It seems than an analogue of the twin prime conjecture for polynomials in finite fields has been solved: see <https://www.quantamagazine.org/big-question-about-primes-proved-in-small-number-systems-20190926/>


Can one expect the perfectoid spaces introduced by Scholze, which if I understand correctly deal with spaces of "mixed characteristics" be used to translate this solution into the integers world and that way bridge the gap that separates us from a proof of the twin prime conjecture?



Can Scholze's perfectoid spaces bridge the gap for twin prime conjecture?
As far as I know, perfectoid spaces have not been used in a non-trivial way in analytic number theory. Since all the progress to twin prime conjecture so far has been mostly analytic in nature, an application of perfectoids to twin primes would have to be creative. 


The passage from finite characteristic to mixed characteristic is a thing with perfectoids but it only works if you have many roots of $p$ on one side and many roots of the indeterminate variable on the other. It is hard to imagine that this passage can interact well with the norm on the integers and whatever notion of norm you have in finite characteristic to make sense of twin primes.


