# Are manifolds typically taught to undergraduates outside mathematics (and possibly theoretical physics) tracks? [closed]

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.

• I would definitely not assume familiarity with manifolds if you want non-(math PhD) readers to understand it. – usul Jan 14 '17 at 1:23
• how much exposition is necessary for it to be readable by scientists, engineers and related professionals: Basically, all of it. But do you really want to write a textbook on manifolds into your dissertation? If it were me, I would probably just start with "Let $M$ be a manifold. (For an introduction to manifolds, see the textbooks [3], [7], [34].)" – Nate Eldredge Jan 14 '17 at 4:57

## 1 Answer

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.

• I can imagine more computer scientists encountering and working with manifolds (related to data science, to robotic motion planning, and mathematical modelling), but it can be argued that these people would be found attending mathematics conferences as well. Gerhard "Is Mathematics Being Or Doing?" Paseman, 2017.01.14. – Gerhard Paseman Jan 14 '17 at 8:04
• @GerhardPaseman I can't say one way or the other for sure (hence the caveat at the end), but is that something undergraduates are really learning, and are they really learning about manifolds? Presumably they are mostly learning about submanifolds of $\mathbb{R}^n$, which isn't really the same. – Ben Webster Jan 14 '17 at 18:36
• Working with and really learning are two different but related activities. As an undergraduate, I think I never got the understanding of manifolds that my teachers had (even after two courses each of topology, d.g., and PDEs), but after years of non-academic experience, I think I can help a student toward that understanding, especially if it is meant toward a real world application. Gerhard "Even If I Don't Understand" Paseman, 2017.01.14. – Gerhard Paseman Jan 15 '17 at 6:36