22 problems

An infinite-dimensional Banach space $X$ is minimal if every infinite-dimensional subspace of $X$ has a further subspace isomorphic to $X$. Rosenthal's conjecture. Every infinite-d…

Let $K(Gamma,R,g)$ denote the class of functions on $Gamma$ that are convex, where $Gamma$ and the operators $L_n^2$ are as defined in the paper. Eventual domination and monotonici…

Let $X$ be a Banach space. A space is isomorphic to no proper subspace of itself if it is not isomorphic to any proper subspace, and it is isomorphic to its hyperplanes if it is is…

Geometric dynamic programming conjecture. The map $w$ should be related to its geometric dynamic programming equation, conjectured to be

Infinite-dimensional extension conjecture. These results should also hold without the assumption of finite dimensionality.

The paper considers conjugate-gradient prediction-error bounds for statistical inverse problems under a centred noise vector with covariance matrix $\delta^2 I_D$, and under a posi…

Consider the infinite-dimensional embedding $\ell_1\hookrightarrow\ell_\infty$. Its deterministic error for every complexity $n\geq1$ is $\frac{1}{2}$. Randomized non-improvement c…

Let $G$ be a stratified Banach-Lie group with finite-dimensional commutator subgroup. An Aronszajn null set is a set null on the relevant affine lines in the Aronszajn sense, and a…

Coincidence conjecture. In this case, all three Lévy Laplacians coincide on the domain of the covariant Lévy Laplacian.

Let $H_0,\ldots,H_m$ be selfadjoint operators on a separable Hilbert space $\mathcal H$, with $H_0$ bounded or unbounded and having non-degenerate, rationally independent pure poin…

Let $X$ be an infinite-dimensional Banach space, let $f,g$ be Matkowski contractions on $X$, and suppose that $f$ is a compact operator. Let $C_{f,g}$ denote the set of parameters…

Existence conjecture. Two $C^*$-algebras playing the roles of $\mathscr{SC}(U)\rtimes U$ and $\mathscr{SC}(U)\rtimes_\tau U$ can be defined, namely…

Let $I=[0,1]$, let $I^\omega$ be the Hilbert cube, and let $\pi$ be its standard probability measure. For a sequence $a=\{a_i\}$ of positive numbers, let $d_a$ be the induced metri…

Let $\x03bc$ be a measure on $(-1,1)^\x221e$. For $s e(0,ldots)$, define … The measure $\x03bc$ is Carleson for $H^2({\mathbb T}^\x221e)$ if the reproducing-kernel testing conditio…

Let a weak limit mean a metric-measure limit of compact homogeneous Riemannian manifolds, and let an abstract Wiener space with separable Banach norm be an abstract Wiener space wh…

Existence conjecture. $\operatorname{WLO}_{rig}(L)$ is well-defined. In particular, all limits involved exist.

Korotyaev–Kuksin conjecture. The function $H^{*}$ extends analytically to a function on $\ell^{0,2}_{+}(\mathbb{N})$ and is strictly concave near $0$.

Sharp-constant conjecture. We conjecture that $1$ is the sharp constant for $\mathbb R_I^\infty$.

Fractional Poisson characteristic-functional conjecture. An infinite-dimensional fractional Poisson measure should have characteristic functional

Let $d\geq 2$. For every subset $K\subset \mathbb{R}^d$, the main theorem asserts the existence of a connected set $\tilde\Gamma\subset\mathbb{R}^d$ containing $K$, with controlled…

A continuous $m$-homogeneous polynomial $P$ from a Banach space $X$ to a Banach space $Y$ is $r$-dominated if $(P(x_j))_{j=1}^{\infty}$ is $\frac{r}{m}$-summable in $Y$ whenever…

A continuous $m$-homogeneous polynomial $P$ from a Banach space $X$ to a Banach space $Y$ is $r$-dominated if $(P(x_j))_{j=1}^{infty}$ is $\frac{r}{m}$-summable in $Y$ whenever…