I will describe how to recover a Hermitian connection from the source-to-solution map of a cubic non-linear wave equation in Minkowski space; the equation is naturally motivated by the Yang-Mills-Higgs equations. Through a linearization trick, we convert the cubic non-linearity to the nonlinear interaction of three linear waves. The collision of the three waves creates new propagating singularities and makes a broken geodesic. By the microlocal analysis of intersecting Lagrangian distributions, we reduce the recovery to a geometric problem: recovering a connection from its broken non-abelian X-ray transform along light rays. This is joint work with M. Lassas (Helsinki), L. Oksanen (UCL), G. Paternain (Cambridge).

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