Nonlinear Interactions of Multiple Wave Fronts

Several physical phenomena of interest exhibit the nonlinear interaction of different kinds of waves.  In this talk, I will discuss two examples with recent mathematical progress. The first example is motivated by situations in optics where there are multiple possible speeds of light in a crystal. The second example is motivated by astrophysical situations where certain kinds of singularities, called shocks, are expected to be commonplace.  In both cases, the emphasis will be on the important role played by analyzing the geometry of the various wave fronts.