13 problems

Let $T(z)$) be an expanding polynomial of degree $d$ with a real Julia set $E$, where $E\subset[-\xi,\xi]$ and $T^{-1}:[-\xi,\xi]\to[-\xi,\xi]$. For $x\in[-\xi,\xi]$, define the Ja…

Degree-eight distinct-diagonal test case. Every reducible connected degree-eight spectral curve is obtained from a constant branch or from reversal symmetry. In particular, there i…

Primitive codimension-growth conjecture. One has

No-connected-hypersurfaces conjecture. For every $n\ge4$, every codimension-one component of the reducible locus is one of the cutting hyperplanes $b_i=0$. Equivalently, the reduci…

Amended reducibility conjecture. Every reducible factorization of $\chi_n$ is expected to be obtained by iterating the following operations on connected blocks: cutting the chain a…

Let $J$ be a $7 \times 7$ Jacobi matrix with symmetric spectrum that realizes perfect state transfer (PST). Its early state exclusion (ESE) property means that no state is excluded…

Tangent-measure conjecture. For every … $\mu$ has a unique tangent measure at $$ $\xi$, and this tangent measure is Lebesgue measure.

Containment conjecture.

Let $K(\gamma)$ be the Cantor-type set determined by $\gamma$, and let $H_{\mu_{K(\gamma)}}$ have recurrence-coefficient sequence $(a_n)_{n=1}^\infty$. An almost periodic sequence…

Let $\gamma=(\gamma_k)_{k=1}^\infty$ be the parameter sequence defining $K(\gamma)$, set $\varepsilon_k:=1-4\gamma_k$ for $k\in\mathbb{N}$, and let $(a_n)_{n=1}^\infty$ be the recu…

Let $\gamma=(\gamma_k)_{k=1}^\infty$ be the parameter sequence defining $K(\gamma)$, and let $\mu_{K(\gamma)}$ be its associated measure with recurrence coefficients…

Chihara's conjecture. The essential spectrum satisfies

Let $J$ be the discrete Schrödinger operator on the half-lattice with bounded real potential $(v_n)$, represented by the Jacobi matrix with diagonal entries $v_n$ and off-diagonal…