12 problems

Let $G$ be the adjacency matrix of the Erdős–Rényi graph $G(n,1/2)$. Babai's conjecture. The matrix $G$ has no repeated eigenvalues with probability $1-o(1)$. This conjecture conce…

Let $\beta>0$, and let the Gaussian beta ensemble G$\beta$E have joint density … Let $t^n_{k,\beta}$ be its $k$-th smallest gap. Small-gap conjecture. There exists a constant…

Let $A_n$ be the symmetric random matrix considered in the source, with independent entries on and above the diagonal, not necessarily identically distributed. Vu's conjecture. The…

The asymptotic eigenvalue and gap conjecture. For the stated parameter regimes, the eigenvalues satisfy

Let $0<\alpha<2$, let $Omega=(a,b)$, and consider the fractional eigenvalue problem with $V(x)equiv0$. Denote its eigenvalues by $lambda_n^alpha$, and let … be the eigenvalues of t…

One-dimensional sharp lower-bound conjecture. The sharp lower bound for given $\alpha$ and $D$ should correspond to the gap for the corresponding one-dimensional problem. Consequen…

Let $Omega$ be a bounded domain with piecewise smooth boundary $partial\Omega$ in an $n$-dimensional Riemannian manifold $M^n$, and let $lambda_i$ denote the eigenvalues of the Dir…

Let $M_n$ be a random symmetric matrix with entries $\pm 1$ and consider the eigenvalues in the bulk of its spectrum. The largest gap means the maximum spacing between two consecut…

Let $M_n$ be a Wigner matrix, let $i$ lie in the bulk region ${\varepsilon} n \leq i \leq (1-{\varepsilon}) n$, and let $u=u_{i/n}$ be the classical location determined by … Let…

Equilateral-triangle minimizer conjecture. Supported by numerical estimates, the gap minimizer among triangular domains is the equilateral triangle.

Triangular fundamental-gap conjecture.

Fundamental-gap extremum conjecture. The function $\xi$ has no extrema on the interior of the moduli space of triangles, and $\xi$ is monotonic on the non-degenerate boundaries of…