That linked PDF is great
I do love watching writers try to explain complex technical topics.1 From the PDF the article links to:
Riemann surfaces are named after the 19th century mathematician Bernhard Riemann, who was the first to understand the importance of abstract surfaces, as opposed to surfaces arising concretely in some ambient space.
I'm trying to imagine the writer who would be informed by the former clause, and enlightened by the latter. "Reimann, eh? I've always wondered who first got out of that whole ambient-space trap."
The piece actually does a pretty good job of summarizing how topology is connected to algebraic geometry, hyperbolic geometry, &c, but I really must wonder how many readers will stick through it to find out.
1No, really. I have a degree in professional writing. 's not what I do on the day job, but it's still interesting to watch.