The study of affine Deligne-Lusztig varieties (ADLVs) $X_w(b)$ and their certain union $X(\mu,b)$ has been crucial in understanding mod-$p$ reduction of Shimura varieties; for instance, precise information about the connected components of ADLVs (in the hyperspecial level) has proved to be useful in Kisin's proof of the Langlands-Rapoport conjecture. On the other hand, first introduced in the context of enumerative geometry to describe the quantum cohomology ring of complex flag varieties, quantum Bruhat graphs have found recent applications in solving certain problems on the ADLVs. I will survey such developments and report on my work surrounding a dimension formula for $X(\mu,b)$ in the quasi-split case, as well as some partial description of the dimension and top-dimensional irreducible components in the non quasi-split case.

[pre-talk at 1:20PM]