This is a learning seminar on chromatic homotopy theory organized by Azélie Picot, Sangmin Ko and myself.
Chromatic homotopy theory originates from Quillen's work on the relation between complex bordism and formal group laws. It allows us to decompose the category of spectra into "layers" which can be used to help compute various things, such as some stable homotopy groups of spheres.
Location: Room 528
Meeting Time: Thursdays 5:00pm
Syllabus: Found here
1. Introduction
Carlos Andrés Alvarado Álvarez 09/04/2025
We introduce the infinity category of spectra and notions of higher algebra that will come up during the seminar. We also give a brief application of chromatic homotopy theory and why it would be interesting to study.
2. Complex Bordism and MU
Felix Roz 09/11/2025
I will define complex-oriented multiplicative cohomology theories, formal group laws, and their relation. Then I’ll show that the venerable MU is the universal complex orientable cohomology theory corresponding to Lazard’s universal ring of formal group laws. If time permits I’ll partially prove Lazard’s and Quillen’s theorems.
3. The moduli stack of formal group laws and MU
Hechen Hu 09/18/2025
After reviewing the basic theory of stacks, I'll discuss the moduli stack of formal group laws through the lens of algebraic geometry. I'll connect them to complex cobordism via Quillen's theorem and the Adams-Novikov spectral sequence, showing how their geometry affect the structure of stable homotopy category, e.g. how cohomology theories correspond to quasi-coherent sheaves on the moduli stack. Notes available here.
4. More on Formal Group Laws
Ethan Bottomely-Mason 09/25/2025
We aim to understand the stack M_FG. We will obtain a simple answer in characteristic 0. In characteristic p, we will introduce the notions of the invariant differential and the height of a formal group law. We will discuss the relation between them as well as the relation to Frobenius for formal groups and particular generators of the Lazard ring. The height of a formal group law will allow us to obtain a filtration on M_FG in characteristic p with simple open strata. Time permitting, we will discuss modules over M_FG.
5. Landweber Exact Functor Theorem and Lubin-Tate Theories
Rafah Hajjar Muñoz 10/02/2025
In this talk we will discuss modules over the moduli stack of Formal Group Laws, and state Landweber exact functor theorem, which gives conditions for such a module to be flat. This will allow us to use FGLs to construct cohomology theories, of which we will examine some examples, such as Lubin-Tate and Morava E-theory.
6. Morava K-theory and Nilpotence
Azélie Picot 10/9/2025
We construct out of MU another important family of spectra in chromatic homotopy theory: the Morava K-theory spectra K(n). In the Nilpotence theorem, we will see how they detect nilpotent elements in ring spectra. We will also discuss the periodicity theorem and the thick subcategory theorem if time permits.
7. Descendability, Smash Product Theorem and Convergence
Sangmin Ko 10/16/2025
We review localization in stable homotopy theory and discuss the Smash Product Theorem, which states that E(n)-localization is smashing. We then turn to the Chromatic Convergence Theorem, which describes finite p-local spectra as the homotopy limit of their chromatic tower.
8. The Image of J and the K(1)-local Sphere
Azélie Picot 10/23/2025
According to the chromatic convergence theorem, the sphere spectrum can be approximated by its E(n)-localizations. In this talk, we aim to describe the first stage of this tower, i.e. L_E(1)(S). To do so, we will first compute the K(1)-local sphere and explain how the E(1) and K(1)-local sphere are related. We will also identify the homotopy groups of the E(1)-local sphere with the image of the J-homomorphism.
9. Monochromatic Homotopy Theory
Carlos Andrés Alvarado Álvarez 10/30/2025
We introduce telescopic localization and relate it to localizing at the Morava K-theories. We also highlight properties of these categories like the Bousfield-Kuhn functor.
10. Ambidexterity
Carlos Andrés Alvarado Álvarez 11/6/2025
Last time we discussed the K(n)-local and T(n)-local categories. We will now talk about the phenomena of higher semi-additivity that happens in these categories.
11. Galois Descent and K(n)-local category
Sangmin Ko 11/13/2025
We discuss Galois descent and Rogne’s theory of Galois extension of ring spectra. The Devinatz-Hopkins theorem allows for us to study K(n)-local category via Galois theory. If time permits, we discuss applications to the K(n)-local setting.
12. Goodwillie Calculus and Chromatic homotopy theory
Azélie Picot 11/20/2025
Following work of Nick Kuhn, we will discuss the interactions between the calculus of functors and monochromatic homotopy theory. I shall start with explaining the key features of Goodwillie Calculus. Then, I will explain some splitting results about the T(n)-localization of the Taylor tower. In particular, I’ll sketch how they derive from some Tate vanishing in the T(n)-local category.
13. K(1)-localized Algebraic K-theory and Étale Descent
Myungsin Cho 12/04/2025
I will give an introduction to Thomason’s work on étale K-theory, which plays an important role in the development of the Quillen–Lichtenbaum conjecture, the algebraic K-theory analogue of the Atiyah-Hirzebruch spectral sequence. Thomason’s results describe a connection between algebraic K-theory and étale cohomology and show how information can pass between these settings. In this talk, I will outline this relationship and explain how it fits into ideas from chromatic homotopy theory, with a focus on the K(1)-local viewpoint.