SPEAKER:  Shou-Wu Zhang (Princeton University)

TITLE:  "On Modularity of Kudla’s Generating Series of 0-Cycles"

ABSTRACT:  Given a locally symmetric space and a mod p system of Hecke eigenvalues, the weight part of Serre's conjecture seeks to classify the cohomology groups of the locally symmetric space where this system of eigenvalues contribute. Via Taylor-Wiles patching, this problem is known to be closely linked to the Breuil-Mezard conjecture, which predicts a representation-theoretic recipe to build mod p fibers of local Galois deformation rings (and more generally, the Emerton-Gee moduli stack of local Galois representation) out of atomic pieces corresponding to Serre weights. I will describe progress on both questions in many cases, via a theory of local models, which relates (portions of) the Emerton-Gee stack to mixed characteristics versions of (deformed) affine Springer fibers.