I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to what results heavily rely on this machinery (incl. references to the original papers and alternative explanations of the proof of technique).

I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl. references to the original papers and alternative explanations of the proof of technique).