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Functional Differential Geometry (2012) [pdf]

groups.csail.mit.edu

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Re: Functional Differential Geometry (2012) [pdf]

#31

Post by daveloyall »

mrcactu5 wrote: post #5

without scribd http://groups.csail.mit.edu/mac/users/gjs/6946/calculus-inde... how do I read a book like this? do I copy the code and run scheme? do I translate into my favorite lisp variant? I have been recently looking into Coq which is getting a lot of attention at UPenn http://www.cis.upenn.edu/~bcpierce/sf/current/Basics.html

http://groups.csail.mit.edu/mac/users/gjs/6946/linux-install...

I followed the installation instructions exactly and it works fine on my Debian machine. I had to modify the script `/usr/local/bin/mechanics` since I don't have a binary named `xterm` on my path. While I was at it, I made it skip that and just `exec $CMD` as it would do if `$DISPLAY` weren't set.

This `edwin` that it comes with is a weird (faithful) emacs clone that runs under mit-scheme. Emacs18, that is.

I carefully typed in the first few code examples and they ran flawlessly. There even is some gui tool which pops up a rendering of mathematical expressions.

Re: Functional Differential Geometry (2012) [pdf]

#32

Post by sytelus »

This is amazing! I'd read Lagrange's eq in classical mechanics texts many times but my brain is just so tuned in thinking in terms of computations and programming that this approach in this PDF just feels natural. If lot of other things (how about General Relativity?) were written like this in terms of code, I would eat it up like buttered bread.

Re: Functional Differential Geometry (2012) [pdf]

#33

Post by TOMDM »

andyclymer wrote: post #14

What makes this book exciting to me is that "computer programming is used to explain the mathematical formulas". As someone who struggles with some areas of mathematics (the ones I wish I paid more attention to in high school because I could be using them now) but who finds more ease with understanding the code examples used in this book, I would love to read more books like this. What other books on mathematics coul…

This reminds me of a concept I have been toying with for a while now that I would love to see implemented.

What I would absolutely adore is a wiki styled over the idea of covering mathematical topics in the style of a programming API, and teach it in the same manner. Meaning that someone could use the math as a concept to kludge some code together, or teach it to themselves in this style using the material.

I've been too lazy to start the wiki myself, and not nearly knowledgeable enough to throw together to a few topics to get the ball rolling.

If anyone likes this idea, please steal it, it's something I would love to see.

Re: Functional Differential Geometry (2012) [pdf]

#34

Post by kevdevnull »

crasshopper wrote: post #28
ajtulloch wrote: post #6

I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by following just this course. For example, the Gauss-Bonet theorem doesn't appear to be covered, which is an incredibly beautiful result linking the geometry and topology of manifolds. For a more classical introduction to differential geometry requiring only multivaria…

My three cents: Lee, Spivak v1, Stillwell.

Lee's books are great as well, very different perspective, audience, motivation and development, but great. I am working through his first on topological manifolds now.

Also, at about the same level of difficulty but a different perspective:

http://f3.tiera.ru/2/M_Mathematics/MD_Geometry%20and%20topol...

Re: Functional Differential Geometry (2012) [pdf]

#35

Post by kevdevnull »

cowpig wrote: post #19
adolgert wrote: post #2

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…

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...

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

Re: Functional Differential Geometry (2012) [pdf]

#36

Post by [deleted] »

cowpig wrote: post #19
adolgert wrote: post #2

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…

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...

[deleted]

Re: Functional Differential Geometry (2012) [pdf]

#37

Post by [deleted] »

cowpig wrote: post #19
adolgert wrote: post #2

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…

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...

[deleted]

Re: Functional Differential Geometry (2012) [pdf]

#38

Post by kevdevnull »

wuliwong wrote: post #12
adolgert wrote: post #2

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…

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 :)

Re: Functional Differential Geometry (2012) [pdf]

#39

Post by kevdevnull »

crasshopper wrote: post #25
adolgert wrote: post #2

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…

> Enough of you care about this to vote it to the front page? Possibly only because of the author.

which one?

Re: Functional Differential Geometry (2012) [pdf]

#40

Post by mjw »

cowpig wrote: post #19
adolgert wrote: post #2

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…

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...

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...

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