Im an applied biological math and computer science undergrad at a (the) top us public university.All the pure math classmates and professors are saying i need to take topology because like "its important or whatever" which is probably right but also my university doesn't force applied math students to take topology like the pure math kids have do. I get like two general math course electives for the degree but i really think i should spend my tuition on differential geometry instead and leave some room for functional or stochastic analysis.Are topology and differential geometry similar enough that if i take diff geo instead i can avoid the eternal shame and embarrassment of not taking it?I feel like in general they're both about applying basic analysis to sets that may not have basis. But topology is more general in trying to generalize the consequences of set closure, while diff geo is more specific to the consequences analysis has on the local properties manifolds. Because topology is so broad you have to take like a dozen grad courses to get anything that can naturally handle real life constraints while diff geo from the get go just makes standard multivariate calculus more powerful which obviously has application at the cost of not giving you insight into global properties. Am I, like, correct on that?Like im 99% certain that career-wise i would never use point set topology. But also I feel like diff geo does inherently borrow a lot of ideas from topology because they have a lot of overlap, so maybe its not the end of the world to take it in lieu of topo and just get the topology cliff notes from context? I know at some schools topology is actually a prereq to diff geo so.In the end i either>Take topo and functional/stochastic>Take diff geo and functional/stochastic>pay extra 4k to take topology just because>wait to take it during my mastersSomeone please give me clarity on this.
>>17029283Take diff geo, you should be getting an intro to enough topology to be dangerous and you can always deep dive later on with more.
>>17029283Any undergrad level diff geo class is going to be garbage.
>>17029286I tend to trust the educational content of my university's courses since half the profs have their own wikipedia pages. But yeah given the only prereq is just linear algebra and calc 3 i am suspicious about rigor.The course description is>Curves and surfaces in three-space using calculus. Curvature and torsion of curves. Curvature, covariant differentiation, parallelism, isometry, geodesics, and area on surfaces. Gauss-Bonnet Theorem. Minimal surfaces.the course seems quite popular with the physics kids toowhile topology is>Knots, orientable and non-orientable surfaces, Euler characteristic, open sets, connectedness, compactness, metric spaces. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor.fwiw i actually know the professor, they lead the geometry research project i was on.
>>17029292Okay correction, they offer a few "graduate level" courses that are a bit more advanced but i meet the prereqs:Diff geo (50% grad 50% undergrad):>Second fundamental form, Hadamard manifolds, spaces of constant curvature, first and second variational formulas, Rauch comparision theorem, and other topics chosen by the instructorIntro Topology (30% grad 70% undergrad):>Topics include metric spaces, topological spaces, continuous functions and homeomorphisms, separation axioms, quotient and product topology, compactness, and connectedness. We will also cover a bit of algebraic topology (e.g., fundamental groups) as time permits.
>>17029292That's what I thought, I went to the same place and recognize all of those course descriptions.The undergrad DG course uses something like do Carmo's book and just teaches the classical theory of curves and surfaces embedded in 3 dimensions. It's fine if you just want a fun course, but the contents are very dated (basically a snapshot of the subject pre-Riemann) and it doesn't even teach the kind of tools from modern differential geomtry that are needed for physics or anything else.The grad DG course is much more useful, but might be way too hard and I don't know why they decided to not list any prereqs. There are at least 2 versions of past course notes online if you want to see if its doable.IIRC the contents and difficulty of the topology courses are similar, but the undergrad one is an IBL course like 217 while the grad one is lecture-based, so it's more up to personal preference and whether you want to take the differential and algebraic topology courses during a matsters.
>>17029342>I went to the same place >217oh what year?>just teaches the classical theory of curves and surfaces embedded in 3 dimensionsso it's just calc 3 for big kids? Ive proven things like surface developability outside class so like>but might be way too hardim going to take 525 and maybe 547 or 526, if i do well in those can i do well in the other 500 lvl courses, or are they not equivalent in difficulty?>but the undergrad one is an IBL course like 217ewwwww
>>17029665I took 217 in 2019The undergrad version covers basic things like tangent planes, the Gauss map, the fundamental forms etc and it's meant to be doable with just calc 3 going in. The main issue is there's nothing about vector bundles, Riemannian metrics, Christoffel symbols, connection forms, covariant differentiation, or the curvature tensor which are relevant to almost any modern application of differential geomtry.It's less about difficulty than the fact that it's material very different from those other courses. I wouldn't recommend it without at least some familiarity with the earlier parts of the course already. Past notes are available at https://github.com/ARessegetesStery/MATH635-Notes/blob/main/c1/c1.pdf https://drive.google.com/file/d/18tybQ8VuSOCNYIcQDxVQBjy773sdqQHT/view?usp=sharing https://pbb.sh/posts/Notes/RiemGeo.pdf if you want to check it out>not liking IBLThere's nothing more fun than getting up to prove theorems on a whiteboard at 8:30 AM
