The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times.  Such phenomena, also known as the Talbot effect, have been observed in dispersive waves, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory.  I will present recent progress on the analysis, numerics, extensions to nonlinear wave models, and the effect of coupling dispersive equations to Lamb oscillators.