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
rh3101@columbia.edu or
qy2266@columbia.edu .
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.
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 WeilDeligne
representations, the monodromy theorem, and padic 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 prorepresentability 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 HasseLiu 
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 TaylorWiles 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 TaylorWiles
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 GrossKeating
and its applications to height pairings on modular curves.
