1,502 problems

Let the moduli space of curves carry its intersection theory, and let the KdV hierarchy be the integrable hierarchy associated with the Korteweg–de Vries equation. Witten's conject…

Consider the full constrained 3D system studied in the paper, rather than either of its restrictions to an invariant hypersurface. The Painlevé property means that its solutions ha…

Exact-solvability conjecture. All maximally superintegrable systems are exactly solvable.

Integrability conjecture. Any system of Lagrangians given by such a 1-form is integrable in the sense of multidimensional consistency, equivalently in the sense of commuting flows.…

Let a Hamiltonian system possess a complete algebra of first integrals that provides noncommutative, or superintegrable, integrability. Mishchenko–Fomenko conjecture. Noncommutativ…

Invertibility conjecture. $\mathfrak A_k$ is invertible for all parameters $\mu_i\in[0,1]$.

The open intersection theory studies intersection numbers on moduli spaces of Riemann surfaces with boundary. Let the open intersection-number generating function be the generating…

Let ASDYM denote the anti-self-dual Yang–Mills equations, and let an integrable model mean a model obtained through the dimensional-reduction construction under discussion. Ward's…

Let a principally polarized indecomposable abelian variety be given, with its Kummer variety and associated theta functions. A family of flex lines has a fourth-order jet when its…

Let a cohomological field theory be given, and let its double ramification hierarchy be the Hamiltonian integrable system constructed from double ramification cycles. Let its Dubro…

For a positive integer $N$, let $X_j^{(N)}(z)$, for $j=0,\dots,N$, be the tetrahedral $X$-operators depending on the spectral parameter $z$. Commutativity conjecture. For an arbitr…

Frenkel–Reshetikhin conjecture. The eigenvalues of the transfer matrices can be obtained by computing the q-character of an associated representation and performing a suitable subs…

Let $X=\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)$ be the resolved conifold equipped with the anti-diagonal $C^*$-action on the fibers, and let the generati…

Ablowitz–Ramani–Segur conjecture. There exists a connection between nonlinear integrable PDEs and nonlinear ODEs with the Painlevé property such that any ODE emerging from the redu…

Affine Bethe-ansatz completeness conjecture. For generic $(k,\ell)$, the Bethe ansatz provides all eigenstates of the integrable structure in every space…

A Birkhoff billiard is the billiard system inside a smooth strictly convex plane domain, with trajectories reflecting elastically at the boundary. Such a billiard is integrable whe…

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 $y(\lambda)$ be the tritronquée solution of Painlevé I, and let $\alpha\in\mathbb{C}$ be a pole of this solution. Dubrovin's conjecture. If $\alpha$ is a pole of the tritronqué…

Let ${\mathcal H}$ be the $\mathfrak{U}_q(\mathfrak{gl}_n)$-invariant Hamiltonian, and let ${\mathfrak b}_{(N-k)k}$ be a state in the $k$-sector satisfying … For…

The magnetic flows under consideration are defined on spheres $S^{n-1}$, with dimension parameter $n$ and magnetic parameters $\kappa$ as in the preceding construction. Complete-in…

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 $x_1,x_2$.

The equivariant Gromov–Witten theory of $\mathbb P^1$ is governed by the equivariant Toda hierarchy. Toda conjecture. This is an analogue of the Witten conjecture connecting an enu…

Let $w^\alpha$ be the coordinates obtained from the Dubrovin–Zhang quasi-Miura transformation of the dispersionless hierarchy, and consider the transformed equations, Hamiltonians…

In an integrable model, let the long-time asymptotic stationary state be the state reached after the system evolves for arbitrarily long times from a nonequilibrium initial conditi…

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…