10 problems

Toda-to-KdV asymptotic correction conjecture. If $f$ satisfies

Toda spectral-curve approximation conjecture. With $A(n)=2+\epsilon^2g(\epsilon n)$ and $B(n)=-1$, the sets $S_K\cap\calB_g$ and $S_K\cap\calB_{A,B}$ can be made arbitrarily close…

Let $U({\bf p};\hbar)=\partial_{p_1}^2\log\bigl(\mathcal D^{\rm NP}_{\mu,\nu}(\hbar\cdot{\bf t}({\bf p});\hbar;w)\bigr)$ be the stated solution of the KdV equations. Let $G_2$,…

Hierarchy-identification conjecture. The topological deformation of the Principal Hierarchy is related to the $q$-deformed KdV hierarchy and the Volterra hierarchy through these La…

KdV-characterization conjecture. The set of Hirota bilinear equations

Let $F^c$ and $F^o$ be the closed and open generating series of descendent integrals, respectively, and set … Here $u$, $s$, and $t_d$ are the formal variables appearing in the gen…

Let the cubic Hodge hierarchy be the integrable hierarchy associated with the cubic Hodge integrals involving $\Lambda_g^\vee(1)\Lambda_g^\vee(r)\Lambda_g^\vee(-1-r)$, with paramet…

Open Witten KdV conjecture. For every $n\geq 1$,

Let ${\cal D}$ be the ring of commuting KdV derivations, let $A$ be the space of KdV fields, and let $\bar S_{2n}$ be defined by … Using the identification…

Let $A={\mathbb C}[u,u',\ldots]$ be the space of KdV fields, and let ${\cal D}={\mathbb C}[\partial_1,\partial_3,\ldots]$ be the ring of commuting derivations, where…