Applications of Condensed mathematics? I was reading the lectures notes by Peter Scholze about condensed mathematics and I was curious about the applications of the tools that Scholze and Clausen developed.
There are some motivational remarks in the paper but I was wandering whether there is a new result that was established with the use of condensed staff or an old theorem that has a new or simpler proof using the new mechanics.
Actually I would be happy to see any non trivial application of Condensed mathematics.
 A: This question is probably premature. In challenging the formalization community to prove a fundamental theorem in the theory, Scholze admits that there were "no real applications" as of December 2020, which was not so long ago. In a Bonn masterclass, Clausen finishes by lecturing on how to re-prove some basic theorems on Riemann surfaces in the condensed framework, though I have not watched his talks.
A: I am not an expert, but I can tell you that there are at least a few, even that were available at the time the OP asked the question. Most of these are related to the "solid" formalism, for example this paper. There's also the Fargues-Scholze local Langlands paper, which is a major development in its area.
There are a few other applications, which seem to show up in non-archimedean geometry and/or the langlands program. Put simply, condensed mathematics (in particular the solid formalism) solves a certain type of technical issue in these areas, so it's easy for someone who knows what they're looking for to know if they need it or not. I've heard people say things roughly to the effect of "I don't really understand condensed mathematics, but I know it can fix this problem in my research".
As far as applications of the "liquid" theory, I've only heard of the things in Scholze and Clausen's lectures.
