To take part, please, on June 12 after 3:45 pm Novosibirsk time (UTC+7) connect to the Zoom conference via this link: .From the WKB point of view, the question is: starting from a differential operator (ex: $\hatP(x,\hbard/dx,\hbar) = \hbar^2 d^2/dx^2 - V(x)$), is there a geometric and systematic way to compute the terms $S_k$ in the WKB expansion of a solution $\psi(x) = \exp{\sum_{k=-1}^\infty \hbat^k S_k(x)}$. It is obvious that $y=dS_{-1}(x)/dx$ satisfies the algebraic equation $P(x,y,0)=0$ (classical curve). Topological recursion on the other hand is a recursive algorithm that starts from a classical spectral curve (ex: $P(x,y)=0$) and defines a sequence of meromorphic differential forms $\omega_{g,n}$ associated to it, in a geometric manner. The "Topological-Recursion-Quantum-Curve" conjecture is that, if you start from the classical spectral curve which is the character variety of the differential operator, then you get $S_k(x) = \sum_{2g-2+n=k} \int_o^x\dots \int_o^x \omega_{g,n}$ (this simplified formula is in fact suplemented by some subttleties that I will mention). This conjecture has been proved for a variety of cases, but so far proofs were specific to each curve, and no general method has been developped to prove the conjecture in its generality. I will review some known and proved cases, and mention some open ones, for example the application to knot theory. In knot theory, this conjecture is an extension of the famous volume conjecture.
