11 problems

Let $(X,\mathcal{X},\mu,T_1,\ldots,T_\ell)$ be a system, meaning a standard probability space with commuting invertible measure-preserving transformations, and let…

Let $(X,\mathscr{B},\mu)$ be a probability space and let $T,R,S:X\rightarrow X$ be measure-preserving automorphisms. Suppose that $R$ and $S$ commute, $S$ is weakly mixing, and let…

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving system, let $c_1,\ldots,c_k$ be distinct positive non-integers, and let $f_1,\ldots,f_k\in L^{\infty}(\mu)$. The multiple ergodi…

Let $f_1,\ldots,f_k$ be linearly independent functions of the form … Let $(X,\mathcal{B},\mu,T)$ be a totally ergodic measure-preserving system and let…

Let $f_1,\ldots,f_k$ be functions from a Hardy field, and define … Let $(X,\mathcal{B},\mu,T)$ be an ergodic measure-preserving system and let $h_1,\ldots,h_k\in L^\infty(X)$. Fran…

Let $(X,\mathcal{X},\mu,\Gamma)$ be a weakly mixing nilpotent group action, let $f_1,\ldots,f_k\in L^\infty(X,\mu)$, and let $T_1,\ldots,T_d\in\Gamma$ with polynomials $p_{i,j}$. D…

Generalized-polynomial convergence conjecture. Theorems 1 and 3 hold if $p_{i,j}$ are generalized polynomials.

Identification conjecture. The measures $\mu_x^{(d)}$ are the measures defined in that decomposition theorem.

Generic-point model conjecture. The system has a topological model $(\hat{X},\hat{T})$ such that, for almost every $x\in\hat{X}$, $(x,\ldots,x)$ is a generic point for some ergodic…

Random averaging conjecture. Then the random sequence is $\ell$-averaging.

Polynomial multiple ergodic averages conjecture. These averages converge in $L^2(\mu)$ for every choice of $f_1,f_2,\ldots,f_k\in L^\infty(\mu)$. This would extend the known $L^2$-…