Class Field Theory
Student Learning Seminar
Purpose
The main goal of the seminar will be for graduate students to read, learn and present the material in the book A Gentle Course in Local Class Field Theory by Pierre Guillot. Graduate students will take turns presenting sections of the book and solve exercises contained at the end of each chapter.
Why this book?
This book offers a self-contained exposition of local class field theory. The only prerequisite is a first semester of Galois theory. It is written for beginning graduate students. The book is split into four parts:
Part I: Preliminaries
Part II: Brauer Groups
Part III: Galois Cohomology
Part IV: Class Field Theory
This material is useful for a variety of disciplines, including number theory, representation theory, algebraic geometry, and algebraic topology.
Who is this seminar for?
Any graduate student at the CUNY Graduate Center is welcome and encouraged to participate in this seminar! Professors are also welcome to attend but the intended audience for this seminar is graduate students.
Contact Information
This seminar is organized by Ajmain Yamin.
You can contact me at ayamin(at)gradcenter(dot)cuny(dot)edu.
Spring 2022 Schedule
Meeting Details
There will be weekly meetings on Wednesdays at 4:15 PM - 6:15 PM.
The meetings will be at the Graduate Center, Math Thesis Room (4214.03).
Participants may also join the meetings virtually through zoom. Send an email to ayamin(at)gradcenter(dot)cuny(dot)edu for the permanent zoom link.
February 2
Speaker: Ajmain Yamin
Title: First Meeting
Meeting agenda: We will figure out how we intend to run this seminar. We may finish up some exercises from Part I of the book or move on to Part II Brauer groups: Chapter 5 on Skewfields, algebras, and modules.
February 9
Speaker: Connor Stewart
Title: Modules and endomorphisms
Meeting agenda: Connor will continue presenting Chapter 5: Skewfields, algebras, and modules. Last time Connor presented most of Section 1: Skewfields and algebras. This time, Connor will begin presenting Section 2: Modules and endomorphisms and move on to Section 3: Semisimplicity if there is time.
February 16
Speaker: Ajmain Yamin
Title: The radical and matrix algebras
Meeting agenda: Connor will finish presenting Section 3: Semisimplicity and Ajmain will pick up by presenting Section 4: The radical and Section 5: Matrix algebras.
February 23
Speaker: Ajmain Yamin
Title: The radical and matrix algebras
Meeting agenda: Ajmain will continue presenting Section 4: The radical and Section 5: Matrix algebras.
March 2
Speaker: Ajmain Yamin
Title: Matrix algebras
Meeting agenda: Ajmain will finish presenting Section 5: Matrix algebras.
March 9
Speaker: N/A
Title: N/A
Meeting agenda: There will be no meeting this week as there will be the Lightning Talks on Wednesday March 9th 4-6pm in the Math Lounge.
March 16
Speaker: Ajmain Yamin
Title: Review of Chapter 5 - Skewfields, algebras, and modules
Meeting agenda: Ajmain will give a brief review of Chapter 5 - Skewfields, algebras, and modules which will be recorded. After this, we will begin discussing exercises from Chapter 5.
March 23 & 30
Speaker: N/A
Title: Chapter 5 exercises
Meeting agenda: We will spend two meetings discussing exercises from Chapter 5.
April 6 & 13
Speaker: Ajith Nair
Title: Central simple algebras
Meeting agenda: Ajith will begin presenting Chapter 6 - Central simple algebras.
April 20 & 27
Speaker: Ajith Nair
Title: Tensor products and the group law
Meeting agenda: There was no meeting on April 20, as it was Spring break. On April 27, Ajith will finish his presentation on Section 2: Tensor products and perhaps cover Section 3: The group law, if there is time. This will be the last meeting of the Spring 2022 semester.
Summer 2022 Schedule
Meeting Details
The first meeting will be on Wednesday June 22 9 AM- 11 AM Eastern Time.
The meetings will be held virtually through zoom
Send an email to ayamin(at)gradcenter(dot)cuny(dot)edu for the permanent zoom link.
June 22 @ 9 AM - 11 AM Eastern Time
Speaker: Connor Stewart
Title: The fundamental theorems
Meeting agenda: Connor will begin presenting the rest of Chapter 6: Central Simple Algebras starting with Section 4: The fundamental theorems.
June 29 @ 9 AM - 11 AM Eastern Time
Speaker: Connor Stewart
Title: Splitting fields and Separability
Meeting agenda: Connor will finish presenting Chapter 6: Central Simple Algebras by presenting Section 5: Splitting fields and Section 6: Separability. Connor will also start presenting Chapter 7: Combinatorial constructions Section 1: Introduction.
July 6 @ 9 AM - 11 AM Eastern Time
Speaker: Ajmain Yamin
Title: Group extensions and First applications: cyclic groups
Meeting agenda: Ajmain presented Chapter 7: Combinatorial Constructions Section 2: Group extensions and Section 3: First applications: cyclic groups.
July 21 @ 9:30 AM - 11:15 AM Eastern Time
Speaker: Ajith Nair
Title: Crossed product algebras
Meeting agenda: Ajith presented Chapter 7: Combinatorial Constructions Section 4: Crossed product algebras.
