The Breuil-Mézard conjecture

Time: 9:30 -- 11:00, August 07th, 2024

Venue: Room 612, A6

Abstract: Breuil-Mézard conjectured that the Hilbert-Samuel multiplicities of deformation rings of rank n representations of the Galois group of a p-adic field K with p-adic Hodge theoretic conditions are controlled by certain decomposition numbers of the group GL_n(O_K). More recently, this phenomena has been geometrically interpreted as the (conjectural) existence of highly constrained cycles in the Emerton-Gee stack, which is a way to interpolate between different Galois deformation rings. I will give an introduction to the circle of ideas surrounding this, and describe some recent approach to construct these cycles and prove their internal structure. This is based on joint work with T. Feng and Zhongyipan Lin