Short talk: The Fyodorov-Bouchaud formula and Liouville conformal field theory

Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will explain how to prove rigorously this formula by using the techniques of conformal field theory. The key observation is that the moments of the total mass of GMC on the circle are equal to one-point correlation functions of Liouville theory in the unit disk. The same techniques also allow to derive a similar result on the unit interval [0,1] (in collaboration with Tunan Zhu). Finally we will briefly discuss applications to random matrix theory and to the asymptotics of the maximum of the GFF.