# Last Results in Chromatic Homotopy Theory

I started a PhD in Chromatic Homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where are we now with this theory.

Are Lurie's lecture notes still the best way to approach the topic?

Could you please tell me, in your opinion, which are the best/most important papers about Chromatic Homotopy that came out in the last three years?

Thank you very much!

• Lurie's notes are a great introduction to the topic, although they cover only a subset of the material, as well as Ravenel's books. A resource that only recently appeared is Eric Peterson's book, covering the link with formal geometry – Denis Nardin Feb 2 at 16:09
• This survey article came out last week, and looks exactly what you are looking for: arxiv.org/abs/1901.09004 – Charles Rezk Feb 2 at 16:42

In 2017, the long-standing problem of whether the Brown–Peterson spectrum $$\mathrm{BP}$$ admits the structure of an $$E_\infty$$-ring was shown in the negative.

The question dates back to 1975, formulated as Problem 1 of Peter May's “Problems in infinite loop space theory” [May75]:

Problem 1. Does the Brown–Peterson spectrum $$\mathrm{BP}$$ admits a model as an $$E_\infty$$-ring spectrum?

This was answered for the prime $$2$$ in Theorem 5.5.4 of [Law17] and for odd primes in Theorem 1.2 of [Sen17], which we (partly) state below:

Theorem. The $$2$$-local Brown–Peterson spectrum does not admit the structure of an $$E_n$$-algebra for any $$12\leq n\leq\infty$$.

Theorem. Let $$p$$ be an odd prime. Then the $$p$$-local Brown–Peterson spectrum does not admit the structure of an $$E_{2\left(p^2+2\right)}$$-ring .

# References

[Law17] “Secondary power operations and the Brown–Peterson spectrum at the prime $$2$$.” arXiv:1703.00935.

[May75] “Problems in infinite loop space theory”, Conference on homotopy theory (Evanston, Ill., 1974), Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975, pp. 111–125. MR 761724

[Sen17] “The Brown-Peterson spectrum is not $$E_{2 (p^2+2)}$$ at odd primes.” arXiv:1710.09822.

• This is a fantastic paper and I love it! However, I don’t think it’s really “chromatic”. – Dylan Wilson Feb 13 at 16:21
• @DylanWilson I was not very sure to post it, as I'm still studying the basics of the area. (For what it's worth, I thought it was relevant because I saw it in a footnote of Sanath Devalapurkar's notes on chromatic homotopy theory). Do you think I should delete the answer? – Théo de Oliveira Santos Feb 13 at 16:28
• I liked it and it looks relevant to me, I think it's worth to keep it ;) – Alfred Feb 13 at 17:35
• Certainly don't delete the answer! Just wanted to qualify it a bit. – Dylan Wilson Feb 13 at 18:57
• @DylanWilson and Alfred, Oh, I won't delete it then. I'm happy it's (somewhat) inside the scope of the question. Thank you both! – Théo de Oliveira Santos Feb 13 at 20:27