Computing Node Polynomials for Plane Curves

Enumeration of plane algebraic curves has a 150-year-old history. A combinatorial approach to this problem, inspired by tropical geometry, was recently suggested by Brugalle, Fomin, and Mikhalkin. I will explain this approach and its applications to computing Gromov-Witten invariants (or Severi degrees) of the complex projective plane, and their various generalizations.

According to Goettsche's conjecture (now a theorem), these invariants are given by polynomials in the degree d of the curves being counted, provided that d is sufficiently large. I will discuss how to compute these "node polynomials," and how large d needs to be.