The cohomology of Shimura varieties – a survey of recent developments

Abstract: Shimura varieties are highly symmetric algebraic varieties that play an important role in the Langlands program. In the first part of this talk, I will try to give you a sense of what they are like, with a focus on their different kinds of symmetries. In the second part of the talk, I will survey a recent class of results about the vanishing of the cohomology of Shimura varieties with torsion coefficients. To give you a sense of the breadth of the subject, I will mention both connections to the geometric Langlands program and applications to long-standing problems, such as the modularity of elliptic curves.