A smooth cubic fourfold gives rise to two kinds of hyperkähler fourfolds: one is classical --the Fano variety of lines on the cubic; and the other is "non-commutative" --arising from the symmetric square of the Kuznetsov component. Galkin conjectured that these two objects should be derived equivalent. In this talk, I’ll explain a proof of this conjecture using ideas from matrix factorizations and window categories. This is joint work with Ed Segal.