This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about perfectoid spaces. Of course, a general answer is that homotopy theorists know by now to pay attention whenever number theorists get excited about something, but I'm looking for specific examples.

Here are a few examples I know of:

  1. $THH$ and $TC$ are intimately related to the $A\Omega$ cohomology theory introduced by Bhatt-Morrow-Scholze. This is my favorite example.

  2. Perfectoid spaces feature heavily in Scholze-Weinstein's paper on moduli of $p$-divisible groups, and homotopy theorists really like $p$-divisible groups. I don't know if there are specific ideas yet on how to apply their work in homotopy theory, though.

  3. I've talked to some people interested in applying them to the study of Lubin-Tate space and the Gross-Hopkins period map.

  4. There's some speculation that it could help with $TAF$.

I don't know much about any of these examples past the first, so please feel free to expand on these examples in addition to providing new ones. I'd also be happy to hear about applications of related things like $p$-adic Hodge theory in general or the pro-étale site, even if they don't explicitly mention perfectoid spaces.

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    $\begingroup$ It is also worth pointing out that Scholze has started to use more tools from homotopy theory. I would say that my interest would lie in some connections that maybe haven't been articulated (I have no clue what they are) but homotopy theorists are frequently interested when people are start using their tools. $\endgroup$ – Sean Tilson Jun 30 '17 at 8:36
  • $\begingroup$ The paper by Vezzani arXiv:1405.4548. gives further references. I was interested to note the use there of cubical sets with connections for constructing derived functors. . There is a paper on this Patchkoria, I. ‘Cubical approach to derived functors’ Homology, Homotopy and Applications 1 (2014) 133–158. $\endgroup$ – Ronnie Brown Jul 2 '17 at 10:42
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    $\begingroup$ Thanks for asking this question! I'm curious to read responses to this question. This is slightly tangential, but a (more general?) question that I have is whether higher-dimensional formal groups can influence homotopy theory in some way. For instance, the Gross-Hopkins period map is a special case of the crystalline period map (section 6 of the Scholze-Weinstein paper). The former has had enormous success in chromotopy --- can something in chromotopy be deduced from the latter? $\endgroup$ – skd Jul 3 '17 at 15:13
  • $\begingroup$ (contd.) Higher-dimensional formal groups also pop up in homotopy theory via the Ravenel-Wilson computation of K(n)_* K(Q/Z, k); this is an exterior power of K(n)_* K(Q/Z, 1), and so the sum over all positive integers k of the Dieudonn'e modules of Spf K(n)_* K(Q/Z, k) should be the Dieudonn'e module of some abelian variety. What does the geometry of this abelian variety have to do with K(n)-local homotopy theory? (A more concrete problem, maybe, is understanding what happens at height 1.) $\endgroup$ – skd Jul 3 '17 at 15:15
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    $\begingroup$ Googling led me to chromotopy.org/hypothetical-abelian-varieties, which talks about the same thing as my previous comment. $\endgroup$ – skd Jul 3 '17 at 15:24

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