Open and real Gromov-Witten invariants: theory and applications

Gromov-Witten invariants play a central role in interactions of symplectic topology with
algebraic geometry, dynamics, and string theory. I will explain fundamental issues arising
in the construction of open and real analogues of these invariants and describe how to
resolve them in many cases, building on previous work by Fukaya, Solomon, and others.
I will then outline some applications, including Kontsevich's type formula for
higher-dimensional analogues of Welschinger's counts of real curves and Walcher's
mirror symmetry formula for real curves in the quintic threefold.