We will continue our exploration of infinity categories from last semester and will investigate applications of the theory, particularly to derived algebraic geometry. Our ambitious goal is to shore up our foundation in the fundamentals following [H], to learn derived algebraic geometry from [L], and then to understand the results of [B]. However, we are open to focusing on a subset of this material in greater depth. Applications of infinity category theory to other fields may also be covered, depending on participant interest.

- Organizers: Amal Mattoo
- When: Monday 6:00pm - 7:15 pm
- Where: Math 528
- References:
**[H]**Vladimir Hinich,*Lectures on Infinity Categories***[L]**Jacob Lurie,*Higher Algebra*and*Derived Algebraic Geometry***[B]**Bhargav Bhatt,*Algebraization and Tannaka Duality* - Lecture notes (see below for notes for other speakers)

- January 31
- Amal Mattoo
**Organizational and Introductory Meeting**

We will review the basics of infinity categories and provide an overview of the topics in this seminar. Suggestions for topics or volunteers for talks are welcome!- February 7
- Amal Mattoo
**Crash Course on Model Categories**

We will define model categories and prove some of their key properties. Using the apparatus of cofibrantly generated model categories, we will define model structures on the categories of simplicial sets and topological spaces. Then we will examine the homotopy category of a model category and explore Quillen adjunctions and equivalences, culminating in a proof that simplicial sets and topological spaces admit equivalent homotopy categories. We aim to prove most of these results, though a few proofs will be cited from Hovey and Hinich.- February 14
- Amal Mattoo
**Model Categories and Beyond**

We will continue with last week's material, which we did not finish covering—i.e., the homotopy theory of model categories and Quillen adjunctions and equivalences. We will examine applications including homotopy (co)limits and the derived category of an abelian category.- February 21
- No meeting (Presidents' Day)
- February 28
- Amal Mattoo
**More Models of Infinity Categories**

We define models of infinity categories as simplicial categories and as complete Segal spaces. We tie these to our earlier work by introducing concepts including Dwyer-Kan localization and the Reedy model structure. We will roughly follow Sections 5 and 6 of Hinich.- March 7
- Amal Mattoo
**Infinity Categorical Yoneda and Universal Constructions**

We construct the analogue of the Yoneda embedding for infinity categories using the model of Complete Segal Spaces. To do so, we will build the theory of left fibrations. We can then define limits, colimits, and adjoints for infinity categories. We will roughly follow Sections 7 and 8 of Hinich.- March 14
- No meeting (Spring Break)
- March 21
- Amal Mattoo
**General Infinity Categories**

Zoom Link.

We wrap up our discussion of general infinity categories, now employing language independent of a choice of model. We first use Quillen adjunction to discuss maximal subspaces and infinity localization. Then we introduce cocartesian fibrations, which generalize left fibrations, and provide another approach to (co)limits. We roughly follow the end of Section 8 and Section 9 of Hinich.- March 28
- Amal Mattoo
**Introduction to Derived Algebraic Geometry**

Turning away from general infinity categories, we begin our exploration of derived algebraic geometry. We first present a brief runthrough of stable infinity categories, particularly A-infinity and E-infinity rings. Then we will define and build up some theory on simplicial rings and derived schemes. We follow Khan and Lurie.- April 4
- Amal Mattoo
**Derived Modules and Quasi-coherent Sheaves**

We provide some more detail on modules over simplicial rings and their properties. Then we define and present results on quasi-coherent sheaves over derived schemes, in particular descent. We again follow Khan and Lurie.- April 11
- Kevin Chang
**Some Features of Derived Algebraic Geometry**

In this talk, I will discuss some advantages of derived schemes over ordinary schemes. We will see how Serre's intersection formula, flat base change, and the cotangent complex arise naturally and are enhanced in the derived setting. Lecture Notes.- April 18
- Amal Mattoo
**Bhatt's***Algebrization and Tannaka Duality*

We dive into the Bhatt paper, which uses infinity-categorical derived algebraic geometry to prove concrete results about algebraic spaces and schemes. We build up a bit more general theory of derived algebraic geometry and provide a brief introduction to algebraic spaces. Then we present a powerful result giving an equivalence between hom sets and functor categories. From there we state and outline the proofs of results related to formal points, gluing, and products.- April 25
- Roy Magen
**Some Basic Stuff about Motivic Homotopy Theory**

Zoom Link.

I will present fundamental definitions and explain basic ideas from unstable motivic homotopy theory. We will see how to prove a "purity" result whose proof uses A1-homotopy and deformation to the normal cone to replace the tubular neighbourhood theorem from topology. If time permits, we will also discuss the basic ideas of algebraic K-theory (such as Bott periodicity) and algebraic cobordism from a motivic homotopy perspective, or explain some fundamentals of stable motivic homotopy theory. Main Reference: Antieau and Elmanto. Notes.- May 2
- Various speakers
**Derived Categories, Anabelian Geometry, and More!**

We will have an informal discussion sharing what we have learned at recent conferences in Cornell, Georgia, Paris, and Cambridge.