Holomorphic curves in symplectic topology

Symplectic Topology has its roots in global questions in
Hamiltonian dynamics. It aims to answer questions like, does a Hamiltonian
flow (solution to Hamilton's equations in Classical Mechanics) always have
a periodic orbit? Or how many distinct "types of Classical Mechanics" can
exist on a given phase space? What are the obstructions to condensing a
beam in a particle collider? The field has been revolutionized in the mid
80s through the introduction of pseudo-holomorphic curves by Gromov.

In this talk I will given an overview of some aspects of Symplectic
Topology and the theory of pseudo-holomorphic curves. The talk will focus
on applications and will be geared toward a general audience. At the very
end we will explain some recent results in this area.