In this talk I will discuss the notions of an optimal symplectic connection and stability of fibrations. The former is a PDE for a canonical choice of fibrewise constant scalar curvature Kähler (cscK) metric, for a fibration whose fibres admit cscK metrics. The latter is a notion of algebro-geometric stability for such fibrations. The goal of the talk is to explain what these notions are, partly in preparation for Michael Hallam's talk the following week. I aim to explain how the notions arose, and how they specialise to the Hermite-Einstein equation and slope stability, respectively, for the underlying vector bundle, when the fibration is the projectivisation of a vector bundle. I will survey some of the results known about these notions, and how they are related. This is joint work with Ruadhaí Dervan.
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