What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?


2 Answers 2


I am not sure whether it is in the spirit of the original question, but let me add a wordy version of Emily's extensive and excellent bibliography -- a bit more of a road map. Let me divide to this purpose chromatic homotopy theory pseudo-historically in different phases. The years denote the "main phase"; in each case developments have continued after.

Phase 0: Prehistory -- Complex cobordism MU and the image of J (1960 - 1965)

For the computation of $\pi_*MU$, Switzer's Algebraic Topology and Lurie's Chromatic homotopy theory are good, for example. For the image of $J$, the classic paper On the groups J(X) IV by Adams is still good to have a look at.

Phase 1: Adams-Novikov spectral sequence, formal groups and greek letters (1967- 1977) This was started by Novikov's introduction of the Adams-Novikov spectral sequence; note that in the same paper a link to formal groups was already established in an appendix by Mischenko! The main points of this phase were to develop a structure theory of BP using formal groups to do computations in the Adams-Novikov spectral sequence, in particular in relation to the elements produced by Smith-Toda complexes ($\alpha$-, $\beta$- and $\gamma$-family).

  • The most comprehensive reference for this is Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres, Douglas C. Ravenel.
  • For the structure theory of $MU_*MU$ and $BP_*BP$ it is also good to look at Part II of Adams's Stable Homotopy and Generalized Homology and at Wilsons's Brown-Peterson Homology: Introduction and sampler.
  • A shorter overview to the computational aspects is also contained in Ravenel's A novice guide to the Adams-Novikov spectral sequence
  • The definitive article from this era is Miller-Ravenel-Wilson Periodic phenomena in the Adams-Novikov spectral sequence (1977)
  • For Smith-Toda complexes it is also good to look at the much later paper The Smith-Toda complex $V((p+1)/2)$ does not exist by Nave.
  • Also have a look at Goerss's The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres

Phase 1b: Unstable computations of homology with respect to $MU$ and $K(n)$ (1973-1980) This was started in Wilson's thesis and two of the main paper's of this part are the Ravenel-Wilson papers The Hopf ring for complex bordism (1977) and The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture (1980). Later expositions include Wilson's BP-sampler mentioned above and Section 2 of Hopkins-Lurie's Ambidexterity paper A nice addition to the topic was the 1994 Hopkins-Hunton paper On the structure of spaces representing a Landweber exact cohomology theory. Hill and Hopkins have also a project, extending some of the results to a $C_2$-equivariant setting.

Phase 2: Large scale phenomena (1979-1988)

This is dominated by the theory of Bousfield localization and the Ravenel conjectures. This is well explained in Lurie's chromatic homotopy theory notes. Also Ravenel's paper Localization with respect to periodic homology theories is still a very good read. Other sources include Ravenel's "orange book" and the original papers by (Devinatz), Hopkins and Smith: Nilpotence and stable homotopy theory I, II.

Phase 3: Designer spectra and new computations

The crucial step was here the Goerss-Hopkins-Miller theorem about the action of the Morava stabilizer group on the Lubin-Tate spectra (aka Morava E-theory). This allowed to define higher real K-theory and a little later also $TMF$. The higher real K-theories were used by Goerss-Henn-Mahowald-Rezk (and many others) to write down resolutions for $K(2)$-local spheres and many explicit computations were (and are) done for and using higher real K-theories. For higher real K-theories and the like some good sources are:

Phase 3b: Unstable telescopic homotopy theory (1982 - )

There is a notion on $v_n$-periodic unstable homotopy groups. For a survey of the (computational aspects of the) older literature, have a look at Davis's article in the Handbook of Algebraic Topology. Davis, Thompson and especially Mahowald were maybe the founders of the theory, but also Bousfield did an enormous amount of work here, culminating in On the $2$-primary $v_1$-periodic homotopy groups of spaces. I refer also to Kuhn's Guide to telescopic functors for the general theory.

At some point, it was discovered that Goodwillie calculus interacts very nicely with unstable telescopic homotopy theory, with Arone--Mahowald's The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres as the most important early paper. Kuhn's Goodwillie towers and chromatic homotopy: an overview combines an introduction to Goodwillie calculus with an overview For the modern aspects, the notes from the Thursday seminar contain a very good overview: crucial work here is due to Behrens--Rezk (Spectral algebra models of unstable v_n-periodic homotopy theory), Heuts (Lie algebra models for unstable homotopy theory) and Arone--Ching (mostly unpublished). Also the survey article Goodwillie calculus of Arone and Ching contains a good introduction.

Phase 4: Entering (derived) algebraic geometry (?-?)

In principle, this phase begins with the insights of Morava around 1970. These were translated by Wilson and so into more traditional language. But the insight came back by the introduction of the language of stacks into algebraic topology. Some classic sources here are:

  • Hopkins: COCTALOS
  • Naumann: The stack of formal groups in stable homotopy theory

This was much enhanced in the construction of $TMF$, where a sheaf of $E_{\infty}$-ring spectra was constructed on the moduli stack of elliptic curves. Emily has already linked some of the best sources for this, but I want to add Goerss's surveys, in particular his Bourbaki presentation. The derived algebraic geometry perspective was of course taken much more seriously in Lurie's approach, with the relevant articles again linked in Emily's answer.

