The $D$-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves. I will explain how to prove this conjecture for hyperkahler varieties of $K3^{[n]}$ type (i.e. those that are deformation equivalent to Hilbert schemes of $K3$ surfaces). This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.
For more information see: https://personal.math.ubc.ca/~jbryan/Zoominar-UBC-ETH/