16 problems

Product conjecture. The product of the $u_i$-tilted matrices

Let ${\cal{A}}=\{A_1,\ldots,A_m\}$ be a family of $m$ real $d\times d$ matrices, and let ${\cal{Q}}_m$ denote the space of all such families, identified with ${\mathbb R}^{m d^2}$.…

Binary-matrix finiteness conjecture. Every pair of binary matrices has the finiteness property.

Let $S=\{v_i\}_{i=1}^g\subseteq\matrices$ be a collection of matrices, and define its Wielength by … where $S^k=\{v_\sigma:\sigma\in\{1,\dots,g\}^k\}$,…

Undecidability conjecture. It is undecidable to check whether $\rho\geq 1$ for the joint spectral radius $\rho$.

Let $n$ be a positive integer and let $\mathcal{S}$ be an SIA set of $n \times n$ stochastic matrices. Its SIA-index is the length of the shortest SIA product formed from matrices…

Local overline{d}-continuity conjecture. The map

Mixing conjecture. If for some $s>0$ the equilibrium state of $(\mathsf{A},s)$ is ergodic with respect to $\sigma^d$, then it is mixing with respect to $\sigma$. If there exists…

Let $M_2(\mathbb{R})$ be the space of all $2\times2$ real matrices, and let … be the lower spectral radius as a function on pairs of matrices. Discontinuity-dimension conjecture. T…

Bochi–Fayad conjecture. The set of all resistant pairs $(H,R)\in\mathcal{H}\times\mathcal{R}$ has full Lebesgue measure. Partial results are known, and explicit resistant pairs can…

Higher-dimensional singular value pressure conjecture. For each $d\geq 1$, there exist an integer $n(d)$ and a continuous function $\hat{K}_d\colon(0,+\infty)\to(0,+\infty)$ such t…

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 $\mathbb{A}\in\{0,1\}^{K\times K}$ be any matrix, and let … with its natural subspace topology. Here spectral finiteness restricted to $\mathbb{A}$ means that the joint spectra…

Let $(\boldsymbol{S},\Xi)$ be a discrete-time stationary matrix-valued Markovian chain, with consistent convergence and consistent exponential convergence understood in the sense o…

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…