TITLE: What are the potential applications of perfectoid spaces to homotopy theory?
QUESTION [44 upvotes]: 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:

$THH$ and $TC$ are intimately related to the $A\Omega$ cohomology theory introduced by Bhatt-Morrow-Scholze. This is my favorite example.
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.
I've talked to some people interested in applying them to the study of Lubin-Tate space and the Gross-Hopkins period map.
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.

REPLY [3 votes]: As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G/H} g\cdot x$.

$q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)


So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.