In Search of a Hidden Curve

Abstract: Three enumeration problems are discussed in these lectures. One is a detailed account on simple Hurwitz numbers, explaining how the problem was solved by discovering a key curve. This key ignites the generating functions of Hurwitz numbers and drives them into polynomials. The unexpected polynomiality then brings us simple proofs of the Witten-Kontsevich theorem and the $\lambda_g$-theorem of Faber-Pandharipande. An analogous problem associated with Catalan numbers is also presented, which has a simpler feature in terms of analysis. The third enumeration problem is a quest of identifying a hidden curve behind the Ap\'ery sequence, which remains to be discovered. It is examined how a counting problem leads to a \emph{spectral curve} through a differential equation, of Picard-Fuchs type but can be nonlinear. The counting problem is a geometric question associated with the \emph{genus $0$, one marked point} case. Going from the $(0,1)$-case to arbitrary $(g,n)$-case is explained as a process of \emph{quantization} of the spectral curve. This perspective of quantization is discussed in a geometric setting, when the differential equations have holomorphic coefficients, in terms of Higgs bundles, opers, and Gaiotto's conformal limit construction. In this context, however, there are no counting problems behind the scene.