Nakayama-Zariski decomposition and the termination of flips

In this talk we will discuss the termination of flips conjecture, which is one of the main open problems of the Minimal Model Program (MMP). We will first explain the notion of generalised pairs, which is indispensable in all recent progress on the MMP. We will then focus on the termination of flips for pseudoeffective projective pairs and relate the problem to the behaviour of the Nakayama-Zariski decomposition under the operations of a sequence of MMP. In particular, we will introduce a new notion called balanced sequence of MMP, encoding the good properties of the Nakayama-Zariski decomposition. We will show that the termination of a sequence of flips for pseudoeffective pairs is equivalent to the termination of a balanced sequence of MMP. This is joint work with V. Lazić.