Abstract

At the beginning of the 20th century, H. Poincaré and G. D. Birkhoff profoundly contributed to the qualitative theory of dynamical systems motivated by their studies in Celestial Mechanics. Their fixed point theorem eventually inspired many conjectures on Hamiltonian dynamics, culminating in the development of new methods in Symplectic and Contact Topology. In this talk, I will revisit some classical results of Hamiltonian dynamics in light of these new methods, particularly using pseudo-holomorphic curves. I will discuss a generalization of the Poincaré-Birkhoff fixed point theorem for Reeb flows on the tight 3-sphere and also general results on the existence of global surfaces of section and transverse foliations.

Biography