I'd like to get a better picture of how infinity/derived techniques become more important in algebraic/arithmetic geometry.
I'd therefore like to know: What questions/subfields of algebraic/arithmetic geometry seem to (a) benefit the most of using infinity categories and derived AG techniques and (b) which questions/subfields seem to be especially well-suited to get "revolutionized" by the homotopical approach to mathematics, even though it may not yet happening at the moment.
I'm thinking of results like the Exodromy paper and the SHIFTED project in general (and I'd be very interested to get to know what's moreover planned in these cases).
For example: is it to be expected that these techniques could play an important role in anabelian(-ish) questions of arithmetic geometry, like for example some results of Exodromy suggest?