15 problems

Let $A$ and $B$ be complex $n\times n$ matrices. Write the singular values of $A$ as $\sigma_1\geq\dots\geq\sigma_n\geq0$, and define … For $1\leq p<\infty$, let…

Let $A$ be a complex $n\times n$ matrix with singular values … Let $\Sigma_\uparrow(A)=\operatorname{diag}(\sigma_1,\dots,\sigma_n)$ and…

Symmetric-decreasing optimiser conjecture. The minimiser $m(t,x)$ is symmetric decreasing in time and space.

Monotonicity conjecture. Optimisers should always be monotone decreasing.

Let $E$ be a measurable set in the periodic setting, and let $E^{*\operatorname{per}}$ denote its periodic rearrangement, defined by applying Steiner symmetrization in the $x_1$-di…

Hyperbolic fiberwise rearrangement conjecture. The first Dirichlet eigenvalue satisfies

Let $0<\delta<1$. For a measurable set $\Omega\subset\mathbb{R}^d$ of finite measure, let $\Omega^*$ be the centered ball with $|\Omega^*|=|\Omega|$, and let $\alpha[\Omega]$ be it…

Let $A,B\in\mathbb{M}_n^+$ and let $p,q>0$. A matrix norm is symmetric if it is unitarily invariant. Conjecture B. For every symmetric norm, … This is presented as another conjectu…

Let $\mathbb{M}_n$ denote the space of $n$-by-$n$ matrices and $\mathbb{M}_n^+$ its positive semidefinite cone. A pair $(A,B)$ in $\mathbb{M}_n^+$ is monotone if $A=f(C)$ and…

Let $T,\Omega\subset\mathbb{R}$ be measurable sets of finite measure, and let $S_{T,\Omega}=P_TQ_\Omega P_T$ be the time-frequency localization operator, with largest eigenvalue…

The $\ell^p$-ball convolution norm conjecture. For every $r>0$, this map is increasing on $[1,2]$ and decreasing on $[2,+\infty]$. Numerical simulations suggest this conjecture; it…

Burchard–Guo stability conjecture. There is a positive constant $C_\lambda$, depending on $n$, such that

Let $G$ be a bounded open subset of $\mathbb{R}^n$, let $\Phi$ be an Orlicz function, and let $W_0^{1,\Phi}(G)$ be the subspace of the first-order Orlicz-Sobolev space consisting o…

Opposite-phase rearrangement conjecture. For all $\lambda$, $a$, and $\mu$,

Let $m\geq 2$. Consider the higher-dimensional analogue of the theorem for a nondegenerate $n$-tuple of surjective linear maps $L_i:\mathbb R^m\to\mathbb R^1$ and admissible Lebesg…