527 problems

Let a frame be a sequence in a Hilbert space that provides stable expansions, and call it bounded below in norm when its elements have norms uniformly bounded away from zero. Feich…

Let $X$ be an infinite compact Hausdorff space, and let $C(X)$ be the Banach algebra of continuous real-valued functions on $X$. Consider an algebra homomorphism from $C(X)$ into a…

Lipschitz magnitude conjecture. Let $T\subseteq L_1$ be tractable. Then

Let $E$ be a Banach space whose current point is … and let $B\subseteq E$ be closed. The following should hold simultaneously: the elements of $B$ satisfy at $t=0$ the initial cond…

Let $(T(t))_{t\geq 0}$ be a bounded semigroup on a Hilbert space $X$ with infinitesimal generator $-A$ such that $\sigma(A)\cap i\mathbb{R}=\emptyset$. Define … and let…

Let $d\geq 1$ and let $f\in L^2(\mathbb{R}^d)$ be nonzero. A sequence of pure translates of $f$ is a sequence of the form $(f(\cdot-\lambda_n))_{n\geq 1}$ for…

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces extends to a surjective linear or conjugate-linear isometry between the space…

Nieraeth's conjecture. If $M$ is bounded on $X$, then $M$ is bounded on $X'$.

HRT conjecture. The functions

Let $\mathcal{H}_{\operatorname{e}}$ be the function space defined in the paper, and let … A pair $(\Lambda_0,\Lambda_0)$ is a Fourier uniqueness pair for…

Let $\mathcal{X}$ be a Banach space. Let $A\colon\Omega\to\mathcal{L}(\mathcal{X})$ be a generator of a semigroup $e^{At}$ such that $\lVert A\rVert$ is bounded, and let…

Let $G$ be a compact group acting continuously and transitively on a compact Hausdorff space $X$. A space of continuous functions on $X$ is called minimal if it is closed, invarian…

Carlen–Lieb–Reuvers conjecture. The Hilbert–Schmidt norm of the $k$-particle reduced density matrix is maximised by Slater determinants, equivalently,

Let $\mathscr O_\mathrm{def}(\mathbb B^n_r)$ denote the deformed algebra of holomorphic functions on the ball $\mathbb B^n_r$. Theorem establishes that the analogous deformed algeb…

Gilbert's conjecture. There exist $\eta$ and $\mathfrak{H}_0$ such that

Elementary molecule conjecture. Every extreme point of $B_{\mathcal{F}(M)}$ is an elementary molecule.

Let $\boldsymbol{Z}=(Z_1,\ldots,Z_d)$ be a centered Gaussian random vector with $d\in\mathbb{N}$, and let $m\in\mathbb{N}_0$. Gaussian product inequality. One has … The conjecture…

Benítez–Sarantopoulos–Tonge conjecture. One has

Cauchy–Schwarz conjecture. The Cauchy–Schwarz inequality should hold in every complex normed space. The inequality is established in the paper for some classes, including complex r…

Non-constructibility conjecture. The delta functions cannot be constructed.

Karlin's conjecture. For every $\delta>0$,

Mazur-Ulam property conjecture. Every uniform algebra satisfies the Mazur-Ulam property.

Let $G$ be a locally compact group. Write $\operatorname{TLIM}(G)$ for the topological left-invariant means on $L_\infty(G)$ and $\operatorname{TIM}(G)$ for the two-sided topologic…

Let $v_1,\ldots,v_d$ and $x$ be vectors in $\mathbb{R}^d$, and let $\langle v_i,x\rangle$ denote their Euclidean inner product. Real linear polarization constant conjecture. … More…

Frisch–Parisi conjecture. In that Baire functional space, any typical element $f$ obeys some multifractal formalism and satisfies