A meanwhile standard idea for producing geometric invariants (f.e. Donaldson Theory, Gromov-Witten Theory, Symplectic Field Theory) consists of counting solutions of nonlinear elliptic systems associated to the geometric data. Although the basic idea is easy, the implementation can be very difficult and involved, due to a usually large number of technical issues, which in more classical approaches to such type of problems are more than "painful".

In symplectic geometry the partial differential equation in question is the nonlinear Cauchy-Riemann equation. If there weren't these inherent compactness and transversality problems, the solution sets of the nonlinear Cauchy-Riemann operator would be nice manifolds or orbifolds, and the invariants could be achieved by integration of suitable differential forms over them. As it turns out, the arising difficulties can be overcome by a drastic generalization of nonlinear Fredholm theory and new methods for implementing it in concrete problems.