Nicola Pagani: How many rational curves in the plane pass through n given points?

We consider the problem of determining the number of
rational curves of degree d in the plane that pass through n points in general position. The answer is always zero or infinity, unless the number of points is chosen to be equal to 3d-1. In this case we call the result N_d. The number N_1 equals 1, and this is such a basic property of the geometry of the plane that it was singled out by Euclid as his first postulate. The computation of N_d for d up to five was performed by means of clever and increasingly difficult constructions in the course of the last 2300 years. In 1993, Kontsevich found a simple recursive formula for N_d, which allows one to compute all such numbers in terms of N_1. In this talk we will review and prove his formula in elementary terms.