Speaker: Jared Weinstein (Boston University)

Title: On the splitting conjecture of Hopkins

Hopkins’ splitting conjecture predicts the structure of a double localization 𝐿_𝐾(𝑡) 𝐿_𝐾(ℎ) 𝑆 of the sphere spectrum, where 𝐾(ℎ) is Morava 𝐾-theory at a prime 𝑝 and 0 < 𝑡 < ℎ. Perfectoid techniques give powerful evidence for the conjecture while avoiding explicit computation. We show (a) the conjecture is true for (ℎ, 𝑡) = (2, 1) and 𝑝 odd, recovering a difficult result of Shimomura and Yabe, and (b) for ℎ general and 𝑡 = ℎ − 1, the conjecture is true "up to perfection". This is joint work with Lucas Mann, Rin Ray, and Xinyu Zhou.