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.
5. Derived Stacks
Ethan Bottomley-Mason 10/13/26
6. Sheaves on Derived Stacks
Ethan Bottomley-Mason 10/20/26