A lot more could (and should) be said, especially about applications to more classical problems. The Kervaire invariant 1 problem uses some serious Phase-1 chromatic homotopy theory. Topological modular forms have been applied to construct new classes in the cokernel of $J$ and thus to obtain new exotic spheres (see e.g. the article of Wang-Xu and the preprint of Behrens-Hill-Hopkins-Mahowald). $tmf$ has also been used to obtain new results on generalized Smith-Toda complexes and thus to a better understanding of the Adams-Novikov 2-line (see the HHA article by Behrens-Hill-Hopkins-Mahowald and the Hopkins-Mahowald article published in the $TMF$-book).


Preliminaries (i.e. Advanced Algebraic Topology)

General References

  1. Advanced Algebraic Topology, Alexander Kupers;
  2. More Concise Algebraic Topology, J. Peter May;
  3. Introduction to Homotopy Theory, Paul Selick;
  4. Lecture Notes on Homotopy Theory and Applications, Laurentiu Maxim;
  5. Algebraic Topology, volume 2 of notes by Sanath Devalapurkar on lectures by Haynes Miller. See also volume 1;
  6. Modern Classical Homotopy Theory, Jeffrey Strom;
  7. 近代ホモトピー論(1940年代から1960年代まで), in Japanese, Akihiro Tsuchiya and Nakai Hirofumi;

Model Categories

  1. Part 4 of Item 2;
  2. Introduction to Homotopy Theory, nLab;
  3. Sections 2.1–2.3 of Higher Categories and Homotopical Algebra, Denis-Charles Cisinski;
  4. (Parts of) Categorical Homotopy Theory, Emily Riehl;
  5. Model Categories, Mark Hovey;
  6. Homotopical Algebra, notes taken by Sasha Patotski on a course taught by Yuri Berest;
  7. Notes on Homotopical Algebra, Zhen Lin Low;
  8. Chapters 4–6 of Equivariant stable homotopy theory and the Kervaire invariant problem, Michael Hill, Michael Hopkins, and Douglas Ravenel;
  9. Chapter 10 of Item 6;

Spectral Sequences

  1. The Adams Spectral Sequence, John Rognes;
  2. Chapter 5 of Hatcher, Allen Hatcher;
  3. A User's Guide to Spectral Sequences, John McCleary. Also available here;
  4. Spectral Sequences, Fabian Hebestreit, Achim Krause, Thomas Nikolaus;
  5. Chapter 2 of Item 4;
  6. Chapter 6 of Item 5;
  7. Part 6 of Item 6;
  8. Introduction to Spectral Sequences, nLab;
  9. Introduction to the Adams Spectral Sequence, nLab;
  10. The Serre Spectral Sequence, Maximilien Holmberg-Péroux;
  11. 代数的トポロジー, 信州春の学校 1, スペクトル系列, in Japanese;
  12. An Adams Spectral Sequence Primer, Robert R. Bruner.

Stable Homotopy Theory

  1. Notes on stable homotopy theory, Kirsten Wickelgren;
  2. Spectra and Stable Homotopy Theory, Notes by Akhil Mathew on a course delivered by Michael Hopkins;
  3. Introduction to Stable Homotopy Theory, nLab;
  4. Symmetric Spectra, Stefan Schwede;
  5. An Introduction to Stable Homotopy Theory, Maximilien Holmberg-Péroux;
  6. Chapter 2 of Item 52;
  7. Stable Homotopy and Generalised Homology, J. F. Adams;

Chromatic Homotopy Theory

  1. Handbook of Homotopy Theory, in particular the following two references;
  2. Lubin-Tate theory, character theory, and power operations, Nathaniel Stapleton;
  3. Chromatic structures in stable homotopy theory, Tobias Barthel and Agnès Beaudry;
  4. The 2013 Juvitop (on chromatic homotopy theory);
  5. The 2013 Talbot Workshop (on chromatic homotopy theory);
  6. Lurie's notes on chromatic homotopy theory, Jacob Lurie;
  7. Christopher J. Schommer-Pries's notes on Lurie's course on chromatic homotopy theory;
  8. Sanath Devalapurkar's notes on chromatic homotopy theory;
  9. Complex Cobordism andStable Homotopy Groups of Spheres, Douglas C. Ravenel;
  10. Nilpotence and Periodicity in Stable Homotopy Theory, Douglas C. Ravenel;
  11. David Mehrle's notes on the Chromatic Homotopy Theory: Journey to the Frontier workshop. Check also the notes available on the workshop webpage;
  12. Eva Belmont's notes on the 2016 West Coast Algebraic Topology Summer School on chromatic homotopy theory;
  13. The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres;
  14. Cobordism, Cohomology Theories and Formal Group Laws, Lev Kiwi;
  15. Complex Oriented Cohomology Theories and the Language of Stacks (also knows as “COCTALOS”);
  16. Aaron Mazel-Gee's notes on the 2011 West Coast Algebraic Topology Summer School;
  17. On Thom Spectra, Orientability, and Cobordism, Yuli B. Rudyak. Also available here;
  18. A Short Introduction to the Telescope and Chromatic Splitting Conjectures, Tobias Barthel;
  19. Introduction to Cobordism and Complex Oriented Cohomology, nLab;
  20. See also Item 34;
  21. Notes on Cobordism, Haynes Miller;
  22. A Survey of Computations of Homotopy Groups of Spheres and Cobordisms, Guozhen Wang and Zhouli Xu;
  23. Fall 2018 Homotopy Theory Seminar, notes by Arun Debray;
  24. Chapter 6 of Generalized Cohomology, Akira Kono and Dai Tamaki;

