Scattering configuration spaces
  For a compact manifold with boundary $X$ we introduce the   $n$-fold scattering stretched product $X^n_{\text{sc}}$ which is a compact manifold with corners for each $n,$ coinciding with the previously known cases for $n=2,3.$ It is constructed by iterated blow up of boundary faces and boundary faces of multi-diagonals in $X^n.$ The resulting space is shown to map smoothly, by a b-fibration, covering the usual projection, to the lower stretched products. It is anticipated that this manifold with corners, or at least its combinatorial structure, is a universal model for phenomena on asymptotically flat manifolds in which particle clusters emerge at infinity. In particular this is the case for magnetic monopoles on $\mathbb{R}^3$ in which case these spaces are closely related to compactifications of the moduli spaces with the boundary faces mapping to lower charge idealized moduli spaces. 
Richard Melrose and Michael Singer