The Langlands program is a vast network of conjectures that connect many areas of pure mathematics, such as number theory, representation theory, and harmonic analysis. At its heart lies reciprocity, the conjectural relationship between Galois representations and modular, or automorphic forms. A famous instance of reciprocity is the modularity of elliptic curves over the rational numbers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some recent progress in the Langlands program, with a focus on new reciprocity laws for elliptic curves over imaginary quadratic fields.

Given a vector field in R^d, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow (namely, the solution X(t) of the ODE X′(t)=b(t,X(t)) from any initial datum x∈R^d provided the vector field is sufficiently smooth. The theorem looses its validity as soon as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.

The talk presents an overview and new results in the context of the celebrated DiPerna-Lions and Ambrosio’s theory on flows of Sobolev vector fields, including a negative answer to the following long-standing open question: are the trajectories of the ODE unique for a.e. initial datum in R^d for vector fields as in Di Perna and Lions theorem? We will exploit the connection between the notion of flow and an associated PDE, the transport equation, and combine ingredients from probability theory, harmonic analysis, and the “convex integration” method for the construction of nonunique solutions to certain PDEs.

This colloquium talk glimpses into some different aspects of water waves, in particular travelling such, and how the physical settings and solution properties are linked to the mathematical constructions and analysis.

Our main focus will be on travelling waves over finite-depth water, such as long-lasting wave trains propagating out at sea. With the help of examples from my own interest and research into these equations, I will describe some of the questions asked, and the methods to solve them. Some proofs will be presented, but the focus will be on the mechanisms of proof, and how these take inspiration from different parts of mathematics (to the extent that they do).

The talk is based on a number of different projects together with collaborators.