9 problems

Joint ergodicity conjecture. The polynomials $p_1,\ldots,p_\ell$ are jointly ergodic for $(X,\mathcal{X},\mu,U)$ if and only if: (1) the product action…

Let $d,k,n\in\mathbb{N}$, let $T_1,\dots,T_d:X\to X$ be invertible measure-preserving transformations on a probability space $(X,\mathcal{B}(X),\mu)$ that generate a nilpotent grou…

Multiparameter polynomial pointwise ergodic conjecture. For every $f\in L^\infty(X)$, these averages converge for $\mu$-almost every $x\in X$ as $\min\{M_1,\ldots,M_k\}\to\infty$.…

Let $d,k\in\mathbb{N}$, let $(X,\mathcal{B}(X),\mu)$ be a probability measure space with invertible commuting measure-preserving transformations $T_1,\dots,T_d:X\to X$, and let…

Polynomial pointwise convergence conjecture. These averages converge almost surely. This is a natural extension of Bourgain's double recurrence theorem; recent progress is known, b…

Let $(X,\mathcal X,\mu)$ be a probability space, let $T_1,\ldots,T_\ell$ be commuting measure-preserving transformations, and let $p_1,\ldots,p_\ell\in\mathbb Z[n]$. A sequence of…

Multi-parameter linear polynomial convergence conjecture. For every $f\in L^{\infty}(X)$, these averages converge for $\mu$-almost every $x\in X$ as…

Multi-parameter Bellow–Furstenberg conjecture. For every $f\in L^{\infty}(X)$, the limit

Let $(X,\mathcal B,\mu,\mathbb Z^d)$ be a measure-preserving system generated by invertible transformations $T_1,\ldots,T_d$, and let $p_{i,j}(n)\in\mathbb Z[n]$ be the polynomials…