4 problems

Let $k\geq 2$. Choose $k$ points independently according to the volume measure of a compact Brownian surface of fixed genus, and consider the associated $k$ Voronoï cells. Their vo…

Let the Brownian annulus be conformally embedded, and condition its random geometry on the modulus of the annulus. Liouville CFT conjecture. The resulting random geometry is given…

Large-genus Brownian-surface geodesic length spectrum conjecture. As $g\to\infty$, the random point process $c\,\Xi_g$ converges in distribution to $\mathcal{P}$:

The infinite Brownian disk with boundary length $\mathcal{L}_1$ is a random metric space homeomorphic to the complement of the open unit disk in the plane. Shortest-loop conjecture…