21 problems

Let $\Omega$ be a planar polygonal domain, and let $Z(t)$ denote its heat trace for the relevant boundary condition. Brownell's conjecture. The heat trace of a polygon has only thr…

Let $\Omega\subset\mathbb{R}^2$ be a polygon with a fixed number of sides, and define the Riesz-type interaction energy by … where $K(r)=r^{-\alpha}$ with $0<\alpha<2$. Bonnesen–Cr…

Let $L$ be an L-shaped domain with canonically embedded edges $e_1,\ldots,e_6$, and let $\lambda_1^D(L)$ denote the first mixed Dirichlet-Neumann eigenvalue with Dirichlet boundary…

Let $D$ be a polygonal domain that may have reentrant corners, and let Theorem 3.3 denote the theorem asserting the relevant limiting behavior of the first transmission eigenvalue…

Let $\Omega$ be a polygonal domain, let $p$ be the parameter in the Dirichlet-to-Neumann operator, and let $V_k^{(p)}$ be a first Steklov eigenfunction with eigenvalue…

Let a triangle have angles $\alpha_0^{(0)}=\pi/12$, $\alpha_1^{(0)}=\pi/3$, and $\alpha_2^{(0)}=7\pi/12$, and let $c_0$ denote the first asymptotic coefficient for the Dirichlet-to…

Let a regular polygon have $N$ vertices and interior angle … For the asymptotic coefficients $c_k$ governing the first $N$ eigenvalues of the Dirichlet-to-Neumann operator, the reg…

Let $\Omega$ be a simply connected Jordan region in the plane. For $N\in\mathbb N$, let $\rho_N(\Omega)$ denote the squared distance from $\bar z$ to the polynomials of degree at m…

Meta-state minimiser conjecture. For odd $K$, the $Meta$ states are the $z$-invariant energy minimisers, with the splay vertices furthest apart.

Let $P_k\subset\mathbb{R}^2$ be an open $k$-gon, and let $P_k^*\subset\mathbb{R}^d$ be an open regular $k$-gon. For $k\geq 3$, define the fundamental ratio by…

Vertex attainment conjecture. The maximum

Rectangle inequality. For all points $u,v\in R$,

Let $G_k$ be the regular $k$-gon described in the paper, and let $x,y\in\bigcap_k G_k$ be distinct points. Let $\rho_{G_k}$ and $\delta_{G_k}$ denote the Möbius and hyperbolic metr…

Let $G\subsetneq\mathbb{R}^2$ be a bounded convex polygonal domain, and let $\rho_G$ and $\delta_G$ denote the Möbius and hyperbolic metrics in $G$. The convex polygonal-domain ine…

Let $n$-gons be polygons in the plane, and for each such polygon let $\lambda_1$ and $\lambda_2$ denote its first two Dirichlet--Laplacian eigenvalues. Polygonal Ashbaugh-Benguria-…

Let $\Omega$ be a polygon in the plane, and let $u$ be a normalized Dirichlet eigenfunction satisfying … Write $\partial_\nu u$ for its outward normal derivative on…

Fix an integer $n\geq 3$, let $C$ be the class of all $n$-gons with a given area, and let $\mathcal{S}(n,\mathrm{functional},\mathrm{smallest})$ mean that the regular polygon has t…

Contour-flipping conjecture. As matching points are added, $\A_N$ approaches $A$, while the contour flips, analogously to $\sin\to-\sin$, causing $\A_N$ to alternate above and belo…

Contour-through-matching-points conjecture. A more detailed picture requires $N_{\alpha,1}$ to be large enough that such a contour passes continuously through all the matching poin…

Let $L$ be the perimeter of an equilateral triangle, and let $(\sigma_j)_{j\geq 0}$ denote its Steklov eigenvalues, ordered nondecreasingly and normalized so that the products…

Let $\Omega$ be a bounded convex polygonal domain, let $r$ denote the radial weight variable, let $P$ be the Laplace-Leray projection, and let $u$ belong to a suitable space of fun…