"Open Gromov-Witten Theory and the structure of real enumerative geometry"
Abstract: I plan to illustrate the application of open Gromov-Witten theory to real enumerative geometry through examples in the real projective plane. Lines in the projective plane are algebraic curves of degree 1 and genus 0. The beginning of plane geometry is the problem of determining the number of lines through 2 points. In 1870, Zeuthen generalized this problem to the problem of enumerating genus 0 plane curves of degree d through 3d-1 points. Recently, Welschinger introduced an analogous problem for signed counts of real curves. Open Gromov-Witten theory explicitly relates the real enumerative problem with its classical complex analog, simultaneously solving both problems. The formulas are most naturally expressed as a PDE very similar to the WDVV equation. Moreover, like the WDVV equation, the PDE of open Gromov-Witten theory holds for arbitrary target manifolds.