Moduli spaces of representations of Galois groups of p-adic fields with p-adic Hodge theoretic conditions traditionally play a pivotal role in establishing automorphy, and have recently gained even more prominence with the advent of the categorical p-adic Langlands program. Unfortunately, the geometry of these moduli spaces is poorly understood, and conjecturally must be sufficiently complicated to encode the modular representation theory of finite groups of Lie type. In this talk, I will describe some special but important cases where it is possible to probe the geometry of these moduli spaces, by relating them to more amenable (but still complicated) geometric objects.

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