92 problems

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…

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…

Let $D$ be the Dirac operator on a Riemannian manifold, and let the noncommutative residue assign a residue to suitable powers of $D$. Connes's conjecture. The noncommutative resid…

Let $M$ be a closed Riemannian spin manifold of dimension $n\ge 3$, let $S$ denote its scalar curvature, and let $\lambda$ be an eigenvalue of its Dirac operator. The Dirac eigenva…

Let $T^2$ be a two-dimensional torus with a metric $g$ having the symmetry considered in the paper. For an index $l$, denote by $\lambda_1^2(g,l)$ and $\mu_1(g,l)$ the first eigenv…

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…

Let $\Sigma$ be a sphere immersed in $\mathbf{R}^3$, let $\mathcal D$ be a Dirac operator acting on a spinor bundle over $\Sigma$, and let $\mathcal W(\Sigma)$ denote the Willmore…

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

Let $H_a$ be the radial Coulomb–Dirac Hamiltonian … with $a,c<0$, and suppose that the positive-angular-parameter case $\kappa>0$ is under consideration. The point spectrum of…

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.

Generic metrics conjecture. For generic metrics on $M$, the dimension of the space of harmonic spinors is the minimal dimension prescribed by index theory.

Existence conjecture. Every spin manifold $M$ admits a metric with harmonic spinors.

Cylinder-extremizer conjecture. These solutions are exactly those corresponding to the maximizers and minimizers of $\mathcal{F}_{q_D}$, when all spinors are normalized to have…

Nodal-set dimension conjecture. The nodal set of $\varphi$ has Hausdorff dimension at most $n-2$.

Let $N\to S^1$ be a fibration with a family $D$ of twisted Dirac operators on the fibers, and let $\delta_t$ be the associated self-adjoint adiabatic family of operators for $t>0$.…

The generalized eigenfunctions of a one-dimensional Dirac operator with potential $F$ are represented by a multilinear series involving terms $T_n(F,\ldots,F)$. Christ–Kiselev endp…

A spectral triple for a commutative geometry satisfies the order-one condition only up to compact operators, as occurs in certain strictly noncommutative $q$-deformed examples. Sta…

Codimension-two nodal-set conjecture. The nodal set of a solution of a generalized Dirac equation on $M$ has dimension

Let $M$ be a closed spin surface. A harmonic spinor is a nonzero element of the kernel of the Dirac operator for a Riemannian metric and spin structure on $M$. The uniqueness conje…

Let $M^{4m}$ be a compact spin quaternionic Kähler manifold of positive scalar curvature $\kappa$, and let $D$ be its Dirac operator. For an eigenspinor of $D$ with eigenvalue…

Vanishing conjecture. For every positive quaternion-Kähler manifold different from $\mathbb{H}P^n$,

Critical-coupling divergence conjecture. The number of discrete eigenvalues of $\mathscr D_{\eta,\tau}$ diverges as the critical condition $\eta^2-\tau^2=4$ is approached, that is,

Generic-curve accumulation conjecture. The point $z_*$ is an accumulation point of eigenvalues, but is not itself an eigenvalue.

Odd-dimensional upper-bound conjecture. If $n$ is odd, then