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It is my dream to do research on applications of spectral algebraic geometry in homotopy theory one day. Specifically, giving a more uniform treatment for the results proved via scary computations (of which there are plenty, from my impression). There was a post on MO on how to become sufficiently well-versed in DAG so that you can do research yourself, but it reflected the tastes of the answerer to some extent (who is a fine mathematician, but more focused on motives, it appears).

Can somebody give a similar outline, but having applications to (non-motivic) homotopy theory in mind? Since we are homotopists, the local objects for DAG we are using are connective (or $k$-periodic for some $k$, maybe?) $E_{\infty}$-ring spectra.

YCor
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  • Nothing in that answer seems to be focused on motives. Accordingly, I am voting to close as a duplicate. One thing to note is that more resources have emerged since that answer was written, so it might be worth soliciting new answers to that question, but I don't think that asking a duplicate question is the way to do that – Denis Nardin May 17 '19 at 18:29
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    Possible duplicate of Derived algebraic geometry: how to reach research level math? – Denis Nardin May 17 '19 at 18:30
  • @DenisNardin are you sure that "nothing in that answer..."? If I look specifically at the paragraph 5, it does show some bias, I think (like do chromatic guys study algebraic $K$-theory, for example?). –  May 17 '19 at 18:32
  • @schematic_boi Chromatic guys are indeed interested in the chromatic redshift conjecture. Homotopy theory is not as partitioned as you seem (?) to believe. – Denis Nardin May 17 '19 at 18:33
  • @DenisNardin OK, let me rephrase, do chromatic guys study algebraic $K$-theory of non-affine schemes (which is what seems to be featured in that article, to some extent at least)? –  May 17 '19 at 18:36
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    @schematic_boi I don't understand your point. Not everyone is interested in everything but yeah, algebraic K-theory (in all its forms) is of interest to most homotopy theorists, and especially so for those that care about DAG. That said, I'm sure Adeel's list of recommended papers is somewhat influenced by his background, but that does not appear to be an argument for why this question is not a duplicate. I've said my piece, I won't respond further. – Denis Nardin May 17 '19 at 18:38
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    OK, do not respond if you wish so. I won't try to read minds here; it would be nice if OP clarified what exactly they mean by homotopy theory, but they seem to be interested in reproving classical results of homotopy theory using DAG tools ("...more uniform treatment..."). I am not convinced that any of the articles on Adeel's list matches that description. I am therefore strongly against the closure of this as a duplicate. –  May 17 '19 at 18:42

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