29 problems

Let $\mathcal E$ be the spinor bundle over a sphere, let $\mathcal D$ be a Dirac operator on $\mathcal E$, and write $U(z,\bar z)$ for its potential in a Weierstrass representation…

Quadratic Dirac-kernel estimate. For every Dirac operator on a two-sphere, the estimate

Arbitrary finite Dirac-spectrum prescription conjecture. Apart from these algebraic constraints and the index-theoretic constraints on the zero eigenvalue, there is a Riemannian me…

Generic-metric conjecture. For a generic Riemannian metric on any compact spin manifold, the space of harmonic spinors is no larger than forced by the index theorem.

Let $\Lambda_\varphi$ be the one-dimensional Dirac operator on $L^2({\mathbb{R}},{\mathbb{C}}^2)$, with potential $\varphi:{\mathbb{R}}\to{\mathbb{C}}$ satisfying … Call…

Assume that $m\geq0$. For a bounded domain $\Omega\subset\mathbb{R}^2$ with $C^2$ boundary, let $\lambda_\Omega(\theta)$ denote the first nonnegative eigenvalue of the quantum dot…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with $\mathcal C^2$ boundary. Let $D\subset\mathbb{R}^2$ be a disk with the same area as $\Omega$. For $a>0$, let…

Ashbaugh–Benguria-type conjecture. The disk is a solution of the maximisation problem

Non-disk minimiser conjecture. There exist masses $m\geq0$ such that the minimiser of $\lambda_3(\Omega)$ is not the disk.

Krahn–Szegő conjecture. The minimiser of $\lambda_2(\Omega)$ is the union of two disjoint disks, each having area $\frac12$.

Regular polygon conjecture.

Let $(a,b)$ satisfy $0\leq a\leq 1/2$ and $a^2+b^2\geq 1$, let $\mathbb{T}^2=\mathbb{R}^2/\Gamma_{a,b}$ be the corresponding flat torus with flat metric $g_{a,b}$, and let…

Let $\Omega_{a,b}\subset\mathbb{R}^2$ be the right triangle with vertices $O=(0,0)$, $A=(a,0)$ and $B=(0,b)$, where $a,b>0$, and write $\lambda_1(a,b)=\lambda_1(\Omega_{a,b})$. The…

$L^p$ smallness conjecture. There exists a positive constant $d4c$ independent of $V$ such that

Let $\Omega_{a,b}$ be a rectangle, and consider the variational problem defining the lowest positive Dirac eigenvalue under the area constraint, with right-hand side denoted by the…

Let $\Omega_{a,b}$ be a rectangle with side lengths $a,b$, and write … for the lowest positive eigenvalue of the Dirac operator with infinite-mass boundary conditions. Square minim…

Let $m\geq 0$, let $\Omega\subset\mathbb{R}^2$ be an open connected set with locally Lipschitz boundary, and let $\lambda_1(\Omega)$ denote the lowest positive eigenvalue of the Di…

Let $\Omega \subset \mathbb{R}^2$ be a $C^\infty$ simply connected domain, and let $\mu^\Omega(E)$ be the first eigenvalue of the auxiliary operator $H_E^\Omega$. Let $\mathbb{D}$…

Optimality for a delta. The first eigenvalue is minimal when the charge distribution is concentrated at the origin, namely when $\mu=\mu(\mathbb{R}^3)\delta_0$. Equivalently, for a…

Let $M\geqslant2$, let $R_1,\ldots,R_M\in\mathbb{R}^3$, and let $\nu_m\geqslant0$ satisfy $\sum_{m=1}^M\nu_m\leqslant1$. Let $D_0$ be the free Dirac operator and let $\lambda_1$ de…

Tempered positive-energy conjecture. The $K$-group $K_\bullet(\Gamma_0(\tau_G^G)\rtimes G)$ reflects the tempered, positive energy representations of $LG$ in a suitable sense, and…

Complete Lagrangian boundary-condition conjecture. The boundary condition

Let $D_m(b,V)$ be the Dirac operator from Assumption 1.1, with $V=\Phi W$, where $W$ is Hermitian of definite sign and…

Local extension conjecture. There is a unique parabolic monogenic spinor $\Psi$ converging on a neighborhood of $\mathcal V$ in $\mathcal U$ such that

Intrinsic norm conjecture. The $\check{R}$-norm on $R(\operatorname{dom} D_{\rm \max})$ is equivalent to the $\check{H}$-norm. Moreover,