TITLE: Fascinating moments: equivalent mathematical discoveries
QUESTION [63 upvotes]: One of the delights in mathematical research is that some (mostly deep) results in one area remain unknown to mathematicians in other areas, but later, these discoveries turn out to be equivalent!
Therefore, I would appreciate any recollections (with references) to:

Question. Provide pairs of theorems from different areas of mathematics and/or physics, each proven with apparently different methods, and later the two results were found to be equivalent?

The quest emanates from my firm belief that it is imperative to increase our awareness of such developments, as a matter of collective effort to enjoy the value of all mathematical heritage.
There is always this charming story about the encounter between Freeman Dyson and Hugh Montgomery.

REPLY [7 votes]: My personal favorite example of this phenomenon occurred in 1931.
On one hand, Dirac published the paper Quantized Singularities in the Electromagnetic Field which hypothesized the existence of what we would now call a magnetic monopole, and observed using quantum mechanics that the existence of such an object would necessarily imply that electric charge is quantized, meaning it can only occur in integer multiples of some base unit of charge.
On the other hand, Heinz Hopf published the paper Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche which introduced what we now know as the Hopf fibration $S^3 \to S^2$ as a nontrivial element of $\pi_3(S^2)$.  This came as a surprise since most topologists at the time, based on their experience with $\pi_1$ and with homology, assumed that $\pi_k(S^n) = 0$ if $k > n$.
These were both shocking results in their fields that opened up huge areas of future research.  There was no reason to believe at the time that the results were connected in any way, but with the advent of gauge theory much later we now know that they were secretly about the same thing.  The idea is that the monopole that Dirac wrote about would correspond to a connection on a prinicipal $S^1$-bundle over $S^2$.  Monopoles of different charges would correspond to connections on different choices of principal $S^1$-bundle, and so Dirac's observation that electric charge would necessarily be quantized corresponds to the topological fact that principal $S^1$-bundles over $S^2$ are classified up to isomorphism by the group $\pi_3(S^2)$, and by Hopf's results this group is isomorphic to $\mathbb{Z}$ with the Hopf fibration as the generator.

REPLY [2 votes]: Brouwer's fixed point theorem and the "Hex theorem". To quote Wikipedia:
"John Nash was the first to prove (c. 1949) that Hex cannot end in a draw, a non-trivial result colloquially called the 'Hex theorem', which we now know is equivalent to the Brouwer fixed-point theorem. Apparently, he didn't publish the proof. The first exposition of it appears in an in-house technical report in 1952, in which he states that 'connection and blocking the opponent are equivalent acts'. The earliest published proof of the determinacy of Hex, by David Gale in 1979, also showed that it can be used to prove the two-dimensional Brouwer fixed-point theorem, and that the determinacy of higher-dimensional variants proves the fixed-point theorem in general."