Geometry in the reduction modulo p of Shimura varieties

Shimura varieties are higher dimensional analogues of the modular curve, and form an indispensable tool in the Langlands program. The singularities that arise in their reduction moduloa prime number p can be described in terms of linear algebra, in the cases where the level structure at p is not too deep. This goes back to works of Chai and Norman, Deligne and Pappas and de Jong from the early 90’s. In the talk, we focus on the linear algebra side and survey some of the results obtained in the past few years.