We introduce Gromov’s non-squeezing theorem, which is a foundational result in the field of symplectic geometry. We will see an outline of the proof of this result, which establishes why holomorphic curves are important in studying symplectic manifolds.

In the second part of the talk, we will see how holomorphic curves degenerate under a “multiple cut” of the symplectic manifold. In special cases, there is a correspondence between degenerated holomorphic curves and Mikhalkin’s tropical graphs, which are combinatorial objects lying in a plane. Consequently, the problem of counting holomorphic curves is reduced to the combinatorial problem of counting tropical graphs.