Algebraic geometry and its plot to take over the (analytic) world

There's been a flurry of activity recently about using algebraic geometry and category theory to replace topology as the foundations for functional analysis. In particular, Peter Scholze and Dustin Clausen have developed a program of condensed mathematics to do this (and there was a post last week about formalizing one of the main results in Lean).

As someone who is not a functional analyst, but somewhat adjacent to it, I've been reading about this and wrote a blog post with some thoughts. In short, I'm excited to see what new ideas this has to offer but also skeptical that it can efficiently handle questions that are needed for PDE analysis, so I think it is probably better suited as a companion theory rather than a full replacement.

Please let me know if you have any thoughts/ comments.

https://differentialgeometri.wordpress.com/2020/12/12/algebraic-geometry-and-its-plot-to-take-over-the-analytic-world/

Edit: Peter Scholze responded to this post. I've posted his comment as an addendum at the end of the post. It's very much worth reading.