7 problems

Generalized Toda integral formula conjecture. Formulae -- are valid not only for $m=2,3$, for which they were explicitly calculated, but also for an arbitrary value of $m$.

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 $\tau(x,{\bf t},\bar{\bf t})$ be a tau-function of the 2D Toda lattice satisfying the necessary condition … at $t_k+\bar t_k=0$. Higher-flow constrained Toda conjecture. This c…

String equation. The required solution to this reduction is given recursively by

Let $k$ and $m$ be positive integers, let $M_{k,m}$ be the Frobenius manifold of degree-$(k,m)$ Laurent polynomials, and let $D^{M_{k,m}}$ be its Givental total descendent potentia…

Let $Z_{\mathrm{inst}}$ denote the Nekrasov instanton partition function, and let a tau-function of the Toda lattice hierarchy be a function satisfying the hierarchy's tau-function…