16 problems

For $l\in\mathbb N$, let $\mathcal P_l\in\{B=l\omega\}$ be the point with maximal $A$-coordinate for which the corresponding Heun equation has a polynomial solution. Collinearity c…

Consider the three-dimensional parameter space $\mathbb R^2_{B,A}\times(\mathbb R_+)_{\omega}$. For $r\in\mathbb Z_{\neq0}$, the growth points of the phase-lock areas form the $r$-…

Let $\omega_0>0$, $l\in\mathbb N$, $\mu>0$, $B_0=l\omega_0$, and $A_0=2\mu\omega_0$. Let $s\in\mathbb Z_{\geqslant0}$, and suppose $(B_0,A_0)\in\partial L_{s,\pm}(\omega_0)$ for an…

Let $L_\rho$ be the phase-lock area with rotation number $\rho$, and call the separation points between consecutive bounded domains in its chain constrictions when they do not lie…

The no-queer-constrictions conjecture. There are no queer constrictions.

The vertical-ray conjecture.

The positivity conjecture. Every constriction is positive.

The uniqueness conjecture. For every $r\in\mathbb Z\setminus\{0\}$, the simple intersection with positive ordinate is unique.

The right-of-axis conjecture. Each phase-lock area $L_r$ lies to the right of $\Lambda_{r-1}$, namely

The interval conjecture. For every $r\in\mathbb Z$ and $j\in\mathbb N$,

For every $\omega>0$ and every integer $l\geq0$, let … where $\xi_l$ is the functional equation associated with entire Heun solutions and adjacencies. Simplicity conjecture. The ze…

Let $\omega>0$, and let $(B_0,A_0)\in\mathbb R_+^2$ be an adjacency of a phase-lock area. Consider the two branches of the boundary of that phase-lock area meeting at $(B_0,A_0)$.…

Shifted nonintersection conjecture. The phase-lock area $L_r$ does not intersect the line $\Lambda_{r-2}$. The source notes that this conjecture follows from the preceding noninter…

Nonintersection conjecture. The phase-lock area $L_{r+1}$ does not intersect $\Lambda_r$. This conjecture is discussed as a possible consequence of a monodromy/Riccati statement, w…

Small-frequency limit conjecture. As $\omega\to0$, the maximal distance from a point of $L_{r,1}$ to the ray $\{A\geq1\}$ on the $A$-axis tends to zero. This describes the limiting…

For every $r\in\mathbb Z$, let $L_r$ denote the phase-lock area with rotation number $r$, and set … Let $\mathcal A_{r,1},\mathcal A_{r,2},\ldots$ denote the adjacencies in the lin…