Good question!

Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to the effect that many objects of interest in algebraic topology do not admit any ramified covers. For example, for a $K(n)$-local $E_\infty$-ring $A$, the ring $\pi_0 A$ can never be a ramified extension of $\mathbb Z_p$. For $n=1$, this follows from $\pi_0 A$ carrying a canonical structure of a $\delta$-ring.

On the other hand, it seems that the more recent prismatic ideas have a chance of being more directly of interest. One point of overlap is that both $K(1)$-local $E_\infty$-rings and prismatic things are very closely linked to $\delta$-rings. Another point of overlap is that computations in prismatic cohomology are often done in some form via analysis of Drinfeld's stack $\Sigma$, and some kind of descent. This can often be mimicked in algebraic topology by using relative $\mathrm{THH}$ and some kind of Adams spectral sequence. See for example the work of Liu--Wang computing $\mathrm{TC}$ of rings of integers of $p$-adic fields (reproving the results of Hesselholt--Madsen). Another application of prismatic cohomology is the work of Bhatt--Clausen--Mathew showing that $L_{K(1)}K(\mathbb Z/p^n\mathbb Z)$ vanishes (since reproved by Land--Mathew--Meier--Tamme by other means -- and better, as their result applies to all chromatic heights). The idea of using relative $\mathrm{THH}$ has also (in a slightly different context) been used by Hahn--Wilson to study redshift.

So this is all in the spirit of 1) in your question. By the way, here's a weird conjecture about the relation between prisms and algebraic topology. Recall that perfect prisms $(A,I)$ are equivalent to perfectoid rings $R=A/I$, and for such the $p$-completed $\mathrm{THH}$ defines a cyclotomic spectrum concentrated in even degrees with $\pi_\ast \mathrm{THH}(R;\mathbb Z_p) = I^{\ast/2}/I^{\ast/2+1}$. I conjecture that this process can be extended to all prisms, functorially defining a cyclotomic spectrum (concentrated in even degrees) with $\pi_\ast = I^{\ast/2}/I^{\ast/2+1}$ for any prism $(A,I)$. For example, for a Breuil--Kisin prism $(\mathfrak S=W(k)[[u]],I)$ with $\mathcal O_K=\mathfrak S/I$, this ought to give the relative $\mathrm{THH}(\mathcal O_K/\mathbb S[u];\mathbb Z_p)$.

Such a construction would induce some interesting arithmetic structures on any prism. For example, by taking $S^1$-fixed points, one should get an even $E_\infty$-ring with $\pi_0 = A$, and $\pi_2$ some invertible $A$-module; the latter should be the Breuil--Kisin twist $A\{1\}$. Moreover, this even $E_\infty$-ring gives a $1$-dimensional formal group over $A$. Such a natural $1$-dimensional formal group over any prism has actually been constructed by Drinfeld (but I still haven't fully understood its significance).

Regarding 2), 3) and 4), I don't really know. I think a key open question is the computation of the Picard group of the $K(n)$-local category, which is related to the $\mathcal O_D^\times$-equivariant line bundles on the Lubin--Tate space. It's not inconceivable that perfectoid methods can be helpful here!

Good question!

Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to the effect that many objects of interest in algebraic topology do not admit any ramified covers. For example, for a $K(n)$-local $E_\infty$-ring $A$, the ring $\pi_0 A$ can never be a ramified extension of $\mathbb Z_p$. For $n=1$, this follows from $\pi_0 A$ carrying a canonical structure of a $\delta$-ring.

On the other hand, it seems that the more recent prismatic ideas have a chance of being more directly of interest. One point of overlap is that both $K(1)$-local $E_\infty$-rings and prismatic things are very closely linked to $\delta$-rings. Another point of overlap is that computations in prismatic cohomology are often done in some form via analysis of Drinfeld's stack $\Sigma$, and some kind of descent. This can often be mimicked in algebraic topology by using relative $\mathrm{THH}$ and some kind of Adams spectral sequence. See for example the work of Liu--Wang computing $\mathrm{TC}$ of rings of integers of $p$-adic fields (reproving the results of Hesselholt--Madsen). Another application of prismatic cohomology is the work of Bhatt--Clausen--Mathew showing that $L_{K(1)}K(\mathbb Z/p^n\mathbb Z)$ vanishes (since reproved by Land--Mathew--Meier--Tamme by other means -- and better, as their result applies to all chromatic heights). The idea of using relative $\mathrm{THH}$ has also (in a slightly different context) been used by Hahn--Wilson to study redshift.

