A basic problem in number theory is counting the number of mod p points of an algebraic variety over the integers. This is complicated by the phenomena of bad reduction: for example, given a polynomial with integer coefficients, its complex solutions may form a smooth space, but its mod p solutions (upon passing to an algebraic closure) may form a space that is very singular. Such singularities are usually so severe that they prevent an explicit geometric approach, but we want to understand them, as they are often a rich source of arithmetic information. We’ll explain some new methods for studying these singularities by exploiting some hidden group-theoretic structures. For example, it turns out that much like how Euclidean space locally models manifolds and quotients of polynomial rings locally model algebraic varieties, in some cases, these singular spaces can be locally modelled by analogues of Schubert varieties on infinite-dimensional Grassmannians, and by using this additional structure, we can count the mod p points of these singular varieties.