Abstract
Random geometry involves calculating expectations and probabilities on random geometric objects, typically surfaces (hyperbolic surfaces, discrete surfaces, surfaces immersed in a target space, or carrying certain fields, etc.)
Remarkably, the generating functions counting surfaces of fixed topology are often algebraic functions. In addition, there is a universal recurrence called topological recurrence, which relates the enumeration of surfaces of genus g with n edges to that of disks (g=0,n=1) : " if you know how to enumerate disks, topological recurrence tells you how to enumerate all topologies. "
The generating function of the disks is called the spectral curve. This observation allows us to reformulate the enumeration problem in the language of algebraic geometry: once the spectral curve has been specified, all the other generating functions can be derived.
This framework can also be interpreted through the prism of mirror symmetry. From this perspective, an enumeration problem is the " mirror " of an algebraic curve, and enumeration calculations translate into complex analysis calculations on this curve.
