23 problems

Central-invariant conjecture. The central invariants of the bihamiltonian structure of the $(n,1)$-type rational reduction of the 2D-Toda hierarchy are all equal to

Anti-involution conjecture. There are a neighborhood $U\ni m$ and an anti-involution $\alpha$ of the bihamiltonian structure on $U$ such that

Local classification conjecture. There is a local isomorphism of bihamiltonian structures between $M$ near $m$ and $\Phi(\cal W)$ near $m'$.

Metaconjecture. Other integrable systems of mathematical physics should be locally isomorphic to a direct product of several copies of Example $$ .

In the setting of the Casimir-family criterion, let $M$ carry compatible Poisson structures $\{,\}_{1}$ and $\{,\}_{2}$, let $F_{l,\lambda}$ and $W_1(m)$ satisfy the preceding hypo…

Let $M$ be a manifold with compatible Poisson structures $\{,\}_{1}$ and $\{,\}_{2}$. Let $L$ be a finite set with $r$ elements, and let $F_{l,\lambda}$, for $l\in L$ and…

Let $M$ and $M'$ carry homogeneous bihamiltonian structures … For small open subsets $U\subset M$ and $U'\subset M'$, let ${\cal B}_{U}$ and ${\cal B}_{U'}$ be the corresponding we…

The odd-dimensional open Volterra system, the even-dimensional periodic Volterra system, the full Toda lattice, the multidimensional Euler top, and the regular case of the cited bi…

A bihamiltonian structure is studied through its local decomposition into components, with the relevant components expected to be odd-dimensional. Odd-dimensional product meta-conj…

Let $M_{1}$ and $M_{2}$ be manifolds equipped with bihamiltonian structures, and equip $M_{1}\times M_{2}$ with their product bihamiltonian structure. Product-flatness conjecture.…

Let a homogeneous bihamiltonian structure have type $\left(2k_{1}-1,2k_{2}-1,\dots,2k_{l}-1\right)$. Tensorial criterion for Kronecker structures. There exist $N>0$ and natural ten…

The Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bihamiltonian Lie coalgebra are the systems under consideration, together with the kno…

A bihamiltonian deformation of the genus-zero hierarchy is a deformation encoding the recursion relations among correlators of a two-dimensional topological quantum field theory. D…

Consider a multi-component bihamiltonian structure of hydrodynamic type … for $i,j=1,\dots,n$ and $a=1,2$, where $(g^{ij}_1)$ and $(g^{ij}_2)$ form a flat pencil of metrics on a ma…

Let $({\mathcal L}(M);\omega_1,\omega_2)$ be a semisimple bihamiltonian structure of hydrodynamic type. Write ${H}^2({\mathcal L}(M);\omega_1,\omega_2)$ for its second bihamiltonia…

Zakharevich's conjecture. If $w(t)$ is integrable at $n+3$ distinct values of $t$, then the Veronese curve of distributions is integrable.

Gelfand–Zakharevich conjecture. Locally, the Veronese web is complete: it determines the bihamiltonian structure up to an isomorphism.

Equal-central-invariants conjecture. The central invariants of the bihamiltonian structure of the asymmetric RR2T of type $(n,1)$ are all equal to

Let $g\geq0$ and $0\leq j\leq 3g+1$, and let $A^\varphi_{2g,j;\Omega(\boldsymbol{\sigma})}$ be the coefficients of the genus-$g$ part of the first Hamiltonian operator of the abstr…

The paper considers $\tau$-symmetric integrable hierarchies of bihamiltonian evolutionary partial differential equations. Volterra universality conjecture. The universal object for…

Let a semisimple Frobenius manifold have an associated Principal Hierarchy, and let $(P_0,P_1)$ be a tau-symmetric bihamiltonian deformation of that hierarchy. The deformation has…

Main conjecture. The operator $K_2$ is Poisson and compatible with $K_1$. Moreover, the Poisson brackets associated with $K_2$ and $K_1$ give a bihamiltonian structure for the doub…

Let $b\eta_1,\eta_2d$ be a Kronecker bihamiltonian structure, and let $\{\mathcal{S}'_\lambda\}$ be the Kronecker web induced on the local base of its lagrangian foliation. Gelfand…