Some recent progress on integral models of Shimura varieties

Shimura varieties are a rich family of spaces sitting at the intersection of number theory, algebraic geometry, and differential geometry, which generalize the celebrated theory of modular curves. They have furthermore occupied a central role in the study of the widely influential Langlands program, from Wiles's use of modular curves to study Fermat's Last Theorem, to more recent work surrounding innovations of Peter Scholze and his collaborators. Important in these applications are the ability to understand Shimura varieties not only over number fields (where they are originally defined) but also integrally, e.g., over rings of p-adic integers. Starting with the case of modular curves, I will discuss joint work with Naoki Imai and Hiroki Kato which sheds light on a very large class of 'canonical' integral models of Shimura varieties. In particular this allows one to understand the deformation theory theory of these integral models and clarify the meaning of the phrase 'canonical'.