123 problems

Horn's conjecture. The inequalities arising from the recursively characterized triples $(I,J,K)$ are sufficient to characterize all Horn triples $(\lambda,\mu,\nu)$; equivalently,…

Dirac-support conjecture. The measure $\mu$ is minimizing for the $k$-th optimal eigenvalue:

Let $\Sigma^n$ be a smooth closed hypersurface minimally embedded in the unit sphere $\mathbb{S}^{n+1}$ equipped with the round metric, and let $\lambda_1(\Sigma)$ denote the first…

Let $G$ be a $\b\Delta$-regular graph with order $n$, and let $\b\lambda_2(G)$ denote its second largest adjacency eigenvalue. Here $\b\Delta$ denotes the common vertex degree, and…

Let $S$ be a non-compact hyperbolic surface of finite area with $n$ punctures, and let $\bar S$ be the closed surface obtained by filling in its punctures. Write $\chi(S)$ and…

A threshold graph is a graph obtained from an isolated vertex by repeatedly adding either an isolated vertex or a dominating vertex. Let $A_n$ denote the anti-regular graph on $n$…

Li's conjecture. The first positive eigenvalue should satisfy

Jakobson's conjecture. The first eigenvalue of the Laplacian on a surface of genus two was conjectured to be maximized by a singular metric.

Let $T$ be the torus, and let $\Lambda_2(T)$ denote its second topological eigenvalue. Kao–Lai–Osting conjecture. … This conjecture concerns the sharp value of the second eigenvalu…

Let $(X_j,d_j,\mu_j)$ be a sequence in ${\mathcal M}^n_{H,D}$ converging to a metric-measure space $(X_\infty,d_\infty,\mu_\infty)$ in the measured Gromov–Hausdorff sense. Here…

Grassmann eigenvalue conjecture. (i) If $(n,q) eq (2d,2)$, then

Let $S^{n+1}$ be the unit $(n+1)$-sphere with its standard round metric, and let $\Sigma^n\subset S^{n+1}$ be a closed embedded minimal hypersurface. Yau's conjecture. The first ei…

Let $\Omega$ be a bounded domain with smooth boundary in an $n$-dimensional Riemannian manifold, and let $0<\Lambda_1\leq\Lambda_2\leq\cdots$ be the successive eigenvalues of the o…

Weyl's conjecture. The asymptotic expansion of Weyl's law continues, and its second term involves the boundary volume ${\mathfrak a}0text{vol}({\mathfrak a}0partial\beta)$. This co…

Let $H_{+}$ and $H_{-}$ be the block matrices of the operators $E^{+}$ and $E^{-}$, respectively, indexed by families in $Z_n^{0,0}$, and let H^{\raisebox{0.2ex}{\scriptsizepm…

Variable-size eigenvector decomposition conjecture. For any $3\times3$ octonionic Hermitian matrix, the associativity relation should hold when suitably rewritten for a set consist…

Generalized orthogonality conjecture. The correct notion of orthogonality is $\operatorname{NewOrtho}$, rather than $\operatorname{Ortho}$.

Let $\mathfrak{i}_1$ denote the quantity associated with the unique bounded non-trivial solution of equation (1.9), and suppose that this solution exists uniquely. An eigenvalue is…

Inradius upper-bound conjecture. There is no universal constant $C>0$ such that

Mean-distance upper-bound conjecture. There is no universal constant $C>0$ such that

Consider the two cases above for the sequence of subdivided-tree Ricci matrices. In the symmetric case, the Perron eigenvalue $[200~\lambda_\ell$ converges to the unique positive s…

Let $\dot{x}=f(x,\rho)$, with $x\in\mathbb{R}^n$, $f\in\mathcal{C}^\omega$, and parameters $\rho=(\rho_1,\dots,\rho_m)^T\in\mathbb{R}^m$. Suppose that $x_g\in\mathcal{A}$ is a ghos…

Let $\mathcal{R}_a$ denote the rectangle considered in the paper, let $\lambda_2(p;\mathcal{R}_a)$ be its second eigenvalue for the pure $p$-Laplacian, and let…

Leading-block and eigenvalue conjecture. This leading principal diagonal block of $A^{-1}$ is a $\mathbf{Z}$-matrix. Moreover, $A$ has exactly one negative eigenvalue.

Let $T$ be an arbitrary triangle. For each admissible choice of boundary sides, let $\lambda_{1}^{S}$, $\lambda_{1}^{M}$, $\lambda_{1}^{L}$, and $\lambda_{1}^{MS}$ denote the corre…