14 problems

Let $\mathbb D$ be the unit disk, let $b>0$ be the magnetic field strength, and let $\lambda_1(b)$ denote the lowest eigenvalue of the magnetic Neumann Laplacian on $\mathbb D$. Le…

Let $\Omega\subset\mathbb{R}^2$ be a bounded, simply-connected $C^\infty$-smooth domain, let $b>0$ be the intensity of a homogeneous magnetic field, and let $\mu_1^b(\Omega)>0$ den…

Let $(\Sigma,g)$ be a negatively curved Riemannian surface and let $(L,\nabla)$ be a Hermitian line bundle with connection defining the magnetic Laplacian. In the high-energy regim…

Let $\Omega \subset \mathbb R^2$ be a bounded simply-connected planar domain with area $|\Omega|=1$, and let $n\in\mathbb N$. For a homogeneous magnetic field of strength $B\geq 0$…

Square-optimality conjecture. The square is a global minimiser under both the area and perimeter constraints.

Let $B\geq 0$ be the strength of a constant magnetic field. Let $\Omega^*(B)$ be a minimizing domain, and let $h(\Omega^*(B))$ and $R(\Omega^*(B))$ denote its height and radius. Le…

Let $B\geq 0$ be the strength of a constant magnetic field, and let $\Omega^*(B)$ be a domain minimizing the first eigenvalue among the admissible domains of fixed volume. The magn…

Let $n\geq2$, let $B$ be a constant magnetic field, and let $\lambda_n(\Omega,B)$ denote the $n$th eigenvalue of the magnetic Dirichlet Laplacian on a bounded open domain…

Let $B\geq0$ be a constant magnetic field, let $\Omega\subset\mathbb{R}^2$ be a bounded open domain with $|\Omega|=1$, and let $\lambda_2(\Omega,B)$ be the second eigenvalue of the…

Let $\lambda(\beta)$ denote the lowest eigenvalue of the Neumann magnetic Laplacian in the unit disk as a function of the magnetic-field parameter $\beta\geq0$. Strong diamagnetism…

For each $n\in\mathbb N$, let $\beta_n$ be the unique parameter satisfying $\lambda(n,\beta_n)=\lambda(n+1,\beta_n)$, and define … The crossing-sequence monotonicity conjecture. Th…

Explicit-polynomial conjecture. There are explicit polynomials in the metric and its inverse, in the electromagnetic fields and their derivatives in tangential and normal direction…

Let $G_\infty$ be the infinite Sierpinski gasket lattice, let $\mathcal{U}$ be the spectral-decimation map, and let $\mathcal{J}(\mathcal{U})$ denote its Julia set. For parameters…

Let $\alpha_{0}\in(0,\pi)$, and let $\mu^\mathsf{e}(s,\eta)$ denote the lowest eigenvalue of the Neumann realization of … on the sector $\mathcal{S}_{\alpha_{0}}$. Unique-minimum c…