How to count curves: from nineteenth-century problems to twenty-first-century solutions

Find the next term in the sequence 1, 1, 12, 620, 87304. This particular problem belongs to a branch of mathematics called enumerative geometry. This is concerned with curve-counting - counting the number of curves that can be drawn on a particular geometric object. The sequence above is easy to describe: each term represents the number of curves, with increasing complexity, one can draw though a certain number of points on a plane. Despite its simplicity, the problem remained unsolved for most of the twentieth century. The solution - a formula with which one may calculate any term in the series - was discovered only in the century's closing decade. This article will describe the above problem, and some of the unexpected mathematics and physics that was used in finding its solution [corrected]