Hodge-Laplacian $\Delta^{g_{ice}}$ acting on differential forms is essentially self-adjoint and has discrete spectrum.

Thus, by the spectral theorem [@taylor:vol2], there exists an orthonormal basis of $L^2(\Omega^p(M))$ of eigenforms $\Delta^{g_{ice}} \alpha_{j,p} = \lambda^2_{j,p} \alpha_{j,p}$. We also prove that the distribution of eigenvalues satisfies “Weyl asymptotics,” concretely, for fixed degree $p$ $$\label{eq:weyl-asymptotics}
  \# \{ j \mid \lambda^2_{j, p} < \lambda^2 \} = c_n \operatorname{Vol}(M, g_{ice}) \lambda^n +
  o(\lambda^n) \mbox{ as } \lambda \to \infty.$$ See \[sec:spectral-theory\] for the proofs of Theorem \[thm:essentiallyselfadjoint\] and of the asymptotic formula in .

Having established these fundamental properties of the Hodge-Laplacian on such spaces, we turn to the next natural topic: Hodge Theory. Here the object of study is “Hodge cohomology”, or the space of $L^2$ harmonic forms, $$\label{eq:hodge-cohomology}
  \mathcal{H}^p_{L^2}(M, g_{ice}) = {\left\{  \alpha \in L^2(\Omega^p(M), g_{ice}) \mid
    d\alpha = 0 = \delta \alpha \right\} },$$ and one phrasing of the Hodge theory problem is to find a parametrization for $\mathcal{H}^*_{L^2}(M, g_{ice})$ in terms of a topological invariant. As described in [@HHM2004], in analogous settings the relevant topological space for Hodge theoretic statements is not the manifold $M$, but the stratified space $X$ obtained by collapsing the fibration at the boundary over the base, $$\label{eq:collapsed-manifold}
  X 