38 problems

Let $\Gamma$ be a uniform lattice in $SL(2,\mathbb{R})=\operatorname{Isom}(\mathbb{H}^2)$, so that $\mathbb{H}^2/\Gamma$ is a closed orbifold, and let a random walk on $\Gamma$ hav…

Singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to the Lebesgue measure class on $\operatorname{\mathcal{F}}$…

Let an AM-system be a pair of increasing affine interval homeomorphisms $\{f_-,f_+\}$ of $[0,1]$, chosen with probabilities $p_-$ and $p_+$, and let $\mu$ be its stationary measure…

Least-likely-state conjecture. In the stationary distribution, the least likely states are precisely those that are reversed-speed ordered.

Let $\Omega$ be an irreducible component of the configuration space, let $p(x)$ denote its stationary distribution, and let $\alpha$ be the transition parameter. For a configuratio…

Stationary-measure and phase-diagram conjecture. The stated family gives all extremal stationary measures, and the three parameter regimes give the corresponding weak limits of…

Let $Q$ be a compact symplectic manifold, and let $\mu$ be supported on a group $\Gamma_\mu$ of smooth symplectomorphisms of $Q$. Assume that $\mu$ has uniform expansion on every i…

Let $X$ be a complex K3 surface, let $\Gamma\subseteq \operatorname{Aut}(X)$ be a nonelementary subgroup, and let $\mu$ be a probability measure on a generating set of $\Gamma$. Su…

Let $X$ be a complex K3 surface, let $\operatorname{Aut}(X)$ be its automorphism group, and let $\Gamma\subseteq \operatorname{Aut}(X)$ be a nonelementary subgroup, meaning that it…

Stationary-measure decomposition conjecture. If $\mu$ is a stationary measure of a KCM, then there exists a probability measure $\mu_{*}$ on $\mathcal{E}$ such that

Let $\operatorname{\mathsf{G}}$ be a connected semisimple Lie group without compact factors and with finite center. Suppose $\Gamma<\operatorname{\mathsf{G}}$ is a Zariski dense di…

Let $\frac12\log q<0$, with $[?]$?

Mildly dissipative stationary-measure conjecture. The same conclusion holds for arbitrary mildly dissipative expanding on average systems on surfaces.

Measure classification conjecture. The measure $\nu$ is a $\mu$-invariant measure on a smooth submanifold $M$ of $Q$, where the case $M=Q$ is allowed.

Let $B_1$ be the first component of the two-layer process defined by the measure in. For parameters $u,v$, let $\Lambda_1,\Lambda_2$ be distributed according to the two-layer measu…

Let $\mu$ be a probability measure satisfying the Blaschke condition, with support generating a discrete subgroup of $PSU(1,1)$. Let $f_\nu(z)$ be a solution of the equation referr…

Let $\mu$ be a countably supported measure whose support generates a Fuchsian group $\Gamma \subset PSU(1,1)$ of second type. Let a solution be a function satisfying the equation r…

Let $\mathpzc{HSLG}$ denote the half-space log-gamma polymer, let $P$ be a down-right path, and let $\theta$ and $\alpha$ be its model parameters. For each … let $z_{v,P}(\cdot)$ b…

Let $Z(n,m)$ denote the point-to-point log-Gamma polymer partition function, and call a process $(h(k))_{k\geq0}$ horizontal-stationary when the solution with initial data…

Let $u,v\in\mathbb R$. Let $\mathbb{P}_{\operatorname{KPZ}}^{u,v}$ denote the measures on paths constructed from the process $B_1=W+X$, where $W=(B_1+B_2)/2$ and $X=(B_1-B_2)/2$, w…

Let $G=\operatorname{PGL}_n(\mathbb{R})$. Let $X$ and the groups $Q$, $R_0$, $R$, $S\simeq Q/R_0$, $\Lambda\simeq R/R_0$, and $H$ be as defined above. Suppose that we are in Case 2…

KPZ scaling-theory conjecture. The one-parameter family $(P_\lambda)_{\lambda\in\mathbb{R}}$ completely describes the spatially translation-invariant stationary measures for the KP…

Let $u,v e 0$ be boundary parameters, and let $mathcal{h}{u,v}$ denote the stationary measure of the open KPZ equation. For $u,v0$, this measure is represented as the law of $W+X$,…

Let $\tilde\mu$ be a finitely supported distribution on $\mathrm{SL}(d,\mathbb{R})$ satisfying the hypotheses of Guivarc’h’s theorem: the semigroup $S_{\tilde\mu}$ is strongly irre…

Let $H$ denote the stationary height function and let the path-integral representation associated with it be defined by the formulas referenced as … , with boundary parameters…