7 problems

Quarter-sphere spectral conjecture. One has

Let $g_0$ be a round metric of curvature one on $\mathbb{S}^2$. Given $k\geq 2$, let $\mu$ be an admissible measure such that $\lambda_k(\mathbb{S}^2,\mu)=2$. Then there exists a c…

Consider the family of ellipses … For each such ellipse, consider the multiplicities of the eigenvalues of its Laplacian. Generic multiplicity-two conjecture. For a generic class o…

Let $G$ be a compact connected Lie group, and let $\mathcal M^G$ denote the space of left-invariant Riemannian metrics on $G$. For $g\in\mathcal M^G$, write $\lambda_1(G,g)$ for th…

Let $\Lambda_k(\mathbb{RP}^2)$ denote the supremum of the normalized $k$-th Laplace eigenvalue over metrics on the real projective plane. A sequence of metrics is said to degenerat…

Let $\Omega$ be a bounded convex plane domain of area $A$ and moment of inertia $I$, and let $\mu_2,\ldots,\mu_n$ be its positive Neumann Laplace eigenvalues. Neumann sum conjectur…

Let $\Omega$ be a bounded convex plane domain, with area $A$, moment of inertia $I$, and first Dirichlet Laplace eigenvalue $\lambda_1$. Convex-domain bounds conjecture. … Equality…