339 problems

Does there exist a universal constant $C>0$ such that, for every connected graph $G=(V,E)$ with boundary $B$, maximum degree $d_{\max}$, and genus $g$, one has…

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain, and let $\lambda_k(\Omega)$ and $\mu_k(\Omega)$ denote its $k$-th Dirichlet and Neumann Laplacian eigenvalues, respectively.…

Let $h_0$ and $h_1$ be the positive heights from part (2) of Theorem, where the ordering of the symmetric and antisymmetric second Neumann eigenvalues changes as the height varies.…

Let $M^n$ be a complete Riemannian manifold, let $\Omega\subset M^n$ be a bounded domain, and let $\lambda_k$ denote the Dirichlet eigenvalues of the operator $\mathcal{L}$ from Pr…

Let $g$ denote the genus of a hyperbolic surface, let $\sys(M)$ denote its systole, and let $\Nsmall$ and $\lambda_1(M)$ be the quantities used in the paper. The first quantity bel…

Let $(M,g)$ be a fixed compact smooth two-dimensional Riemannian manifold with smooth boundary, and let $\mu^{\text{loop}}$ denote the Brownian loop measure on $(M,g)$. Let $K$ be…

Sublinear multiplicity conjecture. There exist $\alpha<1$ and a constant $C_{\alpha}=C_{\alpha}(\alpha,\gamma)$ such that

Let $(M,\theta)$ be a closed pseudohermitian $(2n+1)$-manifold with $n\geq 2$. Assume, in addition, that the CR-Paneitz operator is nonnegative if $n=1$. Suppose there is a positiv…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$, and let $\mathfrak L_k(\Omega)$ denote the infimum of the maximal first Dirichlet eigenvalue over all $k$-partitions of $\Omega$…

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…

Quantum-layer discrete-spectrum conjecture. If $\varSigma$ is not the plane and $K$ is integrable on $\varSigma$, then the Dirichlet Laplacian on $\Omega$ has non-empty discrete sp…

Triangle eigenvalue bound. The eigenvalue satisfies

Let $\Sigma$ be an embedded asymptotically flat surface in $R^3$ that is not totally geodesic, and suppose its Gauss curvature is integrable. Consider the quantum layer built from…

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

Let $\Sigma$ be a closed surface, and let $I(\Sigma)$ denote its Friedlander–Nadirashvili invariant, defined as the infimum over conformal classes $C$ on $\Sigma$ of the supremum…

Quarter-sphere spectral conjecture. One has

Canonical-model zeta conjecture. If $M_{K_1}$ and $M_{K_2}$ are isospectral, then the Hasse–Weil zeta functions of the canonical models of $M_{tilde{K}_1}$ and $M_{tilde{K}_2}$ are…

Let $Q$ be either of the two isospectral mixed Dirichlet-Neumann problems on a quarter-sphere shown in the source, and let $Q_{a,2}$ be the axisymmetric problem with the Dirichlet…

The genus-2 extremal metric conjecture. There exists a metric on a surface of genus $2$ that attains the upper bound

A bounded planar domain is equipped with a decomposition of its boundary into Dirichlet and Neumann parts, and Dirichlet-Neumann isospectrality means that swapping these boundary c…

Let $S^{2}$ be the two-sphere, let $\mu$ be the measure used in the definitions of the functionals $A_{n}$ and $B_{n}$, and let $A_{n}(\varphi)$ and $B_{n}(\varphi)$ be the corresp…

Let $C$ be a smooth closed curve with canonical constant-curvature metric $g_{0}^{C}$, and let $l$ be a holomorphic line bundle with induced canonical metric $g_{0}^{l}$. For a Her…

Let $C$ be a compact complex curve with Hermitian metric $g^{C}$, and let $l$ be a line bundle over $C$ with associated Hermitian metric $g^{l}$. Write $A(g^{C},g^{l})$ for the log…

Monotonicity conjecture. The angle $\alpha(p)$ is a monotone function for

Let $\mathcal{M}$ be a set of Laplace-isospectral manifolds with a uniform upper bound on diameter. Finiteness conjecture. There are only finitely many distinct covering spectra fo…