15 problems

A real matrix is normal if it commutes with its adjoint; for real matrices this means … Let a sign matrix be a matrix whose entries belong to $\{\pm1\}$. Since every real symmetric…

Consider the random normal matrix ensemble in a strip with the soft/hard edge condition, meaning that the gas is completely confined to the droplet $S_n$ by setting…

Let $T\in M_n$ be a normal matrix, let $1\leq k\leq n$, and let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $T$. For a finite set of complex numbers, write…

Marcus–de Oliveira determinant conjecture. The determinant $\det(A+B)$ belongs to the convex hull of the $n!$ points $z_\sigma$ indexed by $\sigma\in S_n$.

Exponential-error conjecture. The exponentially small error bound in the corresponding bulk kernel asymptotic formula is also valid for more general analytic functions $Q$.

Universality conjecture. The term of order $1/\sqrt N$ in the boundary asymptotic expansion of $\rho_n$ is universally valid for a general droplet with a real analytic boundary.

Interior scaling-limit conjecture. The rescaled point fields converge to $ML_\lambda$:

Boundary Ginibre universality conjecture. The point field $BG$ is universal at regular boundary points for all potentials.

Let $A \in \mathbb{C}^{n\times n}$ and $\tilde{A}\in \mathbb{C}^{(n+1)\times (n+1)}$ with $n \geq 4$ be the matrices defined by … Here $\operatorname{nd}$ is the normal defect and…

Let $A \in \mathbb{C}^{n\times n}$ and $\tilde{A}\in \mathbb{C}^{(n+1)\times (n+1)}$ with $n \geq 4$ be the matrices defined by … Here $\operatorname{nd}$ denotes the normal defect…

Starfish conjecture. The starfish is precisely $B(a)$.

Borcea's normal-matrix spectral-distance conjecture.

Borcea's normal-matrix invertibility conjecture. If every eigenvalue of every $B_\ell$ lies outside the unit disk and

Given $S\subseteq \mathbb{C}$, let $\operatorname{conv} S$ denote the smallest convex subset of $\mathbb{C}$ containing $S$. Let $A\in M_n$ be a normal matrix with eigenvalues…

Let $V$ be a potential, let $p_{n,N}$ be the $n$-th orthogonal polynomial for $V$, and let $(\delta_{\underline{\xi}_{n,N}})_{n=1}^\infty$ be the sequence of point masses at its ze…