19 problems

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

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 $[?]$?

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

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 $\mu$ 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 $G=\operatorname{PGL}_n(\mathbb{R})$. 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…

Let $\alpha>0$, $u>-\alpha$, and let $z_{u,v}$ be the explicitly constructed stationary processes indexed by $v\in(-\alpha,\min(u,0)]$ for the half-space log-gamma polymer. For ini…

Let $\mathcal{H}_{u,v}$ denote the explicitly constructed stationary processes for the half-space KPZ equation with Neumann boundary parameter $u$, indexed by…

Let $({\mathscr T},q)$ be a non-null probabilised context tree. A context tree is totally non-stable when its set of infinite branches ${\mathscr C}^i$ has no shift-invariant subse…

Let $({\mathscr T},q)$ be a non-null probabilised context tree. A context tree is totally non-stable when its set of infinite branches ${\mathscr C}^i$ has no shift-invariant subse…

Let $|\bar{P}\rangle$ be the stationary state in a sector $k$, with configurations $(\sigma_1,\ldots,\sigma_L)$ belonging to $\mathcal{B}(k)$, and let…

Let $\Gamma<\operatorname{PSL}_2(\mathbb{R})$ be a lattice, and let $m$ be a finitely supported measure whose support generates $\Gamma$. Let the corresponding stationary measure b…