Peter Woit quoted Atiyah in his blog ( http://scilogs.spektrum.de/hlf/sir-michael-atiyah-unity-mathematics-physics/ ) : "Tate-Shafarevich conjecture might have something to do with the 4-dimensional smooth Poincare conjecture" - it would be great if one could read more about that idea(s)?
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Do you know if there is video or a transcript of the talk? If Atiyah is just speculating, then it is unlikely that anything substantial is written up. – S. Carnahan Sep 26 '16 at 09:06
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Related: https://mathoverflow.net/questions/278288/exotic-4-spheres-and-thetate-shafarevich-group – Watson Apr 23 '18 at 20:50