July 27 @ 9 AM - 10 AM Eastern Time
Speaker: Ajith Nair
Title: Compatibilities
Meeting agenda: Ajith started presenting Section 5: Compatibilities.
Fall 2022 Schedule
Meeting Details
There will be weekly meetings on Wednesdays at 12:00 PM - 1:45 PM.
The meetings will be at the Graduate Center, Math Thesis Room (4214.03).
Depending on interest, we may stream the talks through zoom. Email ayamin(at)gradcenter(dot)cuny(dot)edu if you are interested in participating in the seminar but cannot attend in-person.
September 14
Speaker: Ajmain Yamin
Title: First meeting - Galois Cohomology
Meeting agenda: Ajmain will initiate new participants to the seminar. I will go over goals and objectives for CUNY CFT Student Learning Seminar. We will draft a schedule of talks for the Fall 2022 semester. Please attend this meeting if you are curious about attending/participating in the seminar. In the Fall semester, we will be starting Part III of Pierre Guillot's book, which is on Galois Cohomology. The main focus of this seminar will be to present the majority of the four chapters which constitute Part III:
Chapter 9: Ext and Tor
Chapter 10: Group cohomology
Chapter 11: Hilbert 90
Chapter 12: Finer structure
Please note: you do not need to have participated in previous semesters of the CUNY CFT Student Learning Seminar to follow along in this semester! Part III of Guillot's book is almost entirely independent from the previous sections of the book.
September 21
Speaker: Susan Rutter
Title: Complexes and The Ext groups
Meeting agenda: Last week Ajmain covered Section 1: Preliminaries and Section 2: Resolutions of Chapter 9: Ext and Tor. This week, Susan will begin presenting Section 3: Complexes and Section 4: The Ext groups.
September 28
Speaker: Valera Sergeev
Title: Long exact sequences and The Tor Groups
Meeting agenda: Valera will be presenting the remaining sections of Chapter 9: Ext and Tor, namely Section 5: Long exact sequences and Section 6: The Tor groups.
October 12
Speaker: None
Title: Exercises from Chapter 9: Ext and Tor
Meeting agenda: This meeting will be a problem solving session for the exercises in Chapter 9: Ext and Tor.
October 26
Speaker: Ajith Nair
Title: Definition of group (co)homology and The standard resolution
Meeting agenda: Ajith began presenting Chapter 10: Group cohomology, Section 1: Definition of group (co)homology and Section 2: The standard resolution.
November 2
Speaker: Ajith Nair and Connor Stewart
Title: The standard resolution, Low degrees, and Profinite groups and Galois cohomology
Meeting agenda: Ajith will finish presenting Chapter 10: Group cohomology, Section 2: The standard resolution and Connor will begin presenting Section 3: Low degrees and Section 4: Profinite groups and Galois cohomology.
November 9
Speaker: Connor Stewart
Title: Low degrees, Profinite groups and Galois cohomology
Meeting agenda: Connor will finish up the presentation of Chapter 10: Group cohomology, Section 3: Low degrees and Section 4: Profinite groups and Galois cohomology.
November 16
Speaker: Ajmain Yamin
Title: Kummer theory, Hilbert 90 in Galois cohomology and More Kummer theory
Meeting agenda: Ajmain will give a brief review of Chapter 1: Kummer Theory and move on to present the first two sections of Chapter 11: Hilbert 90, Section 1: Hilbert 90 in Galois cohomology and Section 2: More Kummer theory. If there is time, Ajmain will prove Proposition 7.18 of Section 3: First applications: cyclic groups of Chapter 7: Combinatorial Constructions, which will be relevant for Valera's presentation next week.
November 23
Speaker: Valera Sergeev
Title: The Hilbert symbol
Meeting agenda: Valera will complete the presentation of Chapter 11: Hilbert 90 by presenting Section 3: The Hilbert symbol. If there is time, we will do exercises.
Spring 2023 Schedule
Meeting Details
There will be weekly meetings on Mondays at 12:00 PM - 1:45 PM.
The meetings will be at the Graduate Center, Math Thesis Room (4214.03).
Depending on interest, we may stream the talks through zoom. Email ayamin(at)gradcenter(dot)cuny(dot)edu if you are interested in participating in the seminar but cannot attend in-person.
January 30
Speaker: Ajmain Yamin
Title: Review of Galois Cohomology
Meeting agenda: Ajmain will give a review of the first three chapters of Part III: Galois Cohomology, namely, Chapter 9: Ext and Tor, Chapter 10: Group Cohomology and Chapter 11: Hilbert 90.
February 6
Speaker: Valera Sergeev
Title: Shapiro's isomorphism and A few explicit formulae
Meeting agenda: Valera will begin presenting Chapter 12: Finer Structure by presenting Section 1: Shapiro's isomorphism and Section 2: A few explicit formulae.
February 13 (Zoom meeting)
Speaker: Connor Stewart
Title: The corestriction
Meeting agenda: Connor presented most of Chapter 12, Section 3: The corestriction.
