Eva Miranda (UPC Barcelona), March 9, 1 pm EST

Singular Hamiltonian and Reeb Dynamics--First Steps

Abstract: Floer theory, which mimics an infinite dimensional Morse approach to the study of critical points of smooth functions, appeared as an attempt to prove Arnold conjecture. The theory is more or less well understood in some compact cases.

Non-compact symplectic manifolds can sometimes be compactified as singular symplectic manifolds where the symplectic form "blows up" along an hypersurface in a controlled way (b^m-symplectic manifolds). In natural examples in Celestial mechanics such as the 3-body problem these compactifications are given by regularization transformations à la Moser/Mc Gehee etc.

I will use the theory of b^m-symplectic/b^m-contact manifolds (introduced by Scott, Guillemin-Miranda Weitsman and Miranda-Oms) as a guinea pig to propose ways to extend the study of Hamiltonian/Reeb Dynamics to singular symplectic/contact manifolds. This, in particular, yields new results for non-compact symplectic manifolds and for special (but, yet, meaningful) classes of Poisson manifolds.

Inspiration comes from several results extending the Weinstein conjecture to the context of b^m-contact manifolds (joint with Cédric Oms) and its connection to the study of escape orbits in Celestial mechanics and Fluid Dynamics (joint with Cédric Oms and Daniel Peralta-Salas). Those examples motivate a model for (singular) Floer homology (joint with Joaquim Brugués and Cédric Oms).

I'll describe the motivating examples/results and some ideas to attack the general questions.