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I believe that if someone is going to continue their studies and doing research on Differential Geometry's topics, would never need advanced Abstract Algebra (or maybe not even undergraduate level of it). But why in any graduate program, students must take advanced Abstract Algebra (for example the book Algebra by Hungerford) as a compulsory and common course for any pure mathematics program?

I don't think even advanced Abstract Algebra is needed for Real Analysis (?)

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    $\begingroup$ I do differential geometry and frequently use abstract algebra. But this is not the point of such a requirement. It is to make you a more rounded mathematician; I absolutely agree that every mathematician should know the rudiments of topology, differential geometry, analysis, algebra... even if they don't use them on a daily basis. $\endgroup$ – user98602 Aug 11 '15 at 15:11
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    $\begingroup$ Lie groups is an important part of differential geometry. $\endgroup$ – nonlinearism Aug 11 '15 at 15:12
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    $\begingroup$ I believe a broad basic knowledge is fundamental to be a good mathematician. You might not use any result from a certain area, but learning those points of view and proof techniques might be extremely helpful in a different context. I believe the best of mathematics comes when two areas meet, and that wouldn't be possible without a good knowledge of different areas. $\endgroup$ – Silvia Ghinassi Aug 11 '15 at 15:20
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    $\begingroup$ I also think that differential geometry uses more advanced abstract algebra than you might think. Also, not only advanced algebra is compulsory, so what about analysis, geometry or topology ? Certainly we can find research areas not using much of it. $\endgroup$ – Dietrich Burde Aug 11 '15 at 15:22
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    $\begingroup$ @MikeMiller: I agree that it's good to be a 'rounded mathematician' but it's in the case that the person wouldn't forget 'the other' learnt fields since they won't be used so they will most probably be forgotten. $\endgroup$ – user231343 Aug 11 '15 at 15:31
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You will most likely be involved in teaching as well as research in all but a few cases. And many fields can overlap considerably! These core areas vary from school to school, but as someone with a graduate degree in math, you are generally expected to know something from various subjects and not just your own. For example, I am a discrete mathematician, and haven't really used analysis much even though I was required to take it. But I have taught Calculus several times, and I'm expected to have a rigorous understanding of the subject even if I'm not teaching delta-epsilon proofs.

Also as the comments have suggested, you will come across quite a bit of algebra in studying certain aspects of differential geometry. There are quite a few connections to algebraic geometry, even an area called algebraic differential geometry.

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