13 problems

A Calogero–Moser–Sutherland model is a Schrödinger operator whose potential is given by the corresponding many-body interaction, with coupling constants as its parameters. A Schröd…

Let $X$ be a rationally connected projective variety over an algebraically closed field of characteristic zero, and let $\mathcal F$ be a regular foliation on $X$. Algebraic integr…

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…

Algebraic Calogero–Moser classification conjecture. The algebraic Calogero–Moser equations exhaust, up to a gauge equivalence, all possible second-order elliptic PDEs whose solutio…

Let $X$ be a rationally connected smooth complex variety, and let $\mathcal F$ be a regular foliation on $X$. The known conclusion for regular foliations on weak Fano complex varie…

Let $X$ be a complex projective manifold and let $\mathscr{G}$ be a regular foliation on $X$. Assume that $K_X$ is pseudo-effective, $c_1(\mathscr{G})\equiv 0$, and…

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 $\ell\ge 2$ and let $L=\partial^\ell+a_2(z)\partial^{\ell-2}+\dots+a_\ell(z)$ be a rational differential operator invariant under $\mathbb Z_\ell$, with coefficients vanishing…

Let $\ell\in\{3,4,6\}$ and consider operators $L$ with the same symmetric elliptic setup as above, but allow the indices of the rational homogeneous operators at the fixed points t…

Consider algebraically integrable third-order differential operators $L$ with one pole, in the notation of the classification subsection, and let $q$ and $r$ be the associated gap…

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…

Let $L$ be an algebraically integrable differential operator of order $n\ge 2$, with gaps $q_1,\dots,q_{n-1}$ in its local indices, and call $L$ fully cyclically symmetric when it…

Let $L$ be an algebraically integrable differential operator on an elliptic curve whose coefficients satisfy the symmetries under consideration. Symmetric-operator conjecture. In t…