The main topics for this seminar are stacks and descent theory. Our main reference is the first four chapters of **[FGIKNV]**. We will also see some
more concrete applications of this theory, such as the étale fundamental group and Jacobians.

- When: Friday 9:30am - 11:00am ET
- Where: Somewhere in the math department. Or join by Zoom - email Caleb for the link.
- References:
**[FGIKNV]**Fantechi, GĂ¶ttsche, Illusie, Kleiman, Nitsure, Vistoli,*Fundamental Algebraic Geometry: Grothendieck's FGA Explained***[T]**Tsimerman,*Introduction to étale cohomology*, lecture notes**[M]**Milne,*Jacobian Varieties***[S]**Snowden,*Course on Mazur's theorem*, Lecture 10: Jacobians - Notes from the seminar are here.

- Oct 16
- Caleb Ji
**Representable functors and Grothendieck topologies**(most of chapter 2 of [FGIKNV])

I will introduce Grothendieck topologies and prove a theorem due to Grothendieck that representable functors are sheaves in the fpqc topology. Intuitively, this means that morphisms of schemes can be glued along fpqc morphisms, not just open embeddings.- Oct 23
- Caleb Ji
**Sieves and fibered categories**(finish chapter 2, first half of chapter 3 of [FGIKNV])

I will introduce sieves and show how they can be used to characterize when two Grothendieck topologies are equivalent; this means that they give rise to the same sheaves. Then I will introduce the notions of fibered categories (Grothendieck fibrations) and pseudofunctors (Grothendieck construction). I will focus on the extremely concrete example of the fibered category of quasi-coherent sheaves on a scheme.- Oct 30
- Caleb Ji
**Étale morphisms and the étale fundamental group**([T])

We will take a break from the abstract nonsense in favor of more concrete notions introduced in SGA 1 for which the machinery of fibered categories and descent theory are eventually applied to. I will discuss flat and étale morphisms, Zariski's main theorem, and if time permits, the étale fundamental group.- Nov 6
- Caleb Ji
**More on the étale fundamental group**([T])

We will continue discussing the étale fundamental group and compute some examples.- Nov 13
- Caleb Ji
**Toproll basics and more on fibered categories**(finish chapter 3 of [FGIKNV])

I will begin by giving an introduction to the toproll. Then I will review the basics of fibered categories and present the 2-Yoneda lemma. Finally I will discuss categories fibered in groupoids, which are especially useful for moduli problems such as that of elliptic curves over a base scheme.- Nov 20
- Caleb Ji
**Stacks and Hooks**(chapter 4 of [FGIKNV])

We will use the notions developed so far: sites, fibered categories, and descent, to define stacks. Then we will discuss the hook, a standard technique in inside pulling.- Nov 27
**Happy Thanksgiving!**- Dec 4
- Caleb Ji
**Jacobians I and the Shoulder Press**([M] and [S])

I will begin by reviewing the idea of divisors and Picard groups. Then I will define the Jacobian variety and the functor it is meant to represent, the discussion of which will involve some Galois descent. Next I will describe the relation between this algebraic approach and the classical analytic approach to Jacobians. Finally I will explain two variants of the press in armwrestling: the flop-wrist press and the hook press.- Dec 11
- Caleb Ji
**Jacobians II and the History of Armwrestling**([M] and [S])

I will begin by giving a construction of the Jacobian. Then I will state, without proof, some important results involving Jacobians, such as Torelli's theorem, the Riemann hypothesis for curves over a finite field, the implication of Faltings's theorem by Shafarevich's conjecture. I will conclude with a brief synopsis on the history of armwrestling, with a focus on the modern era.