16 problems

Let $z=(z_1,\ldots,z_N)$ be a configuration of $N$ equal vortices, each of vorticity $1$, and define the pre-shape set … For almost every $z\in\Gamma_N$, the time average of the em…

Integrability conjecture. These equations are integrable in the Jacobi sense with polynomial first integrals $I_1$ and $I_2$, where

Let ${\mathfrak C}_i\subset{\mathfrak C}_{gen}$ be a connectivity component of the critical manifold, and let $a\in{\mathfrak C}_i$. Suppose the nonzero spectrum of the vertical li…

Let a $\tau$-symmetric Hamiltonian perturbation of the Riemann--Hopf hierarchy be written in generalized standard form, with first Hamiltonian … Here $\mathcal{P}_k$ denotes the se…

Let $A_1$ be a first-order Hamiltonian operator of Ferapontov type for a first-order WDVV system, compatible with the third-order Hamiltonian operator supplied by the Hamiltonian t…

Polynomiality conjecture. If $P$ and $R$ are compatible, meaning that their Schouten bracket vanishes, $[P,R]=0$, then every entry of $Q^{\alpha\beta}$ is a polynomial of degree tw…

Jimbo–Miwa–Ueno differential conjecture. One has

Parallel-curve characterization conjecture. (i) The non-solvability of the group $\pi_1^{\mathcal{O}\text{-ab}}$ is characterized by the presence of a pair of parallel curves in th…

Hamiltonian structure conjecture. There exist an energy $\mathcal{E}:\mathcal{Z}\to\mathbb{R}$ and a skew-symmetric operator…

Let $T^2$ be the two-dimensional torus, and let a natural Hamiltonian system on it have a nonconstant potential. A linear, quadratic, or higher-degree polynomial integral is an int…

A chiral topological phase of matter (TPM) is a TPM whose topological order is a chiral unitary modular category. A CLP Hamiltonian is a commuting local-projector Hamiltonian. Chir…

Let a Levin–Wen (LW) model be the Hamiltonian model determined by a unitary fusion category, and consider its ground-state subspace. Levin–Wen error-correction conjecture. The grou…

Let a $2D$ topological phase of matter (TPM) be an equivalence class of connected stable topological Hamiltonian schemas, and let its topological order be the unitary modular categ…

Let $K_{4,36}$ denote the order-five variational-equation entry obtained by transporting the jet along the cycle $\gamma=\gamma_-^{-1}\gamma_+^{-1}\gamma_-\gamma_+$. Let…

Bidaut-Véron–Szulkin conjecture. System $(SH)$ with $a=b=0$ admits no radial or nonradial global solutions if and only if $(\delta,\mu)$ lies under the hyperbola

Let $\epsilon$ and $\delta$ be the perturbation parameters, and let $\mathcal{H}_{nm}$ denote the Hamiltonian contribution of bidegree $(n,m)$ in the grading with respect to…