18 problems

Let $A$ and $B$ be positive definite matrices, and let $n$ be a positive integer. The BMV conjecture. The polynomial in $t$ … has all positive coefficients. The source presents thi…

Hillar's commuting-minimizers conjecture. Such matrices $A$ and $B$ commute.

Let $A,B\geq 0$. Let $f$ be a nonnegative and nondecreasing function on an interval $\mathcal{J}$ such that $\operatorname{spec}(A)\subseteq\mathcal{J}$. Reverse trace inequality.…

Let $n\ge 2$ and $1<p<n$, and write $\mathbb{R}^n_+=\mathbb{R}^{n-1}\times\{t>0\}$. For $\varphi\in C_0^\infty(\overline{\mathbb{R}^n_+})$, define the Sobolev quotient infimum … wh…

Let $K\colon \mathbb{R}^d\to\mathbb{R}^d$ be homogeneous of order $1-d$ and smooth outside the origin. Let $\nu$ be a measure satisfying … For $f\in C_0^\infty(\mathbb{R}^d)$, cons…

Divergence-free convolution trace inequality conjecture. The inequality

All-order trace derivative positivity conjecture. The function $t\mapsto\operatorname{tr} f(A+tB)$ has non-negative $k$th derivative.

For positive operators $X$ and $Y$, and parameters $s,t$ satisfying … an inequality of Ando, Hiai and Okubo is known to hold when $s+t\leq \frac{3}{2}$. Extended Ando–Hiai–Okubo co…

Let $F$ be an $\mathbb{R}^d$-valued charge with $\operatorname*{div}F=0$, let $u$ be a measure on $mathbb{R}^d$, and let $K1,K2,ldots,Kd$ be sufficiently smooth functions on…

Let $Omega$ be a domain with Lebesgue measure $nu$, and let $BS_{m,vartheta}(Omega)$ denote the operator class used in the source. Suppose that the commuting $d$-tuple…

Let $H$ and $K$ be Hermitian matrices, let $I$ denote the identity matrix, and for positive definite matrices define the weighted geometric mean by … The preceding argument establi…

Extremal uniqueness conjecture. Equality holds in the sharp inequality if and only if $f=h_p$ up to translations, dilations and multiplicative constants.

Let $\Gamma=\langle\xi_1,\xi_2,\dots,\xi_n\rangle$, and let $\alpha_n$ be as described in the generalized Jørgensen conjecture. For $i\ne1$, define … Let $z_1$ and $z_2$ be the mid…

Henrot's conjecture. One has

Let $\mathsfsl{A},\mathsfsl{B},\mathsfsl{C}\in\mathbb{H}^{d}$ be Hermitian matrices, let $q$ be a positive integer, and let $s>0$. For a Hermitian matrix $\mathsfsl{X}$, write…

Let $A$ and $B$ be positive operators and let $q\in\mathbb{R}$. Complementary McCarthy conjecture. … for $-2<q<0$, while the reversed inequality should hold for $3\leq q$. The disp…

Extension conjecture. The inequality also holds for $q>3$ and, in the reversed sense, for $-2<q<0$.

Furuta's conjecture. It was asked whether