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Berlin-Hamburg-Seminar am 04.07.2025

Arthur Limoge (Heidelberg) Poincaré-Birkhoff theorems and the Three-Body Problem

The original Poincaré-Birkhoff theorem (1912-1913), originally stated for the Planar Three-Body Problem, is one of the most fundamental theorems on the dynamics of the annulus. In this talk, we discuss how one can generalise this theorem to higher dimensions (in particular, to the Spatial Three-Body Problem), summarising five years of attempts/refinements of the model.

Michael Hutchings (Berkeley) Reeb orbits frequently intersecting a symplectic surface

Consider a symplectic surface in a three-dimensional contact manifold with boundary on Reeb orbits. We assume that the rotation numbers of the boundary Reeb orbits satisfy a certain inequality, and we also make a technical assumption that the Reeb vector field has a particular ``nice'' form near the boundary of the surface. We then show that there exist Reeb orbits which intersect the interior of the surface, with a lower bound on the frequency of these intersections in terms of the symplectic area of the surface and the contact volume of the three-manifold. No genericity of the contact form is assumed. An application of this result gives a very general relation between mean action and the Calabi invariant for area-preserving surface diffeomorphisms.