Explore the Breuil-Mezard conjecture and its implications for the geometry of local deformation rings with p-adic Hodge theory conditions in this advanced mathematics lecture. Delve into a version of the conjecture on the Emerton-Gee moduli stack of mod p Galois representations and examine its connection to the weight part of Serre's conjecture. Learn about recent collaborative research on both conjectures for specific classes of potentially crystalline substacks, offering insights into cutting-edge developments in algebraic number theory and representation theory.