13 problems

Fix $\gamma\in(0,2)$, a whole-plane Gaussian free field $h$, and distinct points $z_1,z_2,z_3,z_4\in\mathbb{C}$. Let $\Gamma_{z_i,z_j}$ denote a geodesic from $z_i$ to $z_j$ in the…

Fix a whole-plane Gaussian free field $h$ and let $D_h$ be the associated $\gamma$-Liouville quantum gravity metric for $\gamma\in(0,2)$. Let $\mathfrak{S}_3\subseteq\mathbb{C}$ de…

Fix a dynamical last passage percolation model, and let $\mathscr{T}$ be the set of times at which non-trivial bigeodesics exist. Hausdorff-dimension-zero conjecture. The set…

Non-uniqueness-times conjecture. For nice enough initial conditions $h_0$, almost surely $\mathcal{T}_{\geq2}(h_0)$ is dense in $[0,\infty)$, $\mathcal{T}_{\geq5}(h_0)$ is empty, a…

Johansson's conjecture. Almost surely, $x\mapsto\mathcal{A}(x)-x^2$ attains its maximum at a unique point $Y\in\mathbb{R}$.

Let $n$ be the number of points, $d$ the dimension, $\sigma^2$ the noise parameter, and $\mathcal{E}$ the set of errors made by the maximum-likelihood estimator. Write…

Let $({\bf T},r)$ be a rooted uniform spanning tree of a graph, and obtain ${\bf t}^{<n}(r)$ by successively removing outgoing edges at independent uniformly chosen non-root vertic…

Let $\mathfrak{h}_0$ satisfy Assumption, and let $\mathcal{T}_\infty(\mathfrak{h})$ be the set of times at which the KPZ fixed point has non-unique maximizers. Density conjecture.…

Let $d$ be the dimension, let $\text{BI}$ denote the set of bi-infinite branches, and consider their asymptotic directions toward the past, represented by limits…

Let $\delta$ be an initial curve contained in a convex region $R$ of area $A$. For $t>0$, write $\delta'=\delta(t)$ for its affine curve-shortening flow whenever it is defined. For…

Let $S$ be a convex region, let $F_S(\phi)$ denote the distribution function for the shape of a random triangle formed by three points chosen from $S$, and let…

The limiting density conjecture. For $k>1$,

Let $\mathbf{t}$ be a Boltzmann triangulation with a boundary of perimeter $p\geq 1$, let $\mathsf{Height}(\mathbf{t})$ denote the maximal height of a vertex, and let…