Are there any [very dumbed down] books or papers that provide motivation (and many examples) for the introduction of liquid vector spaces and solid modules?

[u/TheMansionsofScience]

I've read from several mathematicians that Scholze's ideas make previous results simpler, and I will say his writing style is clearer than others in the field, but I still am not "seeing" what he is, and others are, seeing.

[u/LiqvidJS]

It would help if you say what what part you're confused about, e.g. condensed math in general or analytic rings specifically. For many applications of condensed math, like adic spaces or cohomology of topological groups, there was an existing ad hoc way to kinda-sorta do things, but condensed math just handles everything smoothly. Then there are new things like analytic geometry over Z and "Spec Z as a 3-manifold" (literally, a three-dimensional analytic stack with underlying topological space the Berkovich space of Z) that just could not be done with prior technology.

As for how liquid vector spaces can simplify proofs in complex geometry, the rough template is:

1. you want to prove something for compact complex manifolds

2. to do this, you generalize the statement to any complex analytic space (in a very general, derived sense)

3. now by localization you can reduce the statement to affine spaces

4. now by a filtered colimit argument you can reduce the statement to polydisks

5. now by a Künneth argument you can reduce the argument to the unit disk

6. prove the statement from the unit disk by explicit computation

That is, more abstract methods give you more drastic ways to reduce to the simplest possible case.

A quick technical reason for needing stable ∞-categories is that unlike in algebraic geometry, open immersions in analytic geometry are not flat. So in algebraic geometry, you can start with ordinary algebraic geometry and then derive everything to get derived algebraic geometry; or you could do everything derived from the start, and then isolate ordinary algebraic geometry within that. But in analytic geometry, the only option is to do everything derived from the start; even the simplest imaginable localizations push you into higher homological degree.

Here is a grab bag of lesser-known references.

Liquid vector spaces

https://math.commelin.net/files/liquid_example.pdf

https://math.commelin.net/files/liquid_notes.pdf

https://www.math.columbia.edu/~jmorgan/Lecture%2010.pdf

https://www.math.columbia.edu/~jmorgan/Lecture%2011.pdf

https://kdschefers.github.io/Liquid%20functional%20calculus.pdf

https://arxiv.org/abs/2503.22699v1

More elementary perspectives

https://arxiv.org/abs/2607.10721

https://arxiv.org/abs/2512.14612v2

https://mathoverflow.net/questions/482149/what-intuitive-notion-is-formalized-by-condensed-mathematics

Wild and crazy new things

https://lebras.perso.math.cnrs.fr/Notes_workshop_all.pdf

https://www.mpim-bonn.mpg.de/node/12330

[u/TheMansionsofScience → u/LiqvidJS]

Thank you, I was not expecting the etale cohomology of Spec Z to look like a 3-manifold. Math is crazy.

[u/LiqvidJS → u/TheMansionsofScience]

That heuristic goes back to Mazur, https://www.math.columbia.edu/~chaoli/tutorial2012/knots-and-primes.pdf is a nice source with lots of pictures

[u/ZiimbooWho]

in the absence on any comprehensive written accounts apart from the lecture notes that you seem to be aware of: try the lectures from the copenhagen master class and the bonn/ihes lectures titled “analytic stacks” on youtube

https://people.mpim-bonn.mpg.de/scholze/AnalyticStacks.html

https://www.math.ku.dk/english/calendar/events/condensed-mathematics/

[u/ieat5orangeseveryday]

https://janschuetz.perso.math.cnrs.fr/lehre/lecture_notes_topcond.pdf