Conformally Invariant Measures on Simple Loops

We will discuss natural families of measures on the space of simple loops drawn on a Riemann surface.

In the first part of the talk, we will present a striking result of Werner, namely that each Riemann surface comes with a canonical measure on its space of simple loops.

In the second part, we will present conjectures and results aiming at understanding all natural families of measures on the space of simple loops living on a Riemann surface.