Eva Miranda (Universitat Politècnica de Catalunya) will speak in the geometry seminar on Tuesday 26 May 2020 at 11ham in the Salle des Profs. Eva’s title is The symplectic and contact geometry of forms with “singularities” and her abstract is below.
We will overview the study of symplectic and contact structures with singularities which appear modelling some problems in Celestial Mechanics and Fluid Dynamics and describe several applications to the study of their Hamiltonian (and Reeb) Dynamics.
In these motivating examples the singularities are associated to the line at infinity or collision set and are an outcome of regularization techniques. These singular symplectic structures ($b^m$-symplectic structures) can be formalized as smooth Poisson structures which are symplectic away from a hypersurface and satisfy some transversality properties. We can desingularize these structures associating a family of symplectic structures (with good convergence properties) to singular symplectic structures with even exponent (the so-called $b^{2k}$-symplectic structures) and a family of folded symplectic structures for odd exponent ($b^{2k+1}$-symplectic structures) yielding, in particular, topological constraints for their existence. This procedure generalizes to its odd-dimensional counterpart (joint work with Cédric Oms) and puts in the same picture different geometries: symplectic, folded-symplectic, contact and Poisson geometry. The applications of this “desingularization kit” include the construction of action-angle coordinates for integrable systems, the study of their perturbation (KAM theory) and the existence of periodic orbits away from the critical hypersurface. Time permitting, we will end up this talk proving an existence problem of contact structures with singularities on a given manifold and with some open questions concerning the study of Reeb Dynamics in the singular case, in particular, the existence of periodic orbits (Weinstein conjecture).