Mathematicians have been interested in the problem of compactifying the Jacobian variety of curves since the mid XIX century. In this talk we will discuss how all 'reasonable' compactified Jacobians of nodal curves can be classified combinatorically. This suffices to obtain a combinatorial classification of all 'reasonable' compactified universal (over the moduli spaces of stable curves) Jacobians. This is a joint work with Orsola Tommasi.

The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.