321 problems

For every solution of the Calogero–Moser derivative nonlinear Schrödinger equation with initial data $u_0\in L^2_+(\mathbb R)$ belonging to the stated natural critical initial-data…

Let $u(t,x)$ solve the KdV equation … Assume that the initial data $u(0,x)$ is almost periodic in $x$. Deift's conjecture. The solution $u(t,x)$ is almost periodic in time $t$ and…

Let $F^o(s,t_0,t_1,\ldots;u)$ be the open free energy, with double brackets denoting its derivatives with respect to the variables $t_i$ and $s$, and let $F^c$ denote the closed fre…

Let $g$ be a genus, and let $\mathcal{H}_\mathcal{C}$ be the Hirota variety associated with the rational nodal curve data, with main component $\mathcal{H}_\mathcal{C}^{M}$ and the…

Let $X_1$ be the resolved conifold $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)$ equipped with the anti-diagonal action, and consider its equivariantly Calab…

Let $(M,g)$ be a compact Riemannian manifold whose geodesic flow is completely integrable. Let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$, and suppose that all of its orthon…

Conjectured coefficient formula. The coefficients satisfy

Let $w^{\alpha}$ be the normal coordinates for the Dubrovin–Zhang hierarchy, let $\tilde u^{\alpha}$ be the normal coordinates for Buryak's double ramification hierarchy, let…

Let $\mathscr{N\!D}_n$ be the class of completely integrable systems on a $2n$-dimensional symplectic manifold with non-degenerate singularities. Let $Q(\mathscr{N\!D}_n)$ be the c…

Let $u$ be the parameter of the quantum transfer matrix, let $r\geq 1$, and let $a\in\{1,2,\dots,r\}$ and $m\in\mathbb{Z}_{\geq 1}$. Denote by $\widetilde{T}^{(a)}_{m}(v)$ the fuse…

Identify the root lattice of $A_{N-1}$ with … For $j\in\mathbb{Z}/N\mathbb{Z}$, define the Fourier-transformed Nekrasov functions by … where $\omega_0=0$ and $\omega_j$ are the fun…

Simplicity conjecture. All nonzero roots of $W_N^{[m,k,l]}(z)$ are simple.

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…

The integrable map corresponding to the linear equation, respectively the nonlinear equation, is denoted by … . A nonlocal diffeomorphism is a diffeomorphism whose action is nonloc…

Real-analyticity conjecture. On the complement of a discrete set of points, the functions $\lambda$, $a_i$, and $b_j$ are real-analytic in the variables $x_1,x_2$.

Let $\Omega$ be a strictly convex domain with smooth boundary, and let its billiard map act on the cylinder parametrized by…

Low-$T$ Bethe root pattern conjecture. All excitations of the quantum transfer matrix at low $T$ and sufficiently large $N$ can be parametrised in this way, and, up to corrections…

Let $n$ be the fixed number of Bethe roots, let $H$ be the parameter in the transfer matrix, and let $I_j$ denote the corresponding Bethe quantum numbers. Bethe quantum-number conj…

Let $L\geqslant 1$, let $u_1,\dots,u_L$ be inhomogeneity parameters, and define the inhomogeneous open-boundary transfer matrix … where $K^-(u)=K(u)$, $K^+(u)=K(u+2\eta)$, … … Set…

Let $M$ be a $2n$-dimensional Poisson manifold with a real polarization $P$ and two integrable systems given by Lagrangian fibrations $\pi_i:M\to B_i$, with generically transverse…

Quantum toroidal $\mathcal E_2$ Bethe ansatz conjecture. For each such eigenvector, there exist polynomials $Q_0$ and $Q_1$ whose roots satisfy the two displayed Bethe ansatz equat…

In a smooth integrable dynamical system, let the system be defined near a nondegenerate singular point, meaning a singular point satisfying the paper's nondegeneracy condition for…

Let $M$ be a real-analytic symplectic manifold of dimension $2n$, and consider a Hamiltonian system … Assume that the system is completely integrable by an infinitely dimensional n…

Let $d>3$ and let $f$ be the function appearing in the discrete-time $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra system. Let $C$ denote its Casimir function. An invariant is called t…

Let $a_n$ be the cluster variables associated with the extending vertex, and let $\tilde{K}_n$ denote the quantity whose period-two behavior is being conjectured. Period-two formul…