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It is known that a physicist must have, at least in general, a good level in mathematics in order to research and understand other people's works. Linear algebra is a good example, as ODEs and calculus.

I suppose that some astrophysicists have deep knowledge of abstract mathematics and there are some applications of it (for example, I've been told that there is a connection between topology and cosmology). I wondered if there are some field in Astrophysics in which abstract algebra (i.e. fields, rings, category theory...) has been applied.

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  • $\begingroup$ I hope this soft-question is not off-topic. I've read the FAQ and it don't seem so, but if it is, I'll delete the question. $\endgroup$ – Javier Dec 16 '15 at 16:07
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    $\begingroup$ This question is definitely on topic, and I would consider it a great question on this site. You may get some comments saying you'll get better answers at Physics, but whether or not that is the case that shouldn't prevent you from asking here. In the long run, it will be to the benefit of this site if more people ask and answer questions like this here. $\endgroup$ – called2voyage Dec 16 '15 at 17:47
  • $\begingroup$ I'm not an expert on the topic, but I do know that abstract algebra is applied to spectral analysis. $\endgroup$ – called2voyage Dec 16 '15 at 17:54
  • $\begingroup$ I'm not bold enough to answer "no", so I'll just comment. Though I took some classes in abstract algebra as an undergrad, I've never seen it anywhere in a scientific presentation or lecture on astrophysics. It depends on what qualifies as "abstract algebra", though. Through their relationship with differential geometry, I'm pretty sure Lie algebras can turn up in general relativity. $\endgroup$ – Warrick Dec 17 '15 at 7:22
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There are certainly applications of number theory (as opposed to group theory/algebra - but you do also mention abstract mathematics in general) to astrophysics, though many would suggest these applications are on the boundary between real science and metaphysics.

One such application is the measure problem - if we assume our universe is one of an infinite number of multiverses how can we draw conclusions about the fundamental physics of the many multiverses, for instance to say that intelligent life is possible in a given proportion.

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For some mysterious symmetry reasons, we can describe particles with a theory that involves the representations of Lie groups (which we call the standard model). Early universe cosmology involves this mathematics to some extent, since it's thought that there should be a unification of forces from the high energies after the Big Bang.

There are also some cosmological implications to various quantum gravity theories. I've heard a lot about something called the AdS/CFT correspondence, which involves connections between algebra and geometry.

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