Abstract

Shimura varieties are highly symmetric algebraic varieties that play an important role in the Langlands program. I will first try to give you a sense for what they are like, with a focus on their different kinds of symmetries. I will then describe powerful tools that are used to study Shimura varieties: the Hodge--Tate period morphism, introduced by Scholze in 2013; and Igusa stacks, introduced in recent years by Zhang, Daniels--van Hoften--Kim--Zhang and Kim. I will aim to provide intuition for the geometry of these objects, which explains and unifies earlier insights into the geometry and cohomology of Shimura varieties. I will also aim to give you a sense of their far-reaching applications.