Non-emptiness of Affine Deligne-Lusztig Varieties

Deligne-Lusztig varieties, which can be thought of as Frobenius-twisted Schubert varieties, were invented for studying the representation theory of finite Chevalley groups. We introduce several affine variants, one lying inside the affine Grassmannian and another in the affine flag variety, and recall the history of progress made in studying these affine versions. In particular, we demonstrate two methods for proving that certain affine Deligne-Lusztig varieties are non-empty as sets. Both methods are combinatorial in nature, the first of which uses the combinatorics of a certain set of Newton polygons, and the second of which involves the combinatorics of lengths of elements in finite Coxeter groups.