# Ian Gleason: The connected components of affine Deligne--Lusztig varieties in mixed characteristic

## Ian Gleason: The connected components of affine Deligne—Lusztig varieties in mixed characteristic (Oberseminar November 11)

Affine Deligne-Lusztig varieties (ADLV) play an important role in the study of characteristic p points of Shimura varieties, since they parametrize $p$-isogenies of abelian varieties with extra structure. Notably Kisin uses information on connected components of ADLV to study Langlands-Rapoport conjecture for his integral models of abelian type Shimura varieties at hyperspecial level. After Chen-Kisin-Viehmann’s seminal work, many authors have used combinatorial techniques and perfect algebraic geometry to compute connected components of ADLV, each of these works increasing the generality of the statement proved. The purpose of this talk is to describe a new approach to computing the connected components of ADLV. This new approach uses $p$-adic analytic geometry a la Scholze and Fontaine’s classical $p$-adic Hodge-Theory, to solve the problem in full generality.