Summary: This paper is based on a talk at a conference “JDG 2017: Conference on Geometry and Topology”. We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the \(K\)-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the \(K\)-inequality to the fully faithful embedding. We consider two kinds of factorizations of birational maps between algebraic varieties into elementary ones; those into flips, flops and divisorial contractions according to the minimal model program, and more traditional weak factorizations into blow-ups and blow-downs with smooth centers. We review major approaches towards the DK hypothesis for flops between smooth varieties. The latter factorization leads to an weak evidence of the DK hypothesis at the Grothendieck ring level. DK hypothesis is proved in the case of toric or toroidal maps, and leads to the derived McKay correspondence for certain finite subgroups of \(\mathrm{GL}(n, \mathbb{C})\).
For the entire collection see [Zbl 1402.14006].