The singularities of integral models of Shimura varieties are encoded in their local models, schemes over the p-adic integers whose special fibers are unions of affine Schubert cells. A fundamental question is whether these local models are Cohen-Macaulay.

In this talk, I will present a solution for local models with arbitrary parahoric level structure, valid uniformly across all residue characteristics. The proof is centered on a combinatorial property of the admissible set, which parametrizes the cells in the special fiber. We prove that the admissible set is dual EL-shellable, thereby resolving 
a conjecture of G\"ortz from over two decades ago. From this purely combinatorial result, we deduce the Cohen-Macaulay property for the corresponding local models.

This work provides a uniform, characteristic-independent approach that contrasts with and complements prior geometric methods. I will explain the key combinatorial ideas and their translation into this geometric consequence.