Front propagation into unstable states plays an important role in organizing structure formation in many spatially extended systems. When a trivial background state is pointwise unstable, localized perturbations typically grow and spread with a selected speed, leaving behind a selected state in their wake. A fundamental question of interest is to predict the propagation speed and the state selected in the wake. The marginal stability conjecture postulates that speeds can be universally predicted via a marginal spectral stability criterion. In this talk, we will present background on the marginal stability conjecture and present some ideas of our recent conceptual proof of the conjecture in a model-independent framework focusing on systems of parabolic equations.

 

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