Yichao Huang: How to choose a (good) random surface

Polyakov introduced Liouville Conformal Field Theory in his theory of integration over 2d Riemann surfaces (1981). In this talk, we will gently explain a rigorous probabilistic approach by David-Kupiainen-Rhodes-Vargas (2014) based on Feynman’s path integral formalism. In the construction of this path integral over surfaces with exponential interaction, a crucial ingredient is Kahane’s Gaussian Multiplicative Chaos (1985), a natural multifractal random measure defined as the exponential of a log-correlated Gaussian field. We will also briefly explain some extensions of Kahane’s theory to the case of the unit disk.