10 problems

Let $(V,G)$ be a polar representation of a compact Lie group $G$ with section $b[?25l\Sigma$, and let $W\subset O(\Sigma)$ be its Weyl group. The correspondence … which sends $E$ t…

Polar spectrahedron conjecture. If $S\subset\mathfrak{a}$ is a $W$-invariant spectrahedron, then $\mathsf{K}\cdot S\subset V$ is a spectrahedron.

A free spectrahedron is a graded matrix set defined by a linear matrix inequality; its interior is denoted by $\mathfrak{P}_A$. A free map $f:\mathfrak{P}_A\to\mathfrak{P}_B$ is bi…

Spectrahedrality conjecture. Every hyperbolicity cone is a spectrahedron.

Let $S$ be a quartic symmetroid in $\mathbb{C}\mathbb{P}^3$, meaning a quartic surface defined as the determinant of a symmetric matrix of linear forms. Suppose that $S$ has a none…

Let $H$ and $K$ be hermitian $2n\times 2n$ matrices, and let … Let $\mathcal{A}$ be the algebra generated by $H$ and $K$. Kippenhahn's conjecture. If there exist $k\in\mathbb{N}$ w…

Arveson's spanning conjecture. The irreducible boundary representations, equivalently the absolute extreme points, span the free spectrahedron when $\mathcal H$ and…

A hyperbolicity region is the connected component containing the reference point of the set where a real homogeneous polynomial is nonzero. A spectrahedron is a set defined by a li…

Let $M\in\mathbb R^{p\times q}_{\geq 0}$ have rank $k+1$ and satisfy … Let $P$ and $Q$ be the two polytopes corresponding to $M$ in the nested-polytope representation. Spectrahedro…

Let $A,B,C \in \operatorname{Sym}_r(\mathbb{R})$ be real symmetric matrices of size $r$, and set … Assume that $f$ is a polynomial of degree at least three. Singularity conjecture.…