Sympletic Geometry Seminar 2020

Joint Symplectic Seminar in Rio

Meetings 2020

14:30 - 15:30 Vladimir Matveev (University of Jena) Nijenhuis Geometry: singularities and global issues.

The talk is devoted to Nijenhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis torsion. This is possibly the simplest geometric condition; for this reason it appears independently and many times in geometry, analysis and mathematical physics, in particular in the theory of projectively equivalent metrics and in the theory of integrable systems. The talk is based on a series of papers in progress aiming at re-directing the research agenda in this area to next level topics": * Singular points: what does it mean for a point to be generic or singular in the context of Nijenhuis geometry? What singularities are non-degenerate? What singularities are stable? How do Nijenhuis operators behave near non-degenerate and stable singular points? * Global properties: what restrictions on a Nijenhuis operator are imposed by compactness of the manifold? And conversely, what are topological obstructions on a manifold carrying a Nijenhuis operator with specific properties (e.g.with no singular points) ? We demonstrate that this research program is realistic by proving a series open problems some of which are easy to formulate.

16:00 -17:00 Alexsandro Schneider (Universidade Estadual do Centro-Oeste / Unicentro) Global surfaces of section for dynamically convex Reeb flows on Lens spaces.

In this talk, I will show some results that use pseudo-holomorphic curves in simplectizations to find global surfaces of sections. In our main theorem, we proved that a dynamically convex Reeb flow on the standard tight lens space (L(p,1), \xi_{std}), p > 1, admits a p-unknotted closed Reeb orbit P which is the binding of a rational open book decomposition with disk-like pages.