4 problems

Gau–Wang–Wu conjecture. If the numerical range $W(A)$ is a circular disc centered at the origin, then $A$ is rotationally invariant.

Let $\operatorname{Hilb}_n[\mathbb{C}^m]$ denote the Hilbert scheme of $n$ points on $\mathbb{C}^m$, and let $\mathcal{M}$ be the moduli space of extended partial isometries in hig…

Gau et al.'s conjecture. The centre of $W(A)$ must be the origin; equivalently, $a=0$. This extends the known result for $n\leq 4$ to arbitrary dimension. The paper gives a partial…

Let $T$ be a partial isometry on a $4$-dimensional complex Hilbert space, and suppose that $\operatorname{Rank} T=2$. A matrix $T$ is unitarily equivalent to a complex symmetric ma…