February 27
Speaker: Connor Stewart
Title: The conjugation action
Meeting agenda: Connor will present the rest of Section 3 and present Section 4: The conjugation action.
March 20
Speaker: Ajmain Yamin
Title: The five-term exact sequence
Meeting agenda: Ajmain will begin presenting Section 5: The five-term exact sequence.
March 27
Speaker: Ajith Nair
Title: Cup-products
Meeting agenda: Ajith will begin presenting Section 6: Cup-products.
April 3
Speaker: Ajith Nair
Title: Cup-products
Meeting agenda: Ajith will continue presenting Section 6: Cup-products.
April 17
Speaker: Ajith Nair and Valera Sergeev
Title: Cup-products, Milnor and Bloch-Kato
Meeting agenda: Ajith will finish presenting Section 6: Cup-products and Valera will present Section 7: Milnor and Bloch-Kato. If there is time remaining, we will do some exercises from Chapter 12: Finer Structure.
April 24
Speaker: N/A
Title: Exercises from Chapter 12: Finer Structure
Meeting agenda: This will probably be the last meeting of the semester. We will do exercises from Chapter 12: Finer Structure.
Fall 2023 Schedule
Meeting Details
There will meetings every other Friday from 10:00 AM - 11:00 AM.
The meetings will be at the Graduate Center, Math Thesis Room (4214.03).
Email ayamin(at)gradcenter(dot)cuny(dot)edu if you wish to be put on the mailing list.
September 8
Speaker: Ajmain Yamin
Title: Statements of Local Class Field Theory
Meeting agenda: Ajmain started presenting Chapter 13: Local Class Field Theory of Part IV: Class Field Theory of Guillot's book.
September 22
Speaker: Connor Stewart
Title: Cohomological triviality and equivalence
Meeting agenda: Connor will present Section 2: Cohomological triviality and equivalence of Chapter 13. The goal is to prove Tate's theorem which will be used to deduce the reciprocity theorem of local class field theory.
October 6
Speaker: Connor Stewart and Ajmain Yamin
Title: Cohomological triviality and equivalence
Meeting agenda: Connor finished presenting Section 2: Cohomological triviality and equivalence of Chapter 13. Then Ajmain presented the proof of Proposition 8.4, showing that the norm subgroups corresponding to the degree n unramified extensions of a local number field are precisely the kernels of the valuation map mod n.
October 20
Speaker: Ajmain Yamin
Title: The Brauer group of a local number field
Meeting agenda: Ajmain will begin the meeting by presenting Chapter 8: The Brauer group of a local number field. Ajmain will present Section 8.2: The Hasse invariant of a skewfield and Section 8.3: Naturality.
November 3
Speaker: Ajmain Yamin
Title: The Brauer group of a local number field
Meeting agenda: Ajmain continued presenting Chapter 8: The Brauer group of a local number field.
November 17
Speaker: Valera Sergeev
Title: Norms subgroups
Meeting agenda: Valera will present Section 14.2: The reciprocity isomorphisms and Section 14.3: Norm subgroups.
Spring 2024 Schedule
Meeting Details
There will meetings every Wednesday from 12:00 PM - 1:00 PM.
The meetings will be at the Graduate Center, Math Thesis Room (4214.03).
Email ayamin(at)gradcenter(dot)cuny(dot)edu if you wish to be put on the mailing list.
February 21
Speaker: Ajmain Yamin
Title: Review and the Existence theorem
Meeting agenda: Ajmain will review what has been covered of Chapter 13: Local Class Field Theory so far. Namely, the first four sections. Then, Ajmain presented Section 13.6: The Existence theorem.
March 13
Speaker: Ajith Nair
Title: Tate duality
Meeting agenda: Ajith will present Section 13.5: Tate duality.
March 20
Speaker: Ajmain Yamin
Title: Tate duality
Meeting agenda: Ajmain will finish presenting Section 13.5: Tate duality.
March 28
Speaker: Valera Sergeev
Title: The local Kronecker-Weber theorem
Meeting agenda: Valera will present Section 13.6: The local Kronecker-Weber theorem.
April 3
Speaker: Valera Sergeev
Title: The local Kronecker-Weber theorem
Meeting agenda: Valera will finish presenting Section 13.6 and some relevant lemmas from Part I of the book.
April 17
Speaker: Connor Stewart
Title: A concise reformulation
Meeting agenda: Connor will present Section 13.7: A concise reformulation. If there is time remaining, Connor will start presenting Chapter 14: Global Class Field Theory.
May 1
Speaker: TBD
Title: Number fields, their completions, the discriminant & the (global) Kronecker-Weber theorem
Meeting agenda: TBD
May 8
Speaker: Joseph DiCapua
Title: Formal Group Laws and Explicit Local Class Field Theory
Meeting agenda: Joe will give an introduction to Lubin-Tate formal group laws and the application to explicit local class field theory.
May 15
Speaker: TBD
Title: The local and global terminology, and some global class field theory
Meeting agenda: TBD