The problem of computing the number N(d) of degreed rational curves through 3d-1 points in the complex projective plane has very classical geometric origins.
The computation of N(1) is the same as counting the number of straight lines through two points in the Euclidean plane, so N(1)=1. The computation of N(2) is the same as counting the number of conics (ellipses, parabolas or hyperbolas) through five points in the Euclidean plane, so N(2)=1.
The problem becomes progressively more difficult as the degree increases. Researchers computed the numbers N(3)=12 and N(4)=620 about 100 years ago, and the demanding computation of N(5)=87304 only occurred in the current decade. Maxim Kontsevich achieved a major breakthrough in 1993 when he realized that the conjectured associativity property of the "quantum cohomology" construction in theoretical physics can be translated to a recursion formula that easily computes N(d) for any desired degree.
The required associativity property of quantum cohomology was soon proven by mathematicians Yongbin Ruan and Gang Tian, establishing the validity of the Kontsevich recursion formula. The associativity property of quantum cohomology is described in terms of a nonlinear partial differential equation known as the WDVV (Witten, Dijkgraaf, Verlinde, Verlinde) equation. The WDVV equation admits a particular solution that is in effect a generating function for the N(d).
Roughly speaking, the N(d) occur as the coefficients in a power series expansion of the generating function, which explains the origin of the Kontsevich recursion relation. It is unfortunate that the generating function appears to be very complicated, and its explicit form is not known.
Boris Dubrovin developed the theory of Frobenius manifolds to provide a differential-geometric description of solutions to the WDVV equations. In particular, the "quantum cohomology Frobenius manifold" corresponding to the generating function of the N(d) contains all the N(d) as covariant derivatives of a certain tensor field, so it is the appropriate geometric reformulation of the generating function.
Maxim Kontsevich and Yuri Manin posed the problem of explicitly constructing such quantum cohomology Frobenius manifolds.
MU's Jan Segert recently succeeded in constructing the quantum cohomology Frobenius manifold for the complex projective plane, completing a project that he started nearly two years ago. This construction, which is the first of its kind, encodes all the enumerative constants N(d) in a single geometric object. The construction is based on a classifying space for resonant isomonodromic deformations and combines techniques of differential geometry and algebraic geometry to obtain an explicit global description.
