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As a student of algebraic geometry (in an advanced stage, but still far from an expert on anything), I am quite excited about learning some condensed mathematics. I have been told that the theory has already been successfully applied, for example by Fargues-Scholze in their work on the geometrization of the local Langlands correspondence and by Lucas Mann in his work on 6-functor formalisms in rigid analytic geometry, and the idea of replacing topological spaces with a category which is more amenable to be treated with algebraic techniques sounds very appealing.

Nevertheless, I find that it is quite difficult for non-experts to get an idea of what the theory is supposed to do without delving deep into the lectures by Clausen and Scholze, and personally I don't have a clear idea of what to expect in terms of possible applications. In particular, in my studies I am leaning towards complex algebraic geometry rather than arithmetic geometry, and I would focus on trying to understand the notes on Condensed Mathematics and Complex Geometry: it is already impressing that the theory provides new "analysis-free" proofs of classical results, but I wonder if anyone expects condensed mathematics to become an useful tool for complex geometers.

From the comments to this question, I gather that Scholze and collaborators are currently thinking about these matters and that all of this is very much a work in progress, and maybe there is no need to "sell" condensed mathematics to non-experts (perhaps it's a tool whose usefulness becomes apparent only when dealing with very specific technical problems), but I believe it would be beneficial if someone could provide here an overview of what the theory is supposed to achieve. In particular:

  1. In the applications the theory has found so far, what was its role? For example, both of the recent developments I mentioned in the first paragraph go way above my head, as I am not familiar neither with p-adic geometry nor with the Langlands program, but maybe it would still be feasible to give an idea of why condensed mathematics was used there.

  2. What are some areas where condensed mathematics is expected to prove useful? What specific situations could arise where condensed mathematics would be the right answer to the problems at hand? I am particularly interested in complex geometry: condensed mathematics should provide a way of doing analytic geometry (both over the complex numbers and non-archimedean fields) which is more akin to algebraic geometry, but what are some concrete (possibly conjectural) benefits of this approach? Similarly, it seems like condensed mathematics should have applications in functional analysis, what are these expected to look like?

There are similar questions here and on MSE, but I believe the answers haven't been particularly illuminating, and given the fast paced development of the subject, it should be worthwhile to ask for an update.

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    $\begingroup$ It is not "analysis-free", but rather, the analysis is organized so that it is easier for algebraists to use (say, there is well-behaved homological algebra). The most important thing, in my opinion, is the concept of quasicoherent sheaves on, say, complex analytic spaces, which does not exist in classical setting. For example, there exists a quasicoherent sheaf on the complex plane whose support is the closed unit disk. I don't know how this could be phrased in classical setting. $\endgroup$
    – Z. M
    Commented Feb 7, 2023 at 13:35
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    $\begingroup$ Sure, the wording "analysis-free" is used in the notes (which is why it was in quotes), but clearly it is to be interpreted in the way you said. And thank you for your observation about quasicoherent sheaves, that's the kind of insight I'm looking for! $\endgroup$
    – user497019
    Commented Feb 7, 2023 at 16:37

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This is much more mundane than the main intended applications of Condensed Mathematics, but still useful. It is often interesting to consider $H^*(G;M)$, where $G$ is a profinite group and $M$ is some kind of topological abelian group with action of $G$. Here $G$ could be the Galois group of an infinite field extension (in algebraic number theory) or the Morava stabiliser group (in chromatic stable homotopy theory). There are various results about vanishing of certain cohomology groups, duality phenomena, special features of the case where $G$ is $p$-adic analytic or uniformly powerful, and the relationship with cohomology of rational Lie algebras. The traditional treatment of this story has many awkward features caused by the fact that topological abelian groups do not form an abelian category. Different theorems have a maze of different technical conditions formulated in different subcategories of coefficient modules, and it is hard to keep everything straight. It should be much cleaner to redevelop the theory using the abelian category of condensed abelian groups. As far as I know, no one has yet written out the details. However, I expect that it should not be too hard, and that a strong student could do it as a masters thesis, for example.

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  • $\begingroup$ There is a subtle difference between (topological) group cohomology and any analytic version of that, when you say $G$ is $p$-adic analytic. The later seems to be covered in this preprint. $\endgroup$
    – Z. M
    Commented Feb 8, 2023 at 15:27
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    $\begingroup$ I want to point out that this answer is more general than just group cohomology. There are other situations, especially in p-adic geometry, where you have a "topological" this-or-that and something fails to be an abelian category. E.g. the paper that @Z.M linked. Many people know that condensed math can "fix" such issues, so more applications should come out as various people need to overcome these specific technical issues. $\endgroup$
    – Jack J. Garzella
    Commented Feb 20, 2023 at 20:30

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