Talk information will be posted on the Pure Mathematics department page as well as on this page.

The three of us are planning to alternate giving talks on class field theory. Class field theory is a major branch of algebraic number theory that describes the structure of

We will be assuming standard background in Galois theory, algebraic number theory, and likely valued fields, etc. So far, it looks like we will be basing the seminar off Neukirch's excellent book

More references (motivation/"big picture"): Class Field Theory Summarized; A Brief Summary of the Statements of Class Field Theory.

Thorough notes will be posted. You can expect them to be posted here each week, usually sometime Friday evening (after both talks have been given).

**Thu Jan 15**: Michael Baker, "Global theory of algebraic number fields" (notes shall be edited/expanded soon)

In this talk, I will embark on an informal review of the global theory of algebraic number fields: integrality, discriminants and integral bases, Dedekind domains, fractional ideals and the ideal class group, Minkowski theory, Dirichlet's unit theorem, splitting of primes, as time permits. Motivation for class field theory will be discussed.**Fri Jan 16**: Raymond Cheng, "Local theory of algebraic number fields"

Having reviewed the global aspects of algebraic number theory, I will begin to develop the local aspects of the theory. The goal here will be to define and hopefully get to grips with objects known as local fields. To that end, I will speak about absolute values on fields, completions of fields and properties of local fields. Depending on the audience, I may also talk more about the algebraic constructions required for our efforts.**Thu Jan 22**: Raymond Cheng, "Local theory of algebraic number fields II"

We will continue our discussion of the local aspects of algebraic number theory. After a brief look at what completions of fields actually look like, we will figure out how absolute values extend in finite field extensions. This will allow us to classify the completions of algebraic number fields. Once all this is done, we will finally discuss local fields and their properties in a little more depth.**Fri Jan 23**: cancelled due to Grad Student Colloquium.**Thu Jan 29**: Michael Baker, "Group cohomology (and why)"

After a brief recap of the first two weeks of material, I will give a more refined "road map", or big picture, of where we are going. I will then introduce group cohomology,~~after a hopefully satisfactory explanation of how it will move us closer to our ultimate goals~~.**Fri Jan 30**: Ritvik Ramkumar, "Group cohomology II" (this talk and the previous one followed this treatment)

Definition of $G$-modules. The integral group ring $\mathbb{Z}[G]$ and some important ideals. Exact sequences. Free $G$-modules. Hom functors and their exactness properties.

I could have easily gone on for an hour about any one of these topics (there's a concept of "comma category" which would have made my exposition of limits a little more slick, but I knew I didn't really have time to go quite that far). If you keep your eyes peeled, you will start seeing examples of these concepts in all the mathematics you learn (not just in the talks next week).

**Sat Jan 31**: Michael Baker, "Category theory: a framework for reasoning" (somewhat abriged notes courtesy of Ilia Chtcherbakov; video courtesy of Amanda Chan)

Categories, initial and terminal objects, mono and epi morphisms, universal properties, functors, natural transformations, limits, adjunctions, the Yoneda embedding, and monoidal structures.

**Thu Feb 5**: Ritvik Ramkumar, "Group cohomology and its properties" (switching to Milne's CFT notes as of now; Ritvik will post notes here eventually)

We will start by discussing injective $G$-modules. Then given a $G$-module $M$ we will define the $k$th cohomology group of $G$ with coefficients in $M$ i.e. $H^k(G,M)$ via injective resolutions and discuss some of its functorial properties. We will also give an explicit construction by means of cochains.~~We will end by computing various examples of cohomology groups. As a consequence we will prove Hilbert's theorem 90~~.**Fri Feb 6**: Ritvik Ramkumar, "Tate cohomology and local class field theory"

Continuing from where we left off on Thursday, we will define the crucial Tate cohomology groups and discuss its properties. We will then compute the cohomology of finite cyclic groups.~~Finally we will prove Tate's theorem. If time permits I will state the main theorems of local class field theory and describe how the theory we developed can be used to prove them~~.**Thu Feb 12**: Raymond Cheng, "Tate's theorem and local class field theory" (video)

We shall finally make contact with number theory again. I will begin off by stating the main theorems of local class field theory and by briefly sketching how the theorems will be proved with the tools we have developed. With that in mind, I will review the construction of the Tate cohomology groups and then prove some important properties about them. In particular, I shall discuss Tate's Theorem.**Fri Feb 13**: cancelled.**Thu Feb 19**: cancelled due to Reading Week.**Fri Feb 20**: cancelled due to Reading Week.**Thu Feb 26**: Raymond Cheng, "TBA" (video)

TBA (I will move towards establishing the existence of the local Artin map).**Fri Feb 27**: cancelled.