Complements and local singularities in birational geometry

The theory of complements was introduced by Shokurov when he investigated log flips of threefolds, and plays an important role in many areas in birational geometry, e.g. boundedness of Fano varieties, log Calabi-Yau fibrations, K-stability theory, etc. In a recent work, we prove a complements conjecture of Shokurov, and we apply this result to the study of local singularities in birational geometry. Part of this talk is joint work with J. Han and V.V. Shokurov.