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There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for a beginner when entering certain subfields of mathematics.

For example, when I first tried to understand the Peter-Weyl theory for representations of compact topological groups, I would certainly prefer someone telling me then that I should learn the representation theory of compact quantum groups from the $C^\ast$-algebraic point of view, as it not only clarifies the theory, it also makes more apparent the links to Hopf algebras, monoidal structures and Tanakian type reconstruction theorems.

As a somewhat less specialised, perhaps more disputable example, one could argue that one should study modern category theory instead of the theory of structures as presented in the first volume in Bourbaki's éléments (although a mathematician nowadays has a much better chance of bumping into category theory instead of the theory of structure when he or she needs to formulate universal constructions).

I ask here a list of mathematical achievements which shape the modern point of view of certain classical topics in mathematics, as the examples above (first one on representation theory of compact groups, second on formulating universal constructions), in the sense that it simplifies or (and) clarifies our previous point of view on the subject, preferably making the modern research literature more accessible in establishing new standard notions, terminologies, techniques, ways of thinking et cetera.

I know this would be too broad a question, but I simply could not think of a better place than here to ask this. If possible, I'd like that each answer focuses on one topic instead of listing various topics.

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  • $\begingroup$ Maybe Grothendieck's work in algebraic geometry qualifies. $\endgroup$ – Sylvain JULIEN Dec 11 '18 at 19:04
  • $\begingroup$ I have heard that it is "better" to define manifolds in terms of sheaves rather than the traditional machinery of charts, atlases, etc. (see mathoverflow.net/questions/88056/…) but I don't know of any textbook that actually takes this approach. $\endgroup$ – Sam Hopkins Dec 11 '18 at 23:31
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    $\begingroup$ @SamHopkins Maybe this one? $\endgroup$ – Rick Sternbach Dec 12 '18 at 0:03

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