Does anyone know how condensed mathematics would fit into the modern theory of PDEs (which is heavily based on functional analysis)? Perhaps it's a relic of the sort of math Scholze works on, but it looks far too abstract to provide an impetus for people in those fields to embrace it. Topology, on the other hand, is relatively easy to define and work with (though there are some quirks with dual spaces of continuous linear functionals I've seen aesthetic objections to). Or does it "contain" topology in some sense, allowing people to continue working with notions of convergence obtained from norms?
I think that right now it is not clear why condensed/liquid mathematics would be useful for PDEs. On the other hand, your question
> Or does it "contain" topology in some sense, allowing people to continue working with notions of convergence obtained from norms?
has a positive answer. You can, if you want, swap out topological spaces, and use condensed sets instead, and just continue with life as usual.
At the same time, all of this is in fast paced development, so hopefully we will see some killer apps in the near future. But I expect them more in the direction of Hodge theory and complex analytic geometry.
Thanks for sharing your expertise. Would you be open to sharing your background? Obviously it's not required, but it would help contextualize what you're saying for the interested non-mathematician; otherwise we're kinda stuck with 'some guy on the Internet said ...' syndrome. :)
Hi! Imagine for a moment that your next project required you to develop a lot of functional analysis and PDE theory in Lean. Would you be tempted to build that on top of what you've done (or will have done) with condensed sets?
Right now, I think I would go for that classical approach, simply because there is more supporting material for that in the library, and there are more people who understand that approach and can help contributing. (I'm talking specifically about functional analysis and PDEs here.)
On the other hand, it should be a lot of fun to see if we can formalize the new proof by Clausen--Scholze of Serre duality. Using their machinery, the proof should simplify a lot. But this requires building complex analytic manifolds on top of condensed mathematics. So we would first need to set up those foundations.
