The goal of this ERC is to study geometric properties of random planar surfaces such as hyperbolic surfaces or random planar maps.
We are also broadly interested in understanding large scale properties of random objects (random trees, random graphs...).
Two canonical models of random continuum surfaces have been introduced in the past decade, namely the Brownian sphere obtained as the scaling limit of uniform random planar triangulations, and the Liouville Quantum Gravity metric obtained formally from the exponential of the Gaussian free field on the sphere. Our objective is to broaden our understanding of random planar metrics to the case of metrics with ``holes'' or ``hubs'', and to the causal (when a time dimension is singled out) paradigm. We also plan on studying random maps in high genus and to connect to models of 2-dimensional hyperbolic geometry such as the Brook-Makover model, random pants decompositions or Weil-Petersson random surfaces.