UBC Mathematics Department
http://www.math.ubc.ca
Even in the case of the complex projective plane, the result is stunning. Let N_d denote the number of rational curves of degree d passing through 3d-1 general points in {\bf CP}^2. Then the ``potential function'': \Phi(y_0,y_1,y_2) := \frac 12(y_0^2y_2 + y_0y_1^2) + \sum_{d \ge 1} N_d \frac{y_2^{3d-1}}{(3d-1)!} e^{dy_1} satisfies the following partial differential equation: \Phi_{222} = \Phi_{112}^2 - \Phi_{111}\Phi_{122} which allows one to solve for the N_d.
Such a result, unimagined in the course of more than a hundred years of intermittent study of such numbers by enumerative algebraic geometers, is just one of many corollaries of the newly developed quantum cohomology theory. This theory, which is a wonderful blend of ``physical'' insights and proofs coming from algebraic geometry and symplectic topology, has been extensively developed and interpreted over the past several years. In this talk, I will explain how potential functions and PDE's are associated to smooth complex projective varieties by counting rational curves, and present some of my favorite examples.