20 problems

The WDVV equations are the associativity equations for a Frobenius manifold prepotential, and a polynomial WDVV solution means a polynomial prepotential satisfying these equations.…

A WDVV solution is a solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations, and it has the required good analytic properties when it satisfies the analytic conditions intend…

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…

Let $N$ be the dimension of the WDVV system, let $S_L$ denote its linear subsystem, and let $S_{lambda amu b}$ denote the remaining equations indexed by $2<\lambda<\mu\leq N$ and…

Let $S_L$ be the subsystem of WDVV equations that is linear in the $t^p$-free derivatives, and let the remaining equations be the corresponding equations not included in $S_L$. Pas…

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…

A WDVV-type system is a nonlinear PDE system admitting the same bi-Hamiltonian structure as the WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous…

The WDVV equations are equations for the potential governing a Frobenius-manifold structure, and a Hamiltonian operator is a differential operator defining a Poisson bracket for th…

A WDVV system is a system of first-order PDEs considered together with its projective-geometric structures. A quadratic line complex is the associated family of lines in projective…

Let a WDVV system be written as a quasilinear system of first-order PDEs, and let an HHO mean a homogeneous Hamiltonian operator. In the case $\eta=\eta^{(1)}$, consider the local…

Let $F^o$ be a solution of the WDVV equations satisfying the unit and homogeneity properties, together with its calibration. Under possibly additional assumptions, let…

Let $a^{k}$ be the initial coordinates of the six-component WDVV hydrodynamic-type system, and let $\hat{A}_{2}$ denote its reconstructed third-order Hamiltonian operator. A Casimi…

The six-component systems under consideration admit six components and the known examples concern the case $n=N=3$. A Dubrovin–Novikov type third-order Hamiltonian operator is a th…

Conjecture C. For every $(m,\ell)\in\mathbb Z^2\setminus\{0\}$, the series

Conjecture A. There exist unique series $\varphi_{m,\ell}$ for $(m,\ell)\in\mathbb Z^2\setminus\{0\}$ such that the stated symmetries and initial conditions

Let $F(v)$ be a solution of the WDVV equations, let $\hat F(\hat v)$ be its inversion, and let $\tilde{\mathcal F}_g$ and $\hat{\mathcal F}_g$ be the corresponding genus-$g$ free e…

Let $P_2^{[0]}$ be the second Hamiltonian structure of the principal hierarchy, and let $P_2$ denote its image under the quasi-Miura transformation … Here a Hamiltonian structure h…

Let $F$ be a semisimple Frobenius-manifold potential with principal hierarchy, and let $\hat F(\hat v)$ be obtained from $F$ by the type-2 symmetry of the WDVV equations. Denote th…

Let $N$ denote the dimension of the algebra used in the deformation-theoretic construction, and let the WDVV equation be the nonlinear PDE system arising from the reduction $C=1$,…

Let $W$ be a Weyl group, let $\Omega$ be the relevant domain, and let $J(W)$ be the Jacobi group acting on it. Write $\mathcal{R}_W$ for the root system of $W$, let…