8 problems

Let $D_\Phi$ be the $\gamma$-Liouville quantum gravity metric on a $\gamma$-LQG surface, and let $\zeta$ denote its Weyl scaling exponent. Weyl-scaling exponent conjecture. The exp…

Set $\alpha \defeq 1/2+4/\kappa$. Let $(\mathcal{D},\Phi,0,\infty)$ be a $\gamma$-Liouville quantum gravity disk, and let $\Gamma$ be an independent non-nested…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant probability measure…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant measure $\mu_X$, norma…

For $\kappa \in (8/3,4)$, let $\theta(\kappa)$ denote the distance exponent associated with the canonical geodesic metric on the $\operatorname{CLE}_\kappa$ carpet. Distance-expone…

The Calabi metric on the space of Kähler metrics in a fixed Kähler class is the restriction of the deWitt–Ebin metric, and its finite-dimensional approximations are domains in fini…

Loop-cluster metric conjecture. There is a metric on the clusters of $\mathcal{L}_{g,1/2}$ inside $\Sigma$, measurable with respect to $\mathcal{L}_{g,1/2}$, obtained as the limit…

Let $B_g(t)$ be the random metric ball at time $t$, and let $B(t/\mu)$ be the corresponding rescaled limiting shape, where $\mu$ is the time constant. The fluctuations of these two…