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? 



Latest results in chromatic homotopy theory
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 .


See also this [MathOverflow question](https://mathoverflow.net/questions/247526/why-do-homotopy-theorists-care-whether-or-not-bp-is-e-infty).




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References
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[Law17] “Secondary power operations and the Brown–Peterson spectrum at the prime $2$.” [arXiv:1703.00935](https://arxiv.org/pdf/1703.00935.pdf).


[May75] “[Problems in infinite loop space theory](http://www.math.uchicago.edu/%7Emay/PAPERS/16.pdf)”, 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](https://arxiv.org/pdf/1710.09822v1.pdf).


