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While we are very far from being able to completely determine which smooth compact 4-manifolds admit Einstein metrics, the problem becomes much more tractable if we restrict our attention to those 4-manifolds which also admit a symplectic structure. In this context, we now have a complete answer to the question when Einstein constant is also assumed to be non-negative, and we even know know a great deal about the negative case. In this lecture, I will present a new result regarding the question of whether the corresponding Einstein moduli spaces are connected in the positive case. If time allows, I will then survey some interesting open questions regarding the negative case. (1 hour long talk) Link: http://media.scgp.stonybrook.edu/video/video.php?f=20140826_6_qtp.mp4 Simons Center Video Archive