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Kahler-Einstein Metrics, Extremal Metrics and Stability

Simon Donaldson Royal Society Research Professor Imperial College, London ABSTRACT In the first part of the talk we will give a general outline of the two topics in Kähler geometry in the title, both growing out of work of Calabi. We will also discuss the parallels with affine differential geometry which arise when one studies toric manifolds. We will explain the standard conjectures in the field, relating the existence of these metrics to algebro-geometric notions of "stability". In the last part of the talk we will say something about recent work with Chen and Sun which establishes this conjecture in the case of Kähler-Einstein metrics on Fano manifolds (Yau's conjecture). Recorded April 12, 2013 as part of Calabifest: The 28th Geometry Festival. A follow-up lecture series can be viewed here: http://www2.math.umd.edu/~yanir/gas.html