On a (compact and smooth) riemannian manifold $M$ with it's its Hodge-deRham-Laplace operator $\Delta$ \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.$\delta = \text{(some sign)} *d*$and$*$is the Hodge star operator). (The theorem follows from the fact, that$\Delta$is a selfadjointself-adjoint, ellipitc elliptic differential operator of second order, and so it is fredholm 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 represantative 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 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. 1 [made Community Wiki] 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 it's 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.$\delta = \text{(some sign)} *d*$and$*$is the Hodge star operator). (The theorem follows from the fact, that$\Delta$is a selfadjoint, ellipitc 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 represantative$\gamma \in \cal H^p$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.