What networks of oscillators spontaneously synchronize?

Consider a network of identical phase oscillators with sinusoidal coupling. How likely are the oscillators to spontaneously synchronize, starting from random initial phases? One expects that dense networks of oscillators have a strong tendency to pulse in unison. But, how dense is dense enough? In this talk, we use techniques from numerical linear algebra, computational algebraic geometry, and Fourier analysis to derive the densest known networks that do not synchronize and the sparsest ones that do. We will find that there is a critical network density above which spontaneous synchrony is guaranteed regardless of the network's topology, and prove that synchrony is omnipresent for random Erdos-Renyi networks just above the connectivity threshold.