What Hacker News says

6gvONxR4sf7oNov 2020

I just looked through my bookshelves and here’s what I’ve been through since college (in no particular order other than top to bottom on my shelves):

7 books on general ML (highlights: Murphy’s Machine Learning, Hastie et al’s ESL, Koller&Friedman’s PGMs)

5 on more specialized ML (highlights: Agarwal&Chen’s statistical recommender systems book, Manning&Schutze’s statistical NLP, Settles’ active learning)

13 on stats (highlights: Wooldridge’s econometrics of x-section/panel data, Angrist&Pischke’s econometrics)

4 on numerical methods (highlight: Absil et al’s optimization on matrix manifolds)

4 on CS (highlight: CLRS’s intro to algorithms)

10 on calculus/geometry/topology/algebra (highlights: Bachman’s geometric approach to differential forms, Hestenes&Sobzyk's Clifford algebra to geometric calculus)

8 on fiction writing (highlight: Bickham’s Scene & Structure)

And Rosenberg‘s Nonviolent Communication (not a textbook, but still a highlight worth mentioning).

It amounts to between 3 and 4 per year. Looking back and counting them up, my reaction is holy crap that’s a lot, but that’s kinda the point. Each year it is a reasonable amount of self study. Not nothing, but not anything crazy. Over the course of many years, it adds up to a hell of a lever.

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cartepNov 2018

I'm not any of the two you asked, but given that @eigenspace is a physicist, what you are looking for (notations used, etc) is possibly here:

http://geocalc.clas.asu.edu/html/Evolution.html#References

under "Recent Books". Notably "Clifford Algebra to Geometric Calculus" and "New Foundations for Classical Mechanics" by Hestenes.

PD: if you ever attempt to read any of those two, I'd advise to go to the latest (corrected) reprint.

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questerzenJan 2016

Roger Penrose's Road to Reality makes use of Clifford Algebra and Grassmann Products, so at least some serious physicists are using it seriously.

Even though I had come across it before, I had a real revelation when I read David Hestenes short paper on it (from Bret Victor's website) http://worrydream.com/refs/Hestenes-ReformingTheMathematical... It is indeed beautiful to see how complex geometry and vector fields are connected via GA. I found especially revealing the relationship between electric and magnetic fields. I agree with the article that whatever its absolute merits, it would be a good way to teach new physicists. I have already ordered my copy of "Clifford Algebra to Geometric Calculus".

Whether it is actually more practical to do work in General Relativity or Quantum Mechanics, or indeed Quantum Field Theory etc., I'll have to leave for the experts in those fields. From an outsider's perspective, it is a delight to understand some of the connections between different disciplines in a more intuitive way.

In any case, I would imagine that GA is exceptionally useful for computer graphics and physical simulations.

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jacobolusNov 2016

Here’s a comment I made elsewhere recently:

* * *

You might start with teasers like Hestenes’s papers [“Reforming the Mathematical Language of Physics”][oersted], [“Grassmann’s Vision”][gvision], etc.

For a summary from a mathy perspective, try Chisolm’s [book-like thingy on arxiv][chisolm]

If you want a whole book of concrete problems to solve at an advanced undergraduate physics student level, try Hestenes’s [New Foundations for Classical Mechanics][nfcm]. This is the best source I’ve seen anywhere about understanding complex rotations in mechanics problems.

If you want to solve some plane geometry problems, this thing looks fairly accessible [Treatise of plane geometry through the geometric algebra][planegeo] (I haven’t looked too closely).

Two websites are [geocalc.clas.asu.edu][asu] and [geometry.mrao.cam.ac.uk][cambridge]. Also see the [link page at The Net Advance of Physics][netadvance], and this [mirror of Lounesto’s site][lounesto] (he passed away a while back). Lounesto liked to publish [collections of counterexamples][counterexamples].

There are a number of journals, conferences, etc. Try a google search for “Clifford Algebra”.

If you want an introductory undergraduate textbook which tries to teach both geometric algebra and traditional matrix algebra, you could look at [MacDonald][]’s [Linear and Geometric Algebra][laga]; I don’t think Hestenes is really on board with this approach, but there’s not too much else pitched at a similar audience. MacDonald also has a book [Vector and Geometric Calculus][vagc] which I suspect will be substantially easier to work through than Hestenes and Sobczyk’s [Clifford Algebra to Geometric Calculus][cagc]. (I haven’t looked at either of MacDonald’s books)

If you’re interested in geometric modeling for robotics, computer graphics, computer vision, or similar, check out [these papers][invkinematics] and then look at the book [Geometric Algebra for Computer Science][gacs]. The “conformal geometric algebra” model proposed there is pretty neat. Also see [these papers][unifalg] about related topics.

If you’re interested in crystallography, check out the papers [“Point Groups and Space Groups in Geometric Algebra”][crystalsymmetry] and [“The Crystallographic Space Groups in Geometric Algebra”][crystalga].

If you’re interested in Lie theory or representation theory, check out this paper [“Lie Groups as Spin Groups”][lgasg].

If you’re a physicist / physics student / electrical engineer / etc., try the book [Geometric Algebra for Physicists][gap] or perhaps [Understanding Geometric Algebra for Electromagnetic Theory][gaet]

For a slightly different perspective, take a look at [Sobczyk][]’s [New Foundations in Mathematics: The Geometric Concept of Number][nfm]. Sobczyk has some interesting papers about representing geometric algebras using (real or complex-valued) matrices, which might be helpful if you want to write fast numerical code on current computers, which have been optimized to do matrix math.

If you’re interested in the history, I recommend Crowe’s 1967 [History of Vector Analysis][hva].

I believe you can find the collected mathematical papers of Clifford online if you do a google search, or buy a used copy of a nice version published by Chelsea in the 1960s; the recent AMS reprint is awful quality.

Grassmann’s two mid-19th century Ausdehnungslehre books have been relatively recently translated into English, as has Peano’s late-19th century book Geometic Calculus, all three by Kannenberg.

[oersted]: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

[gvision]: http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

[chisolm]: https://arxiv.org/abs/1205.5935

[nfcm]: http://geocalc.clas.asu.edu/html/NFCM.html

[planegeo]: http://web.archive.org/web/20011215062737/http://campus.uab....

[asu]: http://geocalc.clas.asu.edu/

[cambridge]: http://geometry.mrao.cam.ac.uk/

[netadvance]: http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html

[lounesto]: https://users.aalto.fi/%7Eppuska/mirror/Lounesto/

[counterexamples]: https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexampl...

[MacDonald]: http://faculty.luther.edu/~macdonal/

[laga]: http://faculty.luther.edu/~macdonal/laga/index.html