 
Summary: A PERSONAL TOUR THROUGH SYMPLECTIC TOPOLOGY AND
GEOMETRY
MIGUEL ABREU
1. Introduction
In this survey I will present a very personal tour through symplectic topology and geometry,
describing the following three paths and the way most of my work fits in them.
(i) Gromov's Compactness Theorem for pseudoholomorphic curves in symplectic manifolds
([23]) and the topology of symplectomorphism groups of rational ruled surfaces (sections
2 and 3, references [1, 2]).
(ii) AtiyahGuilleminSternberg's Convexity Theorem for the moment map of Hamiltonian
torus actions ([9, 25]) and K¨ahler geometry of toric orbifolds in symplectic coordinates
(sections 4 and 5, references [3, 4, 5]).
(iii) Donaldson's moment map framework for the action of the symplectomorphism group on
the space of compatible almost complex structures ([17]) and the topology of the space
of compatible integrable complex structures of a rational ruled surface (sections 6 and 7,
references [6, 7]).
Acknowledgments. I would like to thank my advisor, Yakov Eliashberg, and the collaborators
I had in the work described in this survey: Dusa McDuff, Gustavo Granja and Nitu Kitchloo. My
mathematical life has been much easier because of them.
2. Pseudoholomorphic Curves in Symplectic Manifolds
