John Bergdall of the University of Arkansas presents a Joint IAS/Princeton Number Theory seminar on a local-global phenomenon connecting automorphic forms and Galois representations. He starts from a known fact: spaces of modular forms are indexed by weights, and those same weights index local deformation rings built from p-adic Hodge theory conditions. Mark Kisin's mid-2000s work showed that modular points appear on every component of such a ring, and Bergdall describes how to go further, giving a method for actually counting how many modular points sit on a given component by reading off the geometry of its special fiber. The result is joint work with Chengyang Bao of Imperial College London and Brandon Levin of Rice University. The talk assumes familiarity with the Langlands program and is aimed at working number theorists rather than newcomers, but it lays out the counting method step by step from Kisin's original theorem.