Yves Andre has proved the direct summand conjecture, using perfectoid methods.
Update (Sept. 7): Andre’s paper is here, and Bhatt’s shorter proof is here.
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totallydisconnected 
I heard that Gabber has doubts about the proof being correct ? On this subject have you heard anything new about Mochizuki’s paper being accepted ?
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My information is up-to-the-minute, and I’m confident in my source. 🙂
Nothing new about Mochizuki.
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