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Differential Geometry

Anonymous Sat 12 Oct 23:44:33 2019 No.11053658 View Reply Original Report

Quoted By: >>11053665 >>11053681 >>11053695 >>11054676 >>11054931 >>11055164

What would I learn in a course on differential geometry? Is it fun or boring math?

For example, https://sites.math.washington.edu/~lee/Courses/442-2013/

For reference, I've taken courses in abstract algebra, topology, real analysis, and combinatorics.
I enjoyed combinatorics and topology the most. Not sure if I should take "differential geometry". I'm not looking for any applications, I'm doing this purely for fun.

Anonymous

Anonymous Sat 12 Oct 2019 23:48:13 No.11053665 Report

Quoted By: >>11054545 >>11055164

>>11053658
>what do you learn in a course on differential geometry
Frenet-Serret theory for curves in R^3, the euler characteristic, genus, stuff in the theory of surfaces in R^3, intrinsic theory of surfaces, the Theorema Egregium and Gauss-Bonnet, usually.
Either that or abstract manifolds and differential forms.

Anonymous

Anonymous Sat 12 Oct 2019 23:49:19 No.11053667 Report

Quoted By: >>11053684

you learn how to dab on normies since DG has the most hieroglyphic notation accessible to undergrad that isnt something like hodge theory or algebraic geometry

Anonymous

Anonymous Sat 12 Oct 2019 23:53:03 No.11053681 Report

Quoted By: >>11054513 >>11055164

>>11053658
differential geometry is kind of a mix of multivariable calculus, real analysis, and topology. you'd learn about manifolds such as curves (1-dimensional) and surfaces (2-dimensional) and you'd learn about how we understand GLOBAL properties of curves and surfaces from LOCAL properties. for example, you can tel that a surface is not compact if you know that it has nonpositive curvature everywhere.
the main tool in differential geometry is curvature. curvature is a simple idea for curves, but for surfaces it's very interesting and deep. you can learn a lot about a surface by looking at its curvature at each point.
differential geometry has some truly beautiful results such as the gauss-bonnett theorem and stokes theorem. you'd probably learn how to prove at least one of these.
i would say it was a very fun class, it had quite a bit of computation when i took it but i learned a lot and got to enjoy some wonderful results.

Anonymous

Anonymous Sat 12 Oct 2019 23:54:13 No.11053684 Report

Quoted By:

>>11053667
it's literally just einstein summation and covariant/contravariant indices
very very intuitive
i would even call it beautiful

Anonymous

Anonymous Sat 12 Oct 2019 23:54:58 No.11053685 Report

Quoted By: >>11053691

I once took a course on curves and surfaces in R^3.
It sucked. I didn't know topology at the time so the real big boy course on Differential Geometry wasn't available to me. Though I guess it could be good for developing intuition.
I think it's better to study this topic at the proper level of abstraction, in terms of smooth manifolds, provided you're comfortable with point set topology.

Anonymous

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Anonymous Sat 12 Oct 2019 23:57:51 No.11053690 Report

Quoted By: >>11053693 >>11053696

Why take something classical like differential geometry when you can take something more contemporary, like discrete geometry?

See https://en.wikipedia.org/wiki/Outline_of_geometry

Anonymous

Anonymous Sat 12 Oct 2019 23:57:53 No.11053691 Report

Quoted By: >>11053819

>>11053685
it's fun. you sound like a pathetic algebraist. go fuck yourself, you don't belong in math.

Anonymous

Anonymous Sat 12 Oct 2019 23:58:54 No.11053693 Report

Quoted By: >>11053697

>>11053690
why take something contemporary like discrete geometry when you can take something more classical, like differential geometry?

Anonymous

Anonymous Sat 12 Oct 2019 23:59:42 No.11053695 Report

Quoted By:

>>11053658
if you enjoy topology, i think you will enjoy diff geo too

Anonymous

Anonymous Sun 13 Oct 2019 00:01:18 No.11053696 Report

Quoted By:

>>11053690
>why enjoy good sex and exquisite fine dining when you could fuck a cactus while being forced to imbibe an endless stream of emetics?
Really makes you think!

Anonymous

Anonymous Sun 13 Oct 2019 00:01:35 No.11053697 Report

Quoted By: >>11053704 >>11053832

>>11053693
What is the difference between classical and contemporary math?

Anonymous

Anonymous Sun 13 Oct 2019 00:06:20 No.11053704 Report

Quoted By:

>>11053697
classical math: soul
contemporary math: soulless

Anonymous

Anonymous Sun 13 Oct 2019 01:03:31 No.11053819 Report

Quoted By:

>>11053691
It wasn't an awful course by any means. I'm not an algebraist, but perhaps my mind has been a bit "poisoned" by abstraction, so that I have a decreased appreciation for classical things.

Anonymous

Anonymous Sun 13 Oct 2019 01:10:21 No.11053832 Report

Quoted By:

>>11053697
It really depends. I consider that there is an "immediate geometry", like Euclid's work or axiomatic projective geometry, and a "built geometry", which requires an understanding of previous things, but can still be comprehended intuitively, like complex geometry or Riemannian geometry.

Anonymous

Anonymous Sun 13 Oct 2019 08:03:02 No.11054513 Report

Quoted By: >>11054634

>>11053681
You are talking about barbarian differential geometry.

Anonymous

Anonymous Sun 13 Oct 2019 08:25:39 No.11054545 Report

Quoted By: >>11054983

>>11053665
In my course there was all of this, plus rigorous tensors and yang-mills theory. I'd recommend it. It opened my eyes as a physicist.

Anonymous

Anonymous Sun 13 Oct 2019 09:28:00 No.11054634 Report

Quoted By:

>>11054513
i'm talking about "differential geometry of curves and surfaces," the undergraduate class which only introduces n-manifolds and maybe forms and doesn't really touch full bundles / sections / sectional curvatures outside of the 2-manifold case

Anonymous

Anonymous Sun 13 Oct 2019 10:10:17 No.11054676 Report

Quoted By:

>>11053658
Differential geometry is very nice and intuitive, but in order to write it all down in symbols you have to spend time developing notation and basic lemmas. This is a big chunk of any first course on the topic, and not all teachers succeed in making it seem exciting. But stick with it and it DOES pay off.

Anonymous

Anonymous Sun 13 Oct 2019 13:36:08 No.11054931 Report

Quoted By:

>>11053658
>Is it fun or boring math?
it's fun and it's absolutely beautiful. BUT sometimes it's hard to see where the actual geometry is and you would need to look for it by yourself. but trust me that it's there and trust me that once you find it, you will be amazed.

Anonymous

Anonymous Sun 13 Oct 2019 13:59:29 No.11054983 Report

Quoted By: >>11055112

>>11054545
>he covered the theory of curves, surfaces, Cartan's differential forms and Yang-Mills theory in one semester
So this is the power of skipping all the proofs.

Anonymous

Anonymous Sun 13 Oct 2019 14:46:41 No.11055112 Report

Quoted By:

>>11054983
It was a 2 semester course.

Anonymous

Anonymous Sun 13 Oct 2019 15:12:39 No.11055164 Report

Quoted By:

>>11053658
I'm taking a course in differential geometry right now. It encompasses basically what >>11053665 and >>11053681 are saying.

I'd describe it like calc 3 done right. Or a calc 3 class where the professor can finally assume that their students are competent at analysis, topology and (linear) algebra.

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