View AbstractAbstract: One of the deepest problems in ecology is in understanding how so many species coexist, competing for a limited number of resources. This motivated much of Darwinâs thinking, and has remained a theme explored by such key thinkers as Hutchinson (âThe paradox of the planktonâ), MacArthur, May and others. A key to coexistence, is in the development of spatial and spatio-temporal patterns, and in the coevolution of life-history patterns that both generate and exploit spatio-temporal heterogeneity. Here, general theories of pattern formation, which have been prevalent not only in ecology but also throughout science, play a fundamental role in generating understanding. The interaction between diffusive instabilities, multiple stable basins of attraction, critical transitions, stochasticity and far-from-equilibrium phenomena creates a broad panoply of mechanisms that can contribute to coexistence, as well as a rich set of mathematical questions and phenomena. This lecture will cover as much of this as time allows.