110 problems

Consider the dynamical systems constructed from instances of the unique games conjecture, with alphabet size $k$, and let the invariant measure assign weight to neighborhoods of op…

Let $f$ satisfy $f(0)=0$ and $f'(0)>0$. For the characteristic equation at $x=0$, … write $\theta=rf'(0)-\gamma$ and let $W_m$, $m\in\mathbb Z$, denote the $m$-th branch of the Lam…

The piecewise deterministic process is defined on the non-compact Poincaré disk, with quantum jumps having place-dependent probabilities and dissipative continuous evolution over r…

Let $g:[0,1]\rightarrow [0,1]$ be a mixing map. For any open set $U\subseteq[0,1]$, suppose there exists an integer $n\in\mathbb N$ such that $g^n(U)=[0,1]$. Let $M_g$ denote the s…

Cesàro-limit conjecture. The limit in Cesàro mean of the sequence

Let $G$ be a definably compact group, and let $\mu\colon \operatorname{Def}(G)\to [0,1]$ be a finitely additive measure on the definable subsets of $G$. Uniqueness conjecture. Ther…

Recurrence criterion. The random dynamical system has a unique invariant Borel probability measure if and only if its associated fundamental Markov system is recurrent.

For $d=1,2$, consider the harness process on ${\mathbb Z}^d\setminus\{0\}$ with generator $L^{{\mathbb Z}^d\setminus\{0\}}$, and let $\eta^{{\mathbb Z}^d\setminus\{0\}}_t$ denote i…

Let $d\geq 3$. The harness process on ${\mathbb Z}^d$ has generator $L^{{\mathbb Z}^d}$, and let $\eta^{{\mathbb Z}^d}_t$ denote its infinite-volume stationary configuration. An in…

Let $\mathcal{I}_e$ denote the set of extremal invariant measures, and let $\hat{\mathcal{H}}_R$ be the relevant class of functions associated with the set $\hat{\mathcal{H}}$ of f…

Let $\sigma\in {\mathcal M}_1((0,1/2])$ and let $\beta_m,\beta_s\in(0,1]$. Let $K_{\sigma,\beta_m,\beta_s}$ denote the Markov kernel governing the coagulation-fragmentation process…

Let ${\mathcal A}$ be the class of probability measures appearing in Theorem, let $\mu$ be an invariant probability measure, and let the theorem's part b) assert uniqueness under t…

$B_1$ distribution conjecture. The distribution of $B_1$ is

Let $\nu$ be a $L_{pol}SU(2)$-invariant probability measure on $\bold L(SL(2,\mathbb C)/SU(1,1))$, and let $g_0$ be the zero-mode component with Cartan and matrix coordinates…

Let $\mu$ be the conjectured unique invariant measure on the relevant loop-group completion, and let $\theta_{-}$ denote its negative-mode coordinate. For every matrix coefficient…

Let $G$ be the complexification of a simply connected compact Lie group, and let $L_{hyp}G$ be the hyperfunction completion of its loop group. Let $\mathcal D$ be the group of anal…

Let $X$ be a simply connected compact symmetric space with $X\simeq U/K$, where $U$ is a simply connected compact group acting isometrically and transitively on $X$, and let $G$ be…

Let $K$ be a simply connected compact Lie group, let $G$ be its complexification, and let $L_{pol}K$ act by left and right multiplication on the formal completion $L_{formal}G$ of…

Let $\hat {\Cal D}_{an}$ be the universal central extension of the group of analytic, orientation-preserving diffeomorphisms of $S^1$, acting on a completion of $\Cal D_{an}$, and…

Consider the stochastic two-dimensional Navier–Stokes equations on the torus with additive noise applied only to a degenerate collection of Fourier modes. The viscosity is denoted…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be a cardinal. Let $\mu$ be an invariant measure on…

Let $\AgeK$ be a strong Fraïssé class satisfying $n$-DAP for every $n$ and having only countably many isomorphism types, where $n$-DAP is the disjoint $n$-amalgamation property. Le…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on…

Full-rank solution conjecture. If $\operatorname{Leb}(D_{\text{AL}})>0$, then there exists a solution $\big(\Omega_v\big)_{v\in V}$ to the system of equations (4) such that every m…