Ana Caraiani – Colloquium “Elliptic curves and modularity”
Abstract: 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.

Ana Caraiani – Seminar talks
“On the cohomology of Shimura varieties with torsion coefficients”
Abstract: I will survey results concerning the cohomology of Shimura varieties with torsion coefficients from the past few years. I will discuss the geometry of the Hodge-Tate period morphism, including a recent generalisation of Igusa varieties to Igusa stacks due to Mingjia Zhang. Then I will contrast the original approach of computing cohomology with torsion coefficients due to myself and Peter Scholze, which relies on the trace formula, with more recent approaches due to Teruhisa Koshikawa, Linus Hamann and Si Ying Lee, who rely on deep local results. Finally, I will explain how, by combining the two approaches, one can obtain a new instance of local-global compatibility. 
“On the modularity of elliptic curves over imaginary quadratic fields”
Abstract: I will survey how to prove the modularity of elliptic curves defined over the rational numbers, as pioneered by Wiles and Taylor-Wiles and completed by Breuil, Conrad, Diamond and Taylor. I will also mention the case of elliptic curves defined over real quadratic fields, more recently completed by Freitas, Le Hung and Siksek. I will then explain why the case of imaginary quadratic fields is qualitatively different from the previous ones. Finally, I will discuss joint work with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. This has an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen, Khare and Thorne.