soft question - Your favorite surprising connections in Mathematics

My favorite connection in mathematics (and an interesting application to physics) is a simple corollary from Hodge's decomposition theorem, which states:

On a (compact and smooth) riemannian manifold $M$ with its Hodge-deRham-Laplace operator $Delta,$ the space of $p$-forms $Omega^p$ can be written as the orthogonal sum (relative to the $L^2$ product) $$Omega^p = Delta Omega^p oplus cal H^p = d Omega^{p-1} oplus delta Omega^{p+1} oplus cal H^p,$$ where $cal H^p$ are the harmonic $p$-forms, and $delta$ is the adjoint of the exterior derivative $d$ (i.e. $delta = text{(some sign)} star dstar$ and $star$ is the Hodge star operator).
(The theorem follows from the fact, that $Delta$ is a self-adjoint, elliptic differential operator of second order, and so it is Fredholm with index $0$.)

From this it is now easy to proof, that every not trivial deRham cohomology class $[omega] in H^p$ has a unique harmonic representative $gamma in cal H^p$ with $[omega] = [gamma]$. Please note the equivalence $$Delta gamma = 0 Leftrightarrow d gamma = 0 wedge delta gamma = 0.$$

Besides that this statement implies easy proofs for Poincaré duality and what not, it motivates an interesting viewpoint on electro-dynamics:

Please be aware, that from now on we consider the Lorentzian manifold $M = mathbb{R}^4$ equipped with the Minkowski metric (so $M$ is neither compact nor riemannian!). We are going to interpret $mathbb{R}^4 = mathbb{R} times mathbb{R}^3$ as a foliation of spacelike slices and the first coordinate as a time function $t$. So every point $(t,p)$ is a position $p$ in space $mathbb{R}^3$ at the time $t in mathbb{R}$. Consider the lifeline $L simeq mathbb{R}$ of an electron in spacetime. Because the electron occupies a position which can't be occupied by anything else, we can remove $L$ from the spacetime $M$.

Though the theorem of Hodge does not hold for lorentzian manifolds in general, it holds for $M setminus L simeq mathbb{R}^4 setminus mathbb{R}$. The only non vanishing cohomology space is $H^2$ with dimension $1$ (this statement has nothing to do with the metric on this space, it's pure topology - we just cut out the lifeline of the electron!). And there is an harmonic generator $F in Omega^2$ of $H^2$, that solves $$Delta F = 0 Leftrightarrow dF = 0 wedge delta F = 0.$$ But we can write every $2$-form $F$ as a unique decomposition $$F = E + B wedge dt.$$ If we interpret $E$ as the classical electric field and $B$ as the magnetic field, than $d F = 0$ is equivalent to the first two Maxwell equations and $delta F = 0$ to the last two.

So cutting out the lifeline of an electron gives you automagically the electro-magnetic field of the electron as a generator of the non-vanishing cohomology class.

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