##### Department of Mathematics,

University of California San Diego

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### Math 209: Number Theory Seminar

## Chengyang Bao

#### UCLA

## Computing crystalline deformation rings via the Taylor-Wiles-Kisin patching method

##### Abstract:

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]*

### October 30, 2024

### 4:00 PM

APM 7321 and online (see https://www.math.ucsd.edu/~nts

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