I hope this question is not too broad to be asked here; if it is, please feel free to close the question.
I'm currently near the end of my masters studies and subsequently search for a particular field of maths I want to focus on in the near future.
Recently, the idea of mingling homotopy theory with arithmetic geometry has been very inspiring to me; be it through the anabelian conjectures of Grothendieck, their natural generalization to étale topoi or étale homotopy theory...
What I feel I'm still missing though is a more general 'overview' of what maths is part of 'arithmetic homotopy theory/homotopical arithmetic geometry', what expectations people have in those fields et cetera.
What (other) parts of mathematics are part of the circle of ideas trying to inject homotopy theoretic machinery into arithmetic questions?
What kind of applications might one hope to achieve by these methods? I.e., what are some interesting (open) conjectures people are or have been working on within the area?
Given that $\infty$-categories (in the sense of lurie) are kind of made to build homotopy theory into the very foundation mathematics, it seems very plausible to me that they should come up in research of homotopical arithmetic geometry; hence I was wondering:
- Are $\infty$-categorical methods entering the field recently and what would be good examples of research in this area using $\infty$-categories.
I'd be really interested in any thoughts on these questions: (other) related papers, descriptions of conjectures, points to the literature, ...