2022 POSTECH-PMI NUMBER THEORY SEMINAR ㅣSome geometry of affine Deligne-Lusztig varieties

ABSTRACT: The mod p points of a Shimura variety have a conjectural description called the Langlands-Rapoport conjecture. In relation to the conjecture, Rapoport defined (generalized) affine Deligne-Lusztig varieties as the (conjectural) p-part of the description. Since then, their basic geometric properties have been studied including nonemptiness, dimensions, irreducible components, and connected components. Depending on what variants of affine Deligne-Lusztig varieties one studies, such questions are completely solved or moderately open. In this talk, I would like to first give some concrete examples and historical background on the name. Then, I will explain what is known, what is conjectured, or what is completely open without any conjectures on the questions of basic geometric properties aforementioned. If time permits, I will introduce my recent works on nonemptiness and the connected components.