4 problems

Positive-measure discontinuity conjecture. The set of discontinuities of $\check{\varrho}$ has positive Lebesgue measure. This is presented as a weaker version of the resist-impuri…

The resist-impurities conjecture. The set of pairs $(H,R)\in\mathcal{H}\times\mathcal{E}$ that resist impurities has full Lebesgue measure in $\mathcal{H}\times\mathcal{E}$. This c…

Let $n,d\geq 1$, let $\mathsf{A}\in\mathit{GL}_d(\mathbb{R})^n$, and let $\ell(\mathsf{A})$ be the smallest index of domination for $\mathsf{A}$. The continuity criterion conjectur…

Let $r$ be an odd integer, and let $A_1,A_2$ be the $(r-1)\times(r-1)$ matrices defined by … and $(A_s)_{ij}=0$ otherwise, for $s=1,2$. A product is an s.m.p. if it is a spectrum-m…