6 problems

Let $K_v^{(3)}$ be the complete $3$-uniform hypergraph on vertex set $\mathbb{Z}_v$, and let $I$ consist of the missing triplets $(i,i+v/3,i+2v/3)$ whenever $3\mid v$. A decomposit…

Let $F_B\in\mathbb{C}[t,x,y]^{C_n}_d$ be the homogeneous polynomial associated with a matrix $B\in\mathbb{C}^{d\times d}$, and let $\mathcal{W}_k(B)$ denote its $k$-th numerical ra…

Let $B\in \mathbb{C}^{d \times d}$, and suppose that its numerical range $\mathcal{W}(B)$ is invariant under multiplication by $e^{2\pi i/n}$. Let $\mathcal{C}_{\mathbb{C}}(n,d)$ d…

Parity conjecture. $M$ has an even number of vertices.

Let $\ell\in\{3,4,6\}$, and let $L$ be an order-$\ell$ operator invariant under the group $\mathbb Z_\ell$ on an elliptic curve with the corresponding symmetry. Assume that its pol…

Let $L$ be a rational algebraically integrable differential operator with gaps $q_1,\dots,q_{n-1}$, and call it fully cyclically symmetric when it has the stated cyclic symmetry. R…