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Comment by adolgert | Hacker News Reader

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic.

Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows.

Grady and Polimeni, Discrete Calculus. This is how you do calculus on graphs, which is how you do scoring algorithms on graphs. You know, for big data.

Discrete exterior calculus in Python! (almost forgot) http://arxiv.org/abs/1103.3076 They have demos that show how extraordinarily powerful it is.

That ought to keep you busy for a few months, you odd ducks.

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Wait, Bamberg wrote a book on differential forms and didn't tell me when I asked him for self-study references (he recommended Hubbard&Hubbard IIRC)? Crazy. (Context: I took a Hilbert Space class from him and only then realized what I had missed by skipping his vector calculus class for the standard intro to analysis.) Maybe he made the generous assumption that I'd already thoroughly reviewed the available books and had found his wanting?

Oh well. Either way, I'll have to check it out. I never did get far in Flanders (which left too much implicit, making it difficult to read) and H&H didn't seem to emphasize the E&M applications I was looking for (admittedly I only skimmed it). It appears that said applications are the focus of Bamberg's book, so it would likely be a better fit.

I hope Bamberg's books are better than his teaching... (I was a math concentrator..)

Where else other than HackerNews am I supposed to find random and interesting PDFs to print out and leave in my bathroom for people to read?

Treating data as a distribution in high-dimensional space is at the core of machine learning, and differentiating across those dimensions is typically how learning is done.

I don't know if this is true, but at a glance it looks to me like differential geometry could be useful to people in machine learning. Am I right about that? I am probably going to look closer tonight...

Yes, you're right about that.

take a look at manifold learning, this should point you in the right direction:

http://web.mit.edu/6.454/www/www_fall_2003/ihler/slides.pdf

It is -- there's a whole area called Information Geometry which treats the parameter spaces of statistical models as Riemannian manifolds under the Fisher information metric.

As an example application, when sampling from the posterior of a Bayesian model it can help to take this natural geometry of the parameter space into account, e.g. via Riemannian Manifold Hamiltonian Monte Carlo [1]

Then again, from what I've seen most Manifold Learning only uses the word manifold in a loose handwavey sense to motivate what they're doing -- the added abstraction level of differential geometry doesn't always add very much you're just interested in learning smooth functions from R^m -> R^n.

[1] http://www.dcs.gla.ac.uk/publications/PAPERS/9149/RMHMC_MG_B...

I thought the same thing when I saw this on the front page! I have a Ph.D. in physics and I still have only had one or two courses which delved much into differential geometry. :)

;) probably just the author :)

Exterior calculus: http://arxiv.org/abs/0907.5356

Discrete diff geom: http://mesh.brown.edu/DGP/pdfs/sg06-course01.pdf

Maybe the reason people are upvoting this is because the book uses software and programming to make the mathematics precise.

This is not a typical math book - just like Sussman's classes in which you program up physics stuff as you go along is not like a typical physics class.

I second the Bamberg and Sternberg recommendation. Those books are great introductions to differential geometry and E&M. You can go directly to part 2 if you already have a strong calculus foundation, but part 1 will give that to you and make a gentle introduction to part 2 where differential geometry is explored in more detail.

Hirani's thesis on Dec is also worth a read. Useful stuff indeed:

http://www.cs.jhu.edu/~misha/Fall09/Hirani03.pdf

What do you think of Burke's Applied Differential Geometry?

Haven't read it unfortunately. The table of contents shows a much more rapid treatment. When working outside of a university course, examples can be very useful. Anyone else read Burke?

I've read Burke. ADG is excellent, though it would be even better if it had more concrete examples. It's also very much aimed at physicists (assuming advanced undergraduate knowledge of mechanics, E&M, etc.) and has less emphasis on mathematical rigor than most differential geometry texts.

I actually helped Burke out with a follow-up to ADG called Div, Grad, Curl are Dead, which looked very promising with lots of examples, but tragically he died from injuries sustained in a car accident before it could be polished for publication.

Thanks for the reference to p. It looks like a good addition to the Thorpe I've done and the do Carmo and Lee that I am currently working through. I appreciate it. This is reasonable?

http://books.google.com/books/about/Curvature_in_Mathematics...

> Enough of you care about this to vote it to the front page?

Possibly only because of the author.

which one?

Gerald Jay Sussman, of SICP fame.

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