In this talk, I will propose a derived Torelli conjecture for hyper-Kähler varieties. Roughly speaking, we expect that the topological K-theory of a hyper-Kähler variety governs its derived category. We prove several examples supporting this Torelli conjecture, including: the D-equivalence conjecture for hyper-Kähler varieties of K3^{[n]}-type, Huybrechts’ conjecture on the derived categories of moduli spaces of sheaves on a K3 surface, and Galkin's conjecture on the derived category of Fano variety of lines on a cubic fourfold.
