Bertrand Eynard
(CEA Saclay)
Topological recursion and WKB formal asymptotic expansion.
From ODE's to loop equations, and the notion of "quantum curve".
1st part: how to get loop equations from an ODE.
2nd part: formal solutions of loop equations are obtained by topological recursion, which gives a formal expansion of solutions of the ODE.
This approach gives a geometric meaning to the coefficients of the WKB expansion, as the TR invariants of the spectral curve.
We shall present some cases in details: Airy function.
If time permits we shall comment the case of Hitchin systems recently done by
Dumitrescu-Mulase.