70 problems

The von Mangoldt function and the higher-order correlation statement referred to as Theorem are as in the source. Green–Tao higher-order conjecture. The higher-order statement of T…

Typical recurrence conjecture. Most two-dimensional Lorentz gases with finite horizon, not necessarily periodic, are recurrent.

Let $M$ be a properly embedded minimal surface in $\mathbb{R}^3$. Recall that a Riemannian surface is recurrent if almost all Brownian paths are dense in $M$. Meeks's recurrence co…

Hurley's conjecture. For every $r\geq 1$, there exists a residual subset ${\cal G}$ of $\operatorname{Diff}^r(M)$ such that, for every $f\in {\cal G}$, the union of the basins of t…

Positivity conjecture. For every irrational $\alpha$,

Let $f$ be a dynamical system and let $\tau(B(x,r))$ denote the first return time of the ball $B(x,r)$ to itself. Let $\mu$ be an ergodic hyperbolic measure with nonzero entropy…

Let $\mu$ be any subsequential limit of the distributions $\mu^t$ of the spin configuration $\sigma^t$, let $R_*(t)$ denote the relevant domain-wall length scale, and call a set of…

Let $M<\infty$, and let $T^M_j$ be chosen uniformly among isomorphism classes of spherical triangulations with $j$ vertices and maximum degree at most $M$; conditional on $T^M_j$,…

Restricted density polynomial Hales–Jewett conjecture. There exists $N=N(d,\delta)\in\mathbb N$ such that, for every $S\subseteq\mathcal P(\{1,\ldots,N\}^d)$ satisfying

VIP-system recurrence conjecture. The VIP-system $v$ is good for nice recurrence.

Let $G$ be a group. A set $R\subset G$ is a set of measurable recurrence if, for every measure-preserving $G$-system $(X,\mathcal{X},\mu,(T_g))$ and every $A\in\mathcal{X}$ with…

Let $G$ be a non-amenable unimodular vertex-transitive graph, and consider the infinite-lifespan frog model. Write $\lambda_{\mathrm{c}}(\mathrm{recurrence})$ for the critical dens…

Brown–Graham–Landman conjecture. If $S$ is not a set of $d$-nilBohr recurrence for some $d$, then $S$ is not $2$-large.

Katznelson's conjecture. There is a set $S\subset\mathbb{N}$ which is a set of Bohr recurrence but not a set of topological recurrence.

Glasner–Huang–Shao–Weiss–Ye conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Consider the $\lambda$-stationary random walk on the lamplighter group $H\wr G$. Critical recurr…

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Let $\lambda_c$ be the critical parameter such that the $\lambda$-stationary random walk on…

Consider a prime walk (PW) on the plane, and let $(x,y)$ denote a visited point. The number of visits to a point is its visit frequency. Prime-walk recurrence conjecture. In the li…

Let $H_0$ denote the first hitting time of $0$ for the one-dimensional Friedman random walk, started after one white draw, with the walk encoded by the difference between the numbe…

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with drift vector $d_\alpha=\sum_{i=1}^4\alpha_i e_i$. Redundancy conjecture. The recurrence hypothesis and the…

Recurrence conjecture. If $d_\alpha=0$, then the RWDE is recurrent.

Unbounded-jump random-walk conjecture. In this setup, the local version of the limit law for $V_n(1)$ is also true, and the random walk defined above is recurrent.

Let $a,b,d$ be integers with $\operatorname{gcd}(a,b,d)=1$, and let $R_{a,b,a,d}$ denote the set of values of the corresponding linear fractional pattern. In this case,…

Let $(X,T)$ and $(X,S)$ be minimal systems on the same compact space $X$. Mixed linear-quadratic nonrecurrence conjecture. There are minimal systems $(X,T)$ and $(X,S)$ such that f…

A topological dynamical system is a pair $(X,T)$ consisting of a compact space $X$ and a continuous map $T:X\to X$; it is minimal if every orbit is dense. For two transformations…