21 problems

Markov field property. The local causal states form a Markov field in space-time.

Let $\mathcal{M}$ be a chaotic smooth compact $d$-manifold, and let $\{(\varphi_{j},\lambda_{j})\}_{j\ge 1}$ be the corresponding sequence of Laplace eigenfunctions and eigenvalues…

Multiplicative-noise extension conjecture. Theorem (ii) and (iii) for the additive-noise equation remain valid in the presence of multiplicative noise, provided that

Uniform dimension conjecture. The uniform dimension theorem stated for $p\geqslant4$ is valid whenever $p\geqslant2$, provided that

Let the KPZ fixed point have a general initial condition that grows sufficiently more slowly than $f(x)=x^2$ as $|x|$ becomes large, so that the KPZ fixed point exists at $(0,1)$.…

Randomly shifted Gumbel convergence conjecture. There exists a random variable $Z>0$ such that

Let $\mathbb{F}_m$ act on its Furstenberg–Poisson boundary $\partial\mathbb{F}_m$, and let the stationary $S\alpha S$ random field associated with this boundary action be the one c…

Let $\mathcal{G}$ be a class of countable groups, let $\mathbb{F}_m$ denote the free group on $m$ generators, and let $W^\ast_m$-rigidity and $W^\ast_R$-rigidity refer to the rigid…

Bourgain–Sarnak–Rudnick-type conjecture. For every $\epsilon>0$,

Let $F$ be a three-dimensional Bargmann–Fock field on $\mathbb{R}^3$. For each $t\in\mathbb{R}$, let $f^{\mathrm{hor}}(t)$ be the two-dimensional Bargmann–Fock field obtained by re…

Let $n=2$ or $n=3$, let $V_N$ denote the finite domain of the membrane model, let $\psi=(\psi_x)_{x\in V_N}$ be its field, let $\operatorname{E}_N$ denote expectation, and let…

Let $\mathcal{D}_p$ denote the excursion domain at level $p$ associated with a centered, normalized, non-degenerate, sufficiently smooth, stationary, isotropic, and positively corr…

Maximal Nazarov–Sodin constant conjecture. For every $\mu\in\mathcal{P}_{\text{symm}}$ that is a weak-$*$ limit of $\{\mu_n\}$, the maximal value $d_{\max}$ is uniquely attained by…

Global infinite-oscillation conjecture. There exist such correlation functions $f$ for which

Global infinite-oscillation conjecture. There exist such correlation functions $f$ for which

Let $f=f_{\alpha;T}$ be a band-limited Gaussian function on an $n$-dimensional compact manifold, with $n\geq 2$ and $\alpha\leq 1$, and let $\Psi_{n;\alpha}$ be the limiting distri…

Let $T$ be a Gaussian eigenfunction on a compact manifold $\mathbb M$. For $u\in\mathbb R$, define the excursion set … For each Lipschitz–Killing curvature of $A_u(T,\mathbb M)$, c…

Universal maximum conjecture. The maximum satisfies, for some constant $C$ and a random variable $Z$ called the derivative martingale,

Let $(X_v,Y_v)_{v\in\mathbb{Z}^2}$ be a hidden Markov field with $E=F=\{-1,1\}$. Suppose … where $(\xi_v)_{v\in\mathbb{Z}^2}$ are independent and identically distributed, independe…

Let $x\in \mathcal{S}^{2}$ and, for $t\in \mathbb{R}$, let $\mathcal{L}_{l;x}^{t}$ denote the random length of the unique component of $f_l^{-1}(t)$ that contains $x$ inside, or…

Tightness conjecture. The sequence of random variables $\{Y_N\}_{N\geq 1}$ is tight.