crystalline deformation rings play an important role in kisin's proof of the fontaine-mazur conjecture for gl2 in most cases. one crucial step in the proof is to prove the breuil-mezard conjecture on the hilbert-samuel multiplicity of the special fiber of the crystalline deformation ring. in pursuit of formulating a horizontal version of the breuil-mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. our approach, based on reverse-engineering the taylor-wiles-kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.

[pre-talk at 3:00pm]