Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples particularly enlightening, and would love to have a list of them. To make the question well-defined, my (arbitrary) cutoff for "elementary" is USA K-12. By "sophisticated" I mean 20th-21st century research mathematics, and the applications should be nontrivial enough to have appeared in research-level publications.

I list two examples I know.

The carrying operation in base 10 addition involves the computation of a group 2-cocycle, as explained in this MO answer and in this ncatlab page following (Daniel C. Isaksen, A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly (109), no. 9 (2002), p. 796--805). See also the $\mathtt{sci.math}$ posts by James Dolan back in January 1994 ($\rm\LaTeX$ version).

The passage from the poset of non-negative integers to the monoid of their differences (

*i.e.*working with expressions of the form $x-y$ where $x$ and $y$ are numbers) is the forgetful functor from a comma category $0/\mathcal{C}$ to $\mathcal{C}$ where $\mathcal{C}$ is a monoid with unique object $0$, as described by Lawvere in Section 4 of Taking Categories Seriously.

Question:What are other examples of sophisticated mathematics elucidating elementary mathematical ideas and concepts?

**Edit:** One example per answer, please. Also, if you have examples which are not research-paper content but are close then that's still interesting for me, but less so as it gets further from research level. The most interesting examples for me are those with citations to research papers, as in the listed examples.

10more comments