Random distances of Liouville quantum gravity: recent and very recent progresses

In this talk I will review some recent and very recent progress on random metric associated with Liouville quantum gravity with focus on the construction and the phase transition. The talk is based on works with Julien Dubédat, Alexander Dunlap, Hugo Falconet, Subhajit Goswami, Ewain Gwynne, Avelio Sepúlveda, Ofer Zeitouni and Fuxi Zhang in various combinations, and especially on a few very recent joint works with Ewain Gwynne.