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I recently came across these books:

Their subject matter really intrigues me, as I really enjoy topology/geometry/analysis, but had not planned to pursue them since I also want to work in an area with very concrete application. However, I am skeptical. At one point I thought topological data analysis (TDA) was the perfect marriage of my interests, but I have found very little evidence of that field actually being used in computer science, much less in industrial or otherwise more 'practical' settings. It seems like TDA makes mathematicians feel more relevant to the data science world, but I'm not convinced that it makes them so (feel free to contradict me if you think I'm wrong on this point, but note that I want a concrete use case, not an abstract argument about its relevance). I have similar stories about coding theory, certain aspects of set theory, etcetera. They may have theoretical relevance, but is there any situation where, in the process of developing software, one might need to consult theses fields? I don't know of any.

So now my question: is there any practical field of computer science that makes advanced use of differential geometry? Medical imaging, other imaging, computer graphics, virtual reality, and some other fields come to mind as potential application areas. In my (admittedly limited) experience, however, these areas seem to use basic 3D geometry, numerical linear algebra, and sometimes numerical analysis of PDEs. Those are all very nice topics, but they do not require anything as abstract as differential geometry.

Thanks in advance.

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I mainly see differential geometry applied to computer science in the following applied subfields:

  • Computer Graphics / Geometry processing
  • Machine Learning / Signal Processing

For Computer Graphics / Geometry processing, recommend looking for:

For Machine Learning /Signal Processing recommend looking for:

Also check this answer in Math exchange, and this conference Differential Geometry meets Deep Learning

Btw the Functional Differential Geometry is a great book.

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  • $\begingroup$ This is really helpful - thanks! $\endgroup$
    – user37344
    Dec 18 '20 at 4:16
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If you found Structure and Interpretation of Computer Programs interesting, you might like Functional Differential Geometry (It's from the same authors) .

Differential geometry is deceptively simple. It is surprisingly easy to get the right answer with unclear and informal symbol manipulation. To address this problem we use computer programs to communicate a precise understanding of the computations in differential geometry. Expressing the methods of differential geometry in a computer language forces them to be unambiguous and computationally effective. The task of formulating a method as a computer-executable program and debugging that program is a powerful exercise in the learning process. Also, once formalized procedurally, a mathematical idea becomes a tool that can be used directly to compute results.

Taken from Sussman, Wisdom: Functional Differential Geometry

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    $\begingroup$ This seems to be more teaching differential geometry by making use of computer science than the other way around. $\endgroup$
    – Discrete lizard
    Aug 20 '20 at 17:26
  • $\begingroup$ I agree with Discrete lizard, this doesn’t seem to be relevant to the original question. But it looks interesting nonetheless! $\endgroup$
    – user37344
    Aug 24 '20 at 11:40
  • $\begingroup$ Yeah, I just happen to stumble upon it a while back. Maybe I should add a disclaimer :) $\endgroup$
    – Bhishmaraj
    Aug 25 '20 at 7:07
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Nowadays, every field which has the name "differential" in it, is somehow applied in neural networks. For differential geometry for example you can think about differential rendering in computer graphics.

At the moment I am working with the paper "A Differential Theory of Radiative Transfer" by Zhang et al.

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  • $\begingroup$ Thanks, but do you happen to have a specific example? Differential equations (the other most prominent field that I know with differential in it) can be studied at a very “applied” level, which is quite different from studying differential geometry (at least from what I’ve seen). $\endgroup$
    – user37344
    Aug 16 '20 at 23:05
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    $\begingroup$ That's not really what differential geometry is. $\endgroup$
    – Pseudonym
    Aug 17 '20 at 13:34

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