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).

1$\begingroup$ Hi warzasch, and welcome to the Math.SE. I upvoted your question but nevertheless I voted also to close it: the reason is that symplectic geometry is extremely broad subject, with many applications and result. It would be almost impossible to give an exhaustive answer to such a broad question, so I invite you to consider narrowing its scope a bit, in order to provide a possible answerer the hope to give a nice answer. $\endgroup$– Daniele TampieriCommented Oct 28, 2023 at 12:09

$\begingroup$ I have not asked for an exhaustive answer, only for examples, a starting point. $\endgroup$– warzaschCommented Oct 28, 2023 at 12:20
1 Answer
Symplectic Floer theory is essentially a variational technique. The energy functional is the action functional on the space of loops. Its critical points are periodic orbits of the Hamiltonian used to defined the action functional. The variational theory of periodic solutions of Hamiltonian equation resembles from this point of view the variational theory of geodesics. The similarities stop here since the indices of these critical points are infinite so traditional Morse theory seems inapplicable.
To make things worse, the gradient flow lines of this functional, are described by nonlinear elliptic equations similar to the CauchyRiemann equation. Viewed as evolution equations, the elliptic equations are ill posed. This is a nogo for p.d.e. experts and explains why people have not investigated the gradient flow of this action functional until Andreas Floer did.
To quote Freeman Dyson "the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible".
Symplectic Floer theory is very challenging analytically. As for references, you could go to the original papers of Floer. His writing is challenging but full of great ideas.