A common goal in algebraic geometry is to understand a geometric object as a moduli space. One fundamental difficulty is determining whether the proposed moduli space is reduced, i.e. that the corresponding defining ideals are radical ideals. This talk will be about problems of this sort that arise in the study of affine Grassmannians, their Schubert varieties, and nilpotent orbit closures.
Specifically, I will discuss work on a conjectural moduli description of Schubert varieties in the affine Grassmannian and proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on equations defining type A affine Grassmannians. As an application of our ideas, we prove a conjecture of Pappas and Rapoport about nilpotent orbit closures. This involves work with Joel Kamnitzer, Alex Weekes, and Oded Yacobi.