On Formal Groups, Formal Schemes, and Their Relation to Chromatic Homotopy Theory

  1. Formal Groups, Neil P. Strickland;
  2. Formal Schemes and Formal Groups, Neil P. Strickland;
  3. Formal Geometry and Bordism Operations, Eric Peterson. Also available on GitHub;
  4. Quasi-Coherent Sheaves on the Moduli Stack of Formal Groups, Paul G. Goerss;
  5. Lectures on Formal and Rigid Geometry, Siegfried Bosch. Also available here;
  6. Lectures on Lubin-Tate spaces, Michael Hopkins;

Miscellaneous Advanced Topics

  1. The past Juvitop seminars;$^1$
  2. Lectures on Power Operations, Charles Rezk;
  3. Notes on the Hopkins-Miller Theorem, Charles Rezk;
  4. A Guide to Tensor-Triangulated Classification, Paul Balmer;

Elliptic Cohomology and Topological Modular Forms

  1. Elliptic Homology and Topological Modular Forms, Lennart Meier;
  2. United Elliptic Homology, Lennart Meier;
  3. Supplementary Notes for Math 512 (on topological modular forms), Charles Rezk;
  4. Elliptic Cohomology and Elliptic Curves, Charles Rezk. Video recorded lectures available here;
  5. Constructing Elliptic Cohomology, Matthew Greenberg;
  6. Topological modular and automorphic forms (Part of the Handbook of Homotopy Theory), Mark Behrens;
  7. Topological Modular Forms, book by Christopher L. Douglas, John Francis, André G. Henriques, Michael A. Hill;
  8. Course Notes for Elliptic Cohomology, Michael Hopkins;
  9. $\mathrm{TMF}$ doctoral student seminar notes;
  10. Chapter 9 of Item 16;
  11. See also this reference list;

Topological Automorphic Forms

  1. Topological Automorphic Forms, Mark Behrens and Tyler Lawson;
  2. An Overview of Abelian Varieties in Homotopy Theory, Tyler Lawson;
  3. Ravenel's Seminar;
  4. Paul VanKoughnett's Seminar;

Lurie's Work on Elliptic Cohomology

  1. Spectral Algebraic Geometry, Jacob Lurie;
  2. A Survey of Elliptic Cohomology, Jacob Lurie;
  3. Notes on a Seminar on A Survey of Elliptic Cohomology, nLab;
  4. A Survey of Lurie's A Survey of Elliptic Cohomology, Aaron Mazel-Gee;
  5. Elliptic Cohomology I, Jacob Lurie;
  6. Elliptic Cohomology II, Jacob Lurie.
  7. Spectral Algebraic Geometry, Charles Rezk.

There is also this seminar. While there are no notes, one may use it as a guide, studying the topics of the talks using other sources.

$^1$ When notes are available, one should put notes_YEAR_SEASON/ between /pastseminars/ and the file name. For example, write http://math.mit.edu/juvitop/pastseminars/notes_2016_Fall/Nishida.pdf for http://math.mit.edu/juvitop/pastseminars/Nishida.pdf.

  • 1
    $\begingroup$ You might want to change the link to Schwede's book to the more recent version, especially since it is slightly harder to find. It contains more material and it is generally better $\endgroup$ Feb 25, 2019 at 16:34
  • $\begingroup$ @DenisNardin Thank you very much! I fixed it. $\endgroup$
    – Emily
    Feb 25, 2019 at 16:38
  • 2
    $\begingroup$ Wow --- thanks for this! I didn't know about many of these. (I also didn't know how to access notes from previous Juvitops, but that's due to my own lack of technological prowess...) $\endgroup$
    – skd
    Feb 26, 2019 at 22:00
  • $\begingroup$ @skd Thank you for writing some of these! $\endgroup$
    – Emily
    Feb 26, 2019 at 22:40
  • $\begingroup$ (note to self: add link when finally updating this list) $\endgroup$
    – Emily
    Jul 6, 2021 at 4:50

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