This is a learning seminar on derived algebraic geometry coorganized by Ethan Bottomley-Mason, Peter Moody and myself.
Our hope is for us to first get familiar with the language of Infinity Categories
and then use it to develop the modern theory of Derived Algebraic Geometry. Towards the later part of the semester our goal will be to give applications
based on the participants interests (e.g. p-adic Hodge Theory, Enumerative Geometry, etc.).
Location: Room 622
Meeting Time: Tuesdays 2:00pm-4:00pm EST
1. Infinity Categories: Introduction
Carlos Andrés Alvarado Álvarez 09/15/26
We give an introduction to the theory of infinity categories modeled by Joyal's quasicategories. We will introduce many concepts such as homotopy categories, homotopy (co)limits and talk about different types of fibrations and other tools that often show up.
2. Infinity Categories: Important Theorems
Carlos Andrés Alvarado Álvarez 09/22/26
We introduce but don't prove a lot of important theorems and definitions. In particular, we hope to cover the Straigthening-Unstraightening equivalence, Yoneda Lemma and also we hope to talk about infinity operads and monoidal structures in general.
3. Infinity Categories: Presentability
Carlos Andrés Alvarado Álvarez 09/29/26
We define presentable infinity categories and state the Adjoint Functor Theorem. We then define the infinity category of spectra and give the definition of a stable infinity category. Throughout we will talk about compact objects and Ind-completions and other constructions that will show up in the future.
4. Derived Commutative Algebras
Peter Moody 10/06/26
In this talk we introduce the theory of derived commutative rings in their incarnation as commutative differential graded algeras. As an application we compute examples of derived fibre products.
5. Derived Stacks
Ethan Bottomley-Mason 10/13/26
We will aim towards understanding classes of derived algebraic stacks. We start with affine derived schemes and clarify its infinity categorical structure. We introduce prestacks and their various properties including truncatedness and connectedness along with methods of constructing them. We then consider descent and general stacks before moving on to more specialized stacks (e.g. derived schemes and derived Artin stacks). Along the way we will see examples of genuine non-classical moduli problems.
6. Sheaves on Derived Stacks
Ethan Bottomley-Mason 10/20/26
We will discuss QCoh and IndCoh. We start by constructing QCoh and IndCoh and their t-structures for arbitrary prestacks and almost finite type schemes respectively and see where classical attempts fail. We then discuss functors between QCoh and IndCoh. We will construct the pullback and pushforward where we will see the first hints of IndCoh's niceness. We end by discussing rigidity properties of QCoh and the symmetric monoidal structure.