7 problems

Let $A$ and $B$ be real positive definite matrices, and let a symmetric word equation be an equation obtained by evaluating a symmetric word in $A$ and $B$ at a prescribed matrix.…

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…

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

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