The main goal of this master thesis is to give a self-contained proof of the Gromov compactness theorem for pseudoholomorphic curves and the non-squeezing theorem in symplectic topology. Pseudoholomorphic curves are smooth maps from a Riemann surface into an almost complex manifold that respect the almost complex structures. If the target manifold is a complex manifold, we recover the notion of holomorphic maps, so pseudoholomorphic maps can be seen as the generalization of holomorphic maps to the almost complex setting. Pseudoholomorphic curves were introduced by Gromov in a ground-breaking paper published in 1985, [Gro]. Since then, they have become one of the main tools in the field of symplectic topology.

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2014, Director: Ignasi Mundet i Riera