures, we will present affirmative answers to both questions in the case of non-singular flows and will outline the main ideas behind the proofs. Our approach relies on recent advances in C⁰-symplectic topology, particularly new insights into the algebraic structure of the group of area-preserving homeomorphisms, which we will also review.

Andras Stipsicz (Rényi Institute Budapest)

Exotic 4-manifolds with cyclic fundamental groups

We will examine exotica for closed manifolds with cyclic fundamental group, with special attention to definite examples, especially examples with the same rational homology as the complex projective plane (so called fake projective planes, FPP’s).