72 problems

Let $V$ be the compact complex manifold underlying the forms in the statement, let $R$ be a compact Riemann surface with boundary, and let $\rho:V\times\partial R\to\mathbf{R}$ sat…

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…

Let $(X,\tilde g)$ be the universal cover of a $d$-dimensional smooth Riemannian manifold $(M,g)$. A mediatrix is the locus equidistant from the reference point and one of its nont…

Let $V$ be a Kähler manifold, let $\mathcal{H}$ be the space of Kähler potentials, and let $\mathcal{H}_0$ denote its normalized subspace. For a normalized $\phi\in\mathcal{H}_0$,…

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…

Cut-time conjecture. The cut time $t_{\operatorname{cut}}$ is identically

Conditional geodesic fluctuation conjecture. Conditional on $\mathcal L(aN,bN)=\ell N$,

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

Comparability conjecture. There exists a constant $C$ independent of $n$ such that

The directed-landscape geodesic upper-tail rate conjecture. For $t\in (0,1/2]$, as $a\to\infty$,

Let $\Omega$ be the domain under consideration, let $\mathcal{E}$ be the Cegrell class of negative plurisubharmonic functions on $\Omega$, and let $g_u$ denote the Green–Poisson re…

Let $F\subset \operatorname{PSL}_2(\mathbb{R})$ be a lattice such that the complex algebraic curve $S=F\backslash\mathbb{H}$, or equivalently its Jacobian, is defined over…

Consider a lattice $F\subset \operatorname{PSL}_2(\mathbb{R})$ with invariant trace field $\K=\mathbb{Q}(\{\operatorname{disc}(\gamma)\mid \gamma\in F\})$. Let…

Let $(X,\alpha^{\mathbb{C}})$ be a complexified Kähler manifold in the hypercritical range, and let $\mathcal{H}$ denote the space of admissible potentials. C^{1,1} geodesic conjec…

Delecroix–Goujard–Zograf–Zorich conjecture. The ratio of frequencies of separating over nonseparating simple closed geodesics admits the uniform asymptotics

Let $T_0$ and $T_t$ be the passage times at perturbation times $0$ and $t$, and let $\pi_0$ and $\pi_t$ be the corresponding geodesics. Write $\operatorname{Var}(T)$ for the varian…

Fix $\gamma \in (0,2)$ and consider $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Let $\operatorname{Star}_k$ denote the set of points joined to the origin by at least $k$ geo…

Existence of ODD geodesics conjecture. There exists a unique ODD geodesic $\gamma$ through $p$ in direction $v$.

Let $S$ be a surface, and let $\alpha$ and $\beta$ be simple closed curves that fill $S$. Let $\mathbf{G}$ be a Teichmüller geodesic ending at $\alpha$ and $\beta$. For sequences…

Nonexistence conjecture for analytic geodesics. There exists an open dense subset of $(E\cap\mathcal{H})^2$ such that no pair of elements in this subset is linked by a geodesic in…

Let $(X,\tilde{g})$ be a compact, Ricci-flat Calabi–Yau manifold, where a Calabi–Yau manifold means a Kähler manifold $(X,g)$ with vanishing first Chern class $c_1(X)=0$ and trivia…

Geodesic--argmax rigidity conjecture. For every $\tau\in(0,1)$ and $T>0$, conditionally given $\mathcal L(0,0;0,T)=L$, as $L\to\infty$,

Let $({\bf x}(t),{\bf y}(t))$ be a nonplanar bicycling geodesic with monodromy having rotation angle $\Delta\theta\in[0,2\pi)$ and translation $\Delta z>0$ about its axis. Let $I$…