So this is all in the spirit of 1) in your question. By the way, here's a weird conjecture about the relation between prisms and algebraic topology. [Edit: This conjecture is wrong. See Jacob Lurie's comments below.] Recall that perfect prisms $(A,I)$ are equivalent to perfectoid rings $R=A/I$, and for such the $p$-completed $\mathrm{THH}$ defines a cyclotomic spectrum concentrated in even degrees with $\pi_\ast \mathrm{THH}(R;\mathbb Z_p) = I^{\ast/2}/I^{\ast/2+1}$. I conjecture that this process can be extended to all prisms, functorially defining a cyclotomic spectrum (concentrated in even degrees) with $\pi_\ast = I^{\ast/2}/I^{\ast/2+1}$ for any prism $(A,I)$. For example, for a Breuil--Kisin prism $(\mathfrak S=W(k)[[u]],I)$ with $\mathcal O_K=\mathfrak S/I$, this ought to give the relative $\mathrm{THH}(\mathcal O_K/\mathbb S[u];\mathbb Z_p)$.

Such a construction would induce some interesting arithmetic structures on any prism. For example, by taking $S^1$-fixed points, one should get an even $E_\infty$-ring with $\pi_0 = A$, and $\pi_2$ some invertible $A$-module; the latter should be the Breuil--Kisin twist $A\{1\}$. Moreover, this even $E_\infty$-ring gives a $1$-dimensional formal group over $A$. Such a natural $1$-dimensional formal group over any prism has actually been constructed by Drinfeld (but I still haven't fully understood its significance).

Regarding 2), 3) and 4), I don't really know. I think a key open question is the computation of the Picard group of the $K(n)$-local category, which is related to the $\mathcal O_D^\times$-equivariant line bundles on the Lubin--Tate space. It's not inconceivable that perfectoid methods can be helpful here!

Good question!

Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to the effect that many objects of interest in algebraic topology do not admit any ramified covers. For example, for a $K(n)$-local $E_\infty$-ring $A$, the ring $\pi_0 A$ can never be a ramified extension of $\mathbb Z_p$. For $n=1$, this follows from $\pi_0 A$ carrying a canonical structure of a $\delta$-ring.

On the other hand, it seems that the more recent prismatic ideas have a chance of being more directly of interest. One point of overlap is that both $K(1)$-local $E_\infty$-rings and prismatic things are very closely linked to $\delta$-rings. Another point of overlap is that computations in prismatic cohomology are often done in some form via analysis of Drinfeld's stack $\Sigma$, and some kind of descent. This can often be mimicked in algebraic topology by using relative $\mathrm{THH}$ and some kind of Adams spectral sequence. See for example the work of Liu--Wang computing $\mathrm{TC}$ of rings of integers of $p$-adic fields (reproving the results of Hesselholt--Madsen). Another application of prismatic cohomology is the work of Bhatt--Clausen--Mathew showing that $L_{K(1)}K(\mathbb Z/p^n\mathbb Z)$ vanishes (since reproved by Land--Mathew--Meier--Tamme by other means -- and better, as their result applies to all chromatic heights). The idea of using relative $\mathrm{THH}$ has also (in a slightly different context) been used by Hahn--Wilson to study redshift.

So this is all in the spirit of 1) in your question. By the way, here's a weird conjecture about the relation between prisms and algebraic topology. Recall that perfect prisms $(A,I)$ are equivalent to perfectoid rings $R=A/I$, and for such the $p$-completed $\mathrm{THH}$ defines a cyclotomic spectrum concentrated in even degrees with $\pi_\ast \mathrm{THH}(R;\mathbb Z_p) = I^{\ast/2}/I^{\ast/2+1}$. I conjecture that this process can be extended to all prisms, functorially defining a cyclotomic spectrum (concentrated in even degrees) with $\pi_\ast = I^{\ast/2}/I^{\ast/2+1}$ for any prism $(A,I)$. For example, for a Breuil--Kisin prism $(\mathfrak S=W(k)[[u]],I)$ with $\mathcal O_K=\mathfrak S/I$, this ought to give the relative $\mathrm{THH}(\mathcal O_K/\mathbb S[u])$.

