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Besides the other answers, things also start to get interesting when you add some qualifiers. E.g., asking for a "differentiable manifold" now asks for manifol
by JulianWasTaken 8y ago
Besides the other answers, things also start to get interesting when you add some qualifiers.
E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.
- Topolomancer 8y agoGood point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?
- JulianWasTaken 8y agoTo be honest it's not even my field so I know a lot less than I wish I did but yeah agreed it has no shortage of interesting things to talk about :) Poincaré Conjecture is yeah interesting to explain though to be honest I vaguely recall an article in similar style to your post that explained it. Possibly something like describing gradient descent on a manifold is interesting to this audience? Or maybe a post on Flatland? Many possibilities really on good follow ups.
- amelius 8y ago> Good point Can a point be a manifold?
- Topolomancer 8y agoYes, but it's a simple one of dimension zero. There's not much to do here---every 'neighbourhood' of the point is exactly that: just the point. (not that there are different 'classes' of manifolds out there; I am not sure if the point would qualify as a Riemannian manifold, for example)