Title :

Gaussian Multiplicative Chaos theory and several extensions

Abstract :

The Gaussian Multiplicative Chaos, introduced by Kahane (1985), is a 
random measure defined as the exponential of a log-correlated gaussian 
field. It is a crucial element in the recent rigorous probabilistic 
construction by David-Kupiainen-Rhodes-Vargas (2014) of Liouville 
Conformal Field Theory, introduced by Polyakov (1981) in his theory of 
integration over 2d Riemann surfaces.

The first part of this talk is an introduction to Gaussian 
Multiplicative Chaos and Liouville Conformal Field Theory on the 
Riemann sphere a la David-Kupiainen-Rhodes-Vargas. We also briefly 
explain the case of the unit disk, where the presence of a boundary 
requires careful analysis of the behaviour of a non-conventional 
Gaussian Multiplicative Chaos measure.

In the second part of this talk, we will discuss some extensions of 
these theories to cases with complex parameters. We will derive in 
particular some results on analytic continuation of correlation 
functions using the path-integral formalism.