Student Learning Seminar on Galois Deformations
The purpose of this seminar is to give an introduction to the theory of Galois deformations and discuss its most important applications in number theory. The seminar is intended for graduate students interested in the topic. The organizers will impart the lectures by default, but we encourage participants to prepare and give some of the talks. If you are an undergraduate student or external graduate student and would like to come, please email or .

When: Wednesday, 2:45pm - 3:45pm
Where: TBD
Organizers: Rafah Hajjar Muñoz, Vivian Yu

Remark: Until we have a definite room for the seminar, we will be meeting at the graduate lounge.

Date Speaker Title and Notes
September 13 Rafah Hajjar Introduction. Review of Galois Representations
Abstract: We will introduce the concept of Galois deformations and discuss briefly their relevance and literature. We will also give a brief review of Galois representations, stating their basic properties and discussing how they naturally arise in several areas of Number Theory.
September 20 Rafah Hajjar Galois Representations (cont.) and the representation theory of profinite groups
Abstract: We will discuss some facts about the representation theory of profinite groups that will be useful in our study of Galois deformations. Also, following the discussion of last week, we will finish our review of the literature on the theory of Galois representations. In particular, today we will talk about the representations of local Galois groups, giving brief accounts of Weil-Deligne representations, the monodromy theorem, and p-adic Hodge theory.
September 27 Kevin Chang
Introduction to Galois deformations
Abstract: I will introduce basic notions in deformation theory, such as universal deformation rings and the Zariski tangent space, and state some important properties of Galois deformations to be covered in more depth in future talks.
October 4 Vivian Yu Representability of the Deformation Functor
Abstract: This week we will present a proof for the representability of the deformation functor (defined last week) under certain assumption on the residual representation. We will begin by introducing Schlessinger’s criteria for pro-representability of functors on categories of artinian rings and prove the main result by checking these criteria. If time allows, I will provide some concrete examples of universal deformation rings for certain deformation problems.
October 11 Matthew Hasse-Liu
Zariski tangent space and obstructions
Abstract: We will continue our discussion of the deformation ring and establish some basic functorial properties. These include different perspectives on the tangent space, as well as obstructed/unobstructed deformation problems.
October 18 Sangmin Ko
Deformation conditions
Abstract: We will introduce the concept of deformation conditions to study the subspace of deformation that satisfies desirable properties. Then we will define deformation with fixed determinant, ordinary deformation and flat deformation.
October 25 Rafah Hajjar Deformation conditions of global Galois representations
Abstract: Following last week's discussion on deformation conditions, today we will talk about their application to representations of Galois groups of global fields, and we will discuss some common cases. If time permits, we will review some results of Galois cohomology that will let us study the tangent spaces of these global deformation problems.
November 1 Baiqing Zhu
Explicit construction of Galois deformation rings
Abstract: We prove the existence of the universal deformation ring for a absolutely irreducible Galois representation over a finite field. We also give another explicit construction of the deformation ring when the representation is tame.
November 8 Rafah Hajjar An overview of the Taylor-Wiles method
Abstract: One of the most important applications of Galois deformation theory is modularity lifting. This is, given a residual representation that is modular, we want to build a global deformation problem such that all liftings for this problem are modular. The Taylor-Wiles method gives a way of lifting modularity by means of proving an R=T theorem
November 15 Alan Zhao
Review of Galois cohomology
Abstract: In this talk we will the describe the techniques from Galois Cohomology used in the study of Galois deformation problems.The Goal of the talk is to state Wiles' product formula, relating the cardinality of the tangent space of a global deformation problem to that of its orthogonal (which are given by generalized Selmer groups). If time permits, we will give a sketch of the proof.
November 22
December 6 Baiqing Zhu
Intersection of Hecke correspondence on modular curves
Abstract: I will give a brief introduction to the theorem of Gross-Keating and its applications to height pairings on modular curves.