17 problems

Nearly symmetric word conjecture. A word has positive trace for every pair of positive definite letters if and only if the word is nearly symmetric.

Let $S(X,B)=P$ be a symmetric word equation in positive definite matrices, with $X$, $B$, and $P$ of size $2\times2$. Two-by-two unique-solvability conjecture. Symmetric word equat…

Let $S(X,B_i)=P$ be a symmetric word equation, where $X$ is an $n\times n$ positive semidefinite matrix, the $B_i$ are fixed $n\times n$ positive definite matrices, and $P$ is an…

Let a word be a product formed from two real positive definite matrices. A word is symmetric if it is unchanged by reversal, and a product of two symmetric words means their juxtap…

Hall–Johnson conjecture. The supremum of $R(A)$ over all $4\times4$ positive definite matrices $A$ is $27/16$.

Let $\mathrm{PD}(n)$ be the complex positive-definite $n\times n$ matrices with their affine-invariant Riemannian metric. For $P_0\in\mathrm{PD}(n)$, define the squared distance fu…

Mean-kernel range conjecture. There exists $p_0\in(2.61,2.611)$ such that the power-Wasserstein metric of parameter $p$ is a mean kernel metric if and only if

Nesbitt–Shapiro endpoint conjecture. If $p=12$ or $p=23$, then

Let $A_1,\dots,A_m$ be positive definite matrices, let $w_1,\dots,w_m$ be weights, and let $0<t<1$. Define $P_t(w;A_1,\dots,A_m)$ as the unique positive definite solution of … wher…

Several-variable determinant conjecture. The same determinant inequality should remain valid for several variables, namely

Let $A_1,\dots,A_m$ be positive definite matrices and let $w_1,\dots,w_m$ be weights. Define … and let $\Omega(w;A_1,\dots,A_m)$ denote the Wasserstein mean. For a positive definit…

Toeplitz completion conjecture. Whenever a partially positive definite Toeplitz matrix with pattern $P_3$ is positive definite completable, it admits a Toeplitz completion.

Berndt's conjecture. The following generalization of the Hlawka inequality holds:

The palindrome-or-two-palindromes trace conjecture. If $w$ has positive trace for every pair of real positive definite matrices $A$ and $B$, then $w$ is a palindrome or a product o…

Determinantal positivity conjecture. The matrix $M$ is positive for every $D>0$. The conjecture is immediate for integer $D$ and is known for $q=2$, but the paper notes that positi…

Finite asymptotic-nullity characterization conjecture. Given any $n$, there exists a finite list of polynomial matrices $P_1,\ldots,P_\ell$ such that a homogeneous ratio…

Let $\frac{\beta}{\alpha}$ be a ratio of products of principal minors, and let $\log\left(\frac{\alpha}{\beta}\right)$ denote its formal logarithm. Write $\mathcal B_n$ for the sem…