13 problems

Let $G$ be a simple graph, and let $\delta(G)$ denote its minimum degree. Write $\nu(G)$ for the greatest codimension of a faithful orthogonal representation whose Gram matrix has…

Fix a simple graph $G$ with minimum degree $\delta(G)$. A faithful orthogonal representation assigns vectors to the vertices so that two vertices are represented by orthogonal vect…

Parker's conjecture. If $\mathbb{Q}(\chi)=\mathbb{Q}$, then $\det(\chi)$ is odd.

Let $V$ be a finite-dimensional real Hilbert space, and let $G\leq \operatorname{O}(V)$ be finite. For a generic $x\in V$, consider the Gram graph of the orbit $Gx$. Real one-orbit…

Let $n$ be a positive integer, let $\Gamma$ be a finite group, and let $\rho,\rho':\Gamma\to\operatorname{O}(n)$ be homomorphisms. Two such homomorphisms are $H$-equivalent if they…

Let $1\ne w\in\mathbb{\mathbf{F}}$, let $\mu=(\mu_{1},\ldots,\mu_{k})$ be a partition, and let $e_{w,\mu}$ be the number of stable-square-length-extremal surfaces, orientable or no…

Let $G$ be a finitely generated group. Let $G \curvearrowright^\pi \mathcal{H}$ be an orthogonal representation, and let $G \curvearrowright^{\pi'} \mathcal{H}'$ be an irreducible…

Let $G$ be a graph, let $\overline G$ be its complement, and let $\rho ^\perp$ denote orthogonal representation dimension. The complementary-dimension bounds conjecture. The two di…

Let $G=(V,E)$ be a self-complementary, vertex-transitive graph with $|V|=n$. The self-complementary vertex-transitive conjecture. Its orthogonal representation dimension is … The p…

Let $G$ be a critical graph (not banned), let $M$ be a maximal matching, and suppose that its orthogonal representation dimension satisfies $\rho ^\text{\perp}(G)=d$. The matching-…

Let a graph be critical banned in dimension $d$ when it is banned in dimension $d$ but deleting any vertex lowers the required dimension. Two vertices are isomorphic when deleting…

Let a graph be banned in dimension $k$ when it has no orthogonal representation in dimension $k$. If a vertex of valence $k$ in a graph banned in dimension $k$ is doubled, then the…

Let $d\geq 1$, let $K$ be an arbitrary field, and let $G$ be a graph on vertex set $[n]$. Primality and complete-intersection conjecture. If $G$ is $(n-d)$-connected, then $L_G$ is…