Counting surfaces: A mixed bag of combinatorics, geometry, and physics
Given some polygons, how many ways can you glue their edges together to create a particular surface? This enumeration is governed by two simple objects --- a "spectral curve" and a "quantum curve" --- that are related by a mysterious process called "quantisation". We will discuss exactly what this means and why it is mysterious, before observing the same structure in seemingly unrelated problems that involve permutations, knots and more. The talk will be G-rated, in the sense that almost no prerequisites are required!