Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from Riemann surfaces with boundary to a symplectic manifold, with boundary conditions and various constraints on boundary and interior marked points. The presence of boundary leads to bubbling phenomena that pose a fundamental obstacle to invariance. In a joint work with J. Solomon, we developed a general approach to defining genus zero OGW invariants. For real symplectic manifolds in dimensions 2 and 3, these invariants are strongly related to Welschinger's invariants.
The construction uses the heavy machinery of Fukaya A_\infty algebras. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW invariants has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called the open WDVV equations. For projective spaces, this system of PDE generates recursion relations that allow the computation of all invariants.
No prior knowledge of any of the above notions will be assumed.