Such a construction would induce some interesting arithmetic structures on any prism. For example, by taking $S^1$-fixed points, one should get an even $E_\infty$-ring with $\pi_0 = A$, and $\pi_2$ some invertible $A$-module; the latter should be the Breuil--Kisin twist $A\{1\}$. Moreover, this even $E_\infty$-ring gives a $1$-dimensional formal group over $A$. Such a natural $1$-dimensional formal group over any prism has actually been constructed by Drinfeld (but I still haven't fully understood its significance).

Regarding 2), 3) and 4), I don't really know. I think a key open question is the computation of the Picard group of the $K(n)$-local category, which is related to the $\mathcal O_D^\times$-equivariant line bundles on the Lubin--Tate space. It's not inconceivable that perfectoid methods can be helpful here!

Good question!

Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to the effect that many objects of interest in algebraic topology do not admit any ramified covers. For example, for a $K(n)$-local $E_\infty$-ring $A$, the ring $\pi_0 A$ can never be a ramified extension of $\mathbb Z_p$. For $n=1$, this follows from $\pi_0 A$ carrying a canonical structure of a $\delta$-ring.

On the other hand, it seems that the more recent prismatic ideas have a chance of being more directly of interest. One point of overlap is that both $K(1)$-local $E_\infty$-rings and prismatic things are very closely linked to $\delta$-rings. Another point of overlap is that computations in prismatic cohomology are often done in some form via analysis of Drinfeld's stack $\Sigma$, and some kind of descent. This can often be mimicked in algebraic topology by using relative $\mathrm{THH}$ and some kind of Adams spectral sequence. See for example the work of Liu--Wang computing $\mathrm{TC}$ of rings of integers of $p$-adic fields (reproving the results of Hesselholt--Madsen). Another application of prismatic cohomology is the work of Bhatt--Clausen--Mathew showing that $L_{K(1)}K(\mathbb Z/p^n\mathbb Z)$ vanishes (since reproved by Land--Mathew--Meier--Tamme by other means -- and better, as their result applies to all chromatic heights). The idea of using relative $\mathrm{THH}$ has also (in a slightly different context) been used by Hahn--Wilson to study redshift.

So this is all in the spirit of 1) in your question. By the way, here's a weird conjecture about the relation between prisms and algebraic topology. Recall that perfect prisms $(A,I)$ are equivalent to perfectoid rings $R=A/I$, and for such the $p$-completed $\mathrm{THH}$ defines a cyclotomic spectrum concentrated in even degrees with $\pi_\ast \mathrm{THH}(R;\mathbb Z_p) = I^{\ast/2}/I^{\ast/2+1}$. I conjecture that this process can be extended to all prisms, functorially defining a cyclotomic spectrum (concentrated in even degrees) with $\pi_\ast = I^{\ast/2}/I^{\ast/2+1}$ for any prism $(A,I)$. For example, for a Breuil--Kisin prism $(\mathfrak S=W(k)[[u]],I)$ with $\mathcal O_K=\mathfrak S/I$, this ought to give the relative $\mathrm{THH}(\mathcal O_K/\mathbb S[u];\mathbb Z_p)$.

Such a construction would induce some interesting arithmetic structures on any prism. For example, by taking $S^1$-fixed points, one should get an even $E_\infty$-ring with $\pi_0 = A$, and $\pi_2$ some invertible $A$-module; the latter should be the Breuil--Kisin twist $A\{1\}$. Moreover, this even $E_\infty$-ring gives a $1$-dimensional formal group over $A$. Such a natural $1$-dimensional formal group over any prism has actually been constructed by Drinfeld (but I still haven't fully understood its significance).

Regarding 2), 3) and 4), I don't really know. I think a key open question is the computation of the Picard group of the $K(n)$-local category, which is related to the $\mathcal O_D^\times$-equivariant line bundles on the Lubin--Tate space. It's not inconceivable that perfectoid methods can be helpful here!