I feel like I can do a decent job “dumbing down” a lot of reasonably high-level ideas in mathematics to someone with only the background of a high school student or a recreational interest in mathematics, by black-boxing some less-essential concepts and using analogies, in a way that feels fairly “honest” to me – like the idea of the concept I’ve sketched for them is a decent first approximation of the intuition one should have about the subject.
Some examples of things I think I can present decently well: Godel’s incompleteness theorems, what ZFC is, P vs NP, the Riemann Hypothesis, the Collatz conjecture, group theory, Galois theory, calculus, the idea of rigorous analysis, graph theory, complexity classes and the halting problem, most basic ideas in combinatorics, what linear algebra is and why it’s so useful. Obviously I can’t give a fully rigorous presentation of these topics in a short period of time, but most of the time I’m able to provide true information that provides some clarifying intuition and doesn’t leave the reader with glaring misconceptions.
But when it comes to topology, I’m at a bit of a loss. There are some things I can talk about semi-cogently without requiring much background – I think I did an OK presentation of local vs global properties of topological spaces here – but I don’t think I could nicely explain what a homeomorphism is or how it differs from a homotopy or an isotopy without handwaving about “rubber sheets” in a way that wouldn’t actually leave the reader with much more info than they started out with.
Any of you have thoughts on how best to talk about topology in a way that goes beyond coffee cups and donuts?