7 problems

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 $F_B\in\mathbb{C}[t,x,y]^{C_n}_d$ 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…

Generalized cluster structure conjecture. These seeds determine an upper generalized cluster algebra structure on $[]$.

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

Let $\ell\ge 2$ and $n>1$, and consider rational order-$n\ell$ differential operators with $\mathbb Z_\ell$ symmetry in the preceding setup. Require that the indices at $0$ are tho…

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…