I'm writing my dissertation on symplectic structure-preserving algorithms for Hamiltonian systems simulation, and I'm trying to figure out how much exposition is necessary for it to be readable by scientists, engineers and related professionals.


I'm actually in love with the chapter-opening sentence "Manifolds are possibly the most interesting and powerful mathematical structure to be bypassed in a typical undergraduate course in the sciences". But I'm not sure that is literally true.



Are manifolds typically taught to undergraduates outside mathematics (and possibly theoretical physics) tracks?
I wouldn't even assume that mathematics undergrads understand manifolds. I think the majority our math majors never see the definition of one before graduating, and only a small handful could actually tell you the definition (manifold is actually a really tricky notion!). In fact, one can even finish our "graduate prepatory track" without encountering the notion. We do have a differential geometry elective which probably contains the definition, but is focused on submanifolds of Euclidean space. In the US, I think outside of a few top places, it's not regarded as a standard part of the undergraduate curriculum; manifolds and their basic properties are the first thing covered in our graduate topology class, and I don't think we assume that students have any experience with them. 


Very serious theoretical physics students may actually be more exposed to manifolds, since they take general relativity, but I doubt that actually includes much interesting exposure to the notion (since generally, they'll work in Euclidean space). 


I think it's unlikely that any other large population of undergraduates is getting exposed to them; I certainly have never heard anything that suggests so, though of course, it's hard to rule it out happening somewhere.


