Xuhua He, speaking in the Joint IAS/Princeton University Arithmetic Geometry seminar, tackles a long-standing question about the singularities of integral models of Shimura varieties. These local models are p-adic schemes whose special fibers break into affine Schubert cells, and the central question is whether they are Cohen-Macaulay. He presents a proof covering arbitrary parahoric level structure that works uniformly across all residue characteristics, built on showing that the admissible set parametrizing these cells is dual EL-shellable. This resolves a conjecture posed by Gortz more than twenty years ago and turns a purely combinatorial result into a geometric one. He lays out the shellability argument before tracing how it forces the Cohen-Macaulay property, framing the approach as a characteristic-independent complement to earlier geometric methods. The talk is aimed at specialists in arithmetic geometry and combinatorics of Shimura varieties, delivered at the Institute for Advanced Study's Simonyi Hall.