54 problems

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

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…

Malliavin–Kontsevich–Suhov (MKS) loop measures are natural measures on curves and loops on Riemann surfaces, characterized by conformal restriction covariance and formulated using…

Let $X$ be a closed hyperbolic surface of genus $g$, let $\mathbb{P}_{g}$ be the Weil–Petersson probability measure on the moduli space of such surfaces, and let $\lambda_{1}(X)$ d…

Universality conjecture. The geometry of large random maps should be universal: there should exist ensembles of random metric spaces depending on a small set of data, such as the c…

A Steiner tree on random points is a minimum-length network connecting the points, and the route-length between two points is the length of the path between them in the tree. For p…

Linear growth conjecture. The length $|\ell(R)|$ is linear in $R$ as $R\to\infty$.

Let $r_n$ be the radius of a random quadrangulation with $n$ faces. Schaeffer's radius scaling conjecture. There exists a constant $c$ such that … The paper states this conjecture…

Let $N$ be the degree of a positive line bundle over a Kähler manifold, and let $\mathcal{N}^{\operatorname{crit}}_{N,h}$ denote the expected number of critical points of random ho…

Let $m$ be the complex dimension of ${\mathbb C}{\mathbb P}^m$, and let $\mathcal{N}^{\operatorname{crit}}_{N,h}({\mathbb C}{\mathbb P}^m)$ denote the expected number of critical p…

Let ${\bf Z}^2$ be the square lattice, and let $(a_1,a_2,\dots)$ be an infinite sequence of distinct bond configurations. A doubly-infinite self-avoiding path is a single self-avoi…

BCP's conjecture. In the low-intensity limit, a random tessellation of the hyperbolic plane is obtained, namely an ideal Poisson–Voronoi tessellation.

Positive-definiteness conjecture. The pairing $(\partial\bar{\partial}\varphi,\partial\bar{\partial}\psi)_{\operatorname{Var}}$ is positive definite. Verification is currently limi…

Let $d\ge 3$, let $p\ge 1$, and consider two independent point clouds $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ sampled uniformly from $[0,1]^d$. A matching $\sigma_p$ is $p$-optimal i…

Graph distance is the intrinsic metric induced by connectivity in a random medium. In several models of random media, let the graph distance measure the length of shortest paths be…

Let $A$ be a compact Riemannian manifold-with-boundary of dimension $d\geq 2$ and Riemannian volume $v(A)\in(0,\infty)$. The paper's coverage quantities and theorems are originally…

The KPZ universality class contains many stochastic growth models with universal spatial and temporal fluctuations. The Airy process and related Airy line ensembles describe spatia…

Radon partition probability conjecture. For all $n\ge d+2$, the probabilities satisfy

Imaginary Liouville generating-function conjecture. The function $\mathcal{G}^{\hat{\mathbb{C}}}_{\beta,\boldsymbol{\alpha}}(\mu)$ behaves similarly to the circle-case generating f…

A disordered Ising ferromagnet on $\mathbb{Z}^2$ has Hamiltonian … where the coupling constants $\eta_{\{x,y\}}$ are i.i.d. positive random variables and $\sigma_x\in\{+1,-1\}$. A…

Consider lattice models of self-overlapping polygons and JT gravity in negative curvature, together with analogous random-geometry models. Universal Schwarzian conjecture. The Schw…

Fix $\alpha\in(1,2)$. Let $T_n^{(\alpha)}$ be a critical Bienaymé–Galton–Watson tree with offspring distribution $\mu_\alpha$ satisfying $\mu_\alpha([x,\infty))\sim cx^{-\alpha}$ a…

The paper studies random quadrangulations obtained from critical Bienaymé–Galton–Watson trees whose offspring distribution has a stably decaying tail with exponent $\alpha\in(1,2)$…

Consider a typical projective variety arising from the random algebraic model discussed in the source, with degree parameter $n$. Let $\operatorname{CLE}(6)$ denote the conformal l…

Let $\beta(p,q)$ denote the generalized integral means spectrum, and let $\beta_{tip}$, $\beta_{0}$, $\beta_{lin}$, and $\beta_1$ be the four spectrum phases. Let…