I will report on a program to describe elliptic fibrations on K3 surfaces developed in collaboration with A. Garbagnati (U. Milano). We consider K3 surfaces that admit a non- symplectic involution with non-empty fixed locus. We give a method of classification of elliptic fibrations on such surfaces by means of analizing linear systems on a simpler surface, namely the quotient by the involution. In this talk, I will describe the ideas behind the method and report on a new stage of this project which is directed to some possible applications of both geometric and arithmetic nature.

This talk is based on a joint work with Nick Lindsay. A compact symplectic manifold (M,ω) is called Fano if the classes c1(M) and [ω] coincide in H2(M). We prove that any symplectic Fano 6-manifold M with a Hamiltonian S1-action is simply connected and satisfies c1c2(M) = 24. This is done by showing that the fixed submanifold of M on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a 2-sphere or a point.

Symplectic embedding problems are at the core of the study of symplectic topology. There are many well-known results for so-called toric domains, but very little is known about other kinds of domains. In this talk, I will mostly speak about a different kind of domain, namely a lagrangian product. These domains are of a very different nature and are related to billiards, as discovered by Artstein-Avidan and Ostrover. I will explain how to use in- tegrable systems to see that some of these products are secretly toric domains and how to use symplectic capacities to obtain sharp obstructions to many symplectic embedding problems.