In this text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation):

In any case, contemporary mathematics provides an example of extraordinarily deep and highly studied equivalence which, without having so far been formulated as an equivalence of Morita, still seems very close to the general framework of Caramello's theory: it is the correspondence from Langlands.

I am trying to understand this remark. The entire mathematical world has produced few, if any, correspondences which unify lots of non-trivial mathematics as elegantly as the Langlands program does (other than the Langlands, that is). Several genuises (including L. and V. Lafforgue, A. Wiles, P. Scholze) have contributed to its study and our understanding of it is still far from complete.

I wonder if there were any papers confirming the overall sentiment of the author, i.e. papers where Caramello's theory is applied to Langlands. Were there at least some long-standing problems that historically were not believed to have any serious connection with logic or topoi, and were solved using Caramello's techniques?

There were some similar questions, but this one seemed to be largely about understanding an incorrect translation of a quote of another distinguished mathematician (hopefully, our translation is correct!), and this one is pretty similar but not the same (it asks about producing theorems without any creative effort, while we allow additional creative input but just want to see theorems that rely in an essential way on ideas of Caramello, among other things). The second question did receive a response that might have qualified as an answer to our question, but that response also received criticism from BCnrd (which was not addressed AFAICT).

P.S. The kind of response we are not looking for in this question:

  • some philosophical argument why topoi are great that does not give a specific resut. While such arguments can be of interest in some situations, not right now.
  • something along the lines of "One can produce thousands of deep theorems easily, so easily in fact that I won't even bother producing a single example".

The kind of response we are looking for:

  • a link to a paper proving a result outside of logic or set theory and whose statement does not involve topoi, and an explanation of how Caramello's techniques enter the proof as an important ingredient.
  • $\begingroup$ Caramello's website lists 3 papers joint with Lafforgue by my count. Presumably they contain some kind of working-out of her theoretical framework, if not directly in the area of the Langlands program, then at least somewhere in that area. $\endgroup$ – Tim Campion May 23 '19 at 14:37
  • $\begingroup$ @TimCampion I browsed through some of them and did not manage to find any results of geometric or topological interest (though those that I saw were in French, so I did not try super hard). $\endgroup$ – user140765 May 23 '19 at 14:39
  • $\begingroup$ The first two publications listed under "Publications" on her website are both papers in English written jointly with Lafforgue. In particular, the second appears to do something reasonably concrete in algebraic geometry / number theory. The first appears to be more category-theoretic, but perhaps closer inspection would reveal some algebraic geometry / number theory. $\endgroup$ – Tim Campion May 23 '19 at 14:44
  • 2
    $\begingroup$ @TimCampion OK, maybe somebody who actually read those papers will comment on them. But who knows, maybe logic will give us the motivic t-structure! $\endgroup$ – user140765 May 23 '19 at 14:47

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