Research-Reeb dynamics and pseudo-holomorphic curves

Research

Colloquium

Reeb dynamics and pseudo-holomorphic curves

Speaker: Pedro Salomão (NYU Shanghai)
Time: Apr 13, 2023, 16:30-17:30
Location: M1001, College of Science Building

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

Pedro A. S. Salomão is a Visiting Professor of Mathematics at NYU Shanghai. Salomão is also a faculty member at the University of São Paulo. His research interests are largely toward understanding global properties of conservative dynamical systems, utilizing recently developed methods in symplectic topology. This research area has now been called Symplectic Dynamics. Salomão has been using pseudo-holomorphic curves to study the existence of global surfaces section and transverse foliations, constructing algebraic invariants that detect periodic orbits and, more recently, estimating systolic ratios of Reeb flows.