Boundedness and existence of $n$-complements

This talk is about the theorem of boundedness and existence of $n$-complements of a local relative pair with boundar. The morphism of pair is supposed to be an FT contraction. The local property means that the morphism is defined over a neighborhood of (not necessarily closed) point. For the existence of $n$-complements it is sufficient the existence of numerical or $R$-complements. The boundedness means that for $n$-comlements of pairs of a fixed dimension it is sufficient a finite set of positive integers $n$. Moreover, such sets has some additional properties: divisibility and aproximation properties for irrational numbers and vectors. The latter properties implies important applications to certain questions and results about acc of some well-known invariants of log pairs. E.g., acc of the log canonical thresholds. This allows to give a new more simple proof for the finite generatedness of log canonical ring and for the existence of flips.