Beyond global surfaces of section: some alternative tools to study three dimensional

This talk will be on alternative tools to study global properties of Reeb flows in dimension 3, when global surfaces of section are unexpected to exist. First I will present a version of the Poincaré-Birkhoff fixed point theorem for Reeb flows on the tight three-sphere, joint with U. Hryniewicz and A. Momin, which applies to Finsler geodesic flows on the two-sphere. After that, I will discuss about the existence of rational $3-2-3$ foliations for Reeb flows on connected sums of $RP^3 # RP^3$, joint with N. de Paulo and U. Hryniewicz, which applies to the Euler problem of two fixed centers in the plane.