As is well known, many results in complex geometry "feel" algebraic (and often have statements which are "completely algebraic") but only have "transcendental" proofs (i.e., using analysis in some essential way). One of the promises of the newly-developed condensed mathematics is to put (almost?) all of the analysis needed in one black-box (the fact that the category of liquid vector spaces is abelian). This should allow for new proofs of the classical results which "feel" more algebraic.
In this recent course, Clausen and Scholze prove quite a lot of interesting results in this fashion. One essential omission is some form of the Hodge decomposition, whose usual proof relies on a very hard theorem on elliptic differential operators. I wonder, then, if it's reasonable to expect condensed mathematics to be able to prove something like the Hodge decomposition.
I'm pretty sure that, as of today, there's no published proof of this. So, my hopes are to receive answers on the lines of "this is an essential problem which seems very hard to solve" or "I've talked with [insert a collection of people], and they seem to have a proof in their minds". Perhaps even "oh, but this is a very simple corollary of [insert reference]". (I would be surprised, though...)
