Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two) has been that we possess fewer tools for studying them. For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms. We will show in this talk that using barcodes and persistence homology one can indirectly define (filtered) Floer homology for Hamiltonian homeomorphisms. This talk is based on joint projects with Buhovsky-Humiliére and Le Roux-Viterbo.

A higher genus version of Welschinger invariant was defined by Shustin for del Pezzo surfaces. These invariants count real curves of positive genera with signs. We study the properties of higher genus Welschinger invariants under Morse simplification. When the signs are defined by the number of solitary nodes, we prove that these higher genus Welschinger invariants depend only on the total number of real interpolated points. The result follows from a reduction of the genus and Brugallé's result on the invariance of genus zero Welschinger invariants.

Everything has been said about Stein fillings of lens spaces when they are endowed with their standard (tight) contact structure. Nevertheless, lens spaces support many more tight structures, that are all classified, but for which a complete list of their Stein fillings is still missing. I will present a series of methods and techniques which can be used to investigate the topology of these fillings, giving constraints, for example, on their Euler characteristic and fundamental group.

Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3–sphere. This defines an equivalence relation under which the set of knots becomes an abelian group with the connected sum operation. The importance of this group lies in its strong connection with the study of 4-manifolds. Indeed, many questions pertaining to 4–manifolds with small topology (like the 4–sphere) can be addressed in terms of concordance. A powerful tool for studying the algebraic structure of this group comes from satellite operations or the process of tying a given knot P along another knot K to produce a third knot P(K).

To each coprime pair of natural numbers is associated a rational homology 4-ball B_{p,q}; these are of interest in algebraic geometry and in constructions of smooth 4-manifolds. Evans and Smith have completely determined which of these may be embedded symplectically in CP^2; the answer coincides with an algebraic geometric result of Hacking and Prokhorov, and is described in terms of solutions to the Markov diophantine equation.

Using double branched covers, we exhibit an infinite family of such balls which embed smoothly but not symplectically in CP^2. We also describe an obstruction using Donaldson’s diagonalisation which may be used to show that no two of our examples may be embedded disjointly.

We will give a survey on the main problems concerning rigidity, stability and formation of black holes and some of the recent results.

The simplex Hilbert transform is a singular integral form related to the multilinear Hilbert transform and Carleson’s operator. Boundedness of the simplex Hilbert transform is is one of the major open problems in harmonic analysis. In this talk we discuss bounds for a superposition of simplex Hilbert transforms in low dimensions, and related singular integral forms. Joint work with Joris Roos.

The bidomain and monodomain models are widely used models in simulating cardiac electrical activity. In this talk, we first briefly describe the unfolding homogenization approach to rigorously derive the bidomain equations from a microscopic model with tensorial and space dependent conductivities . Secondly, we present a positive nonlinear control volume finite element (CVFE) scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (Beeler-Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in finite element methods.

At the starting point of this talk are two integrable Hamiltonian systems in infinite space dimension. The first one is the Benjamin--Ono equation and was introduced about forty years ago in Fluid Mechanics. The second one is the Szegö equation and was introduced about ten years ago as a model of a non dispersive Hamiltonian evolution. Both systems admit a Lax pair structure, involving operators on the Hardy space of the disk enjoying special commuting properties with the shift operator : Hankel operators and Toeplitz operators. I will focus on the inverse spectral problems for these Lax operators, on the similarities in the strategy for solving them and on the dramatically different outputs.