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).

Louis Merlin (Université du Luxembourg) will speak in the geometry seminar on Tuesday 3 March 2020 at 11ham in the Salle des Profs. Louis’ title and abstract are to be announced.

Marvin Dippell (University of Würzburg) will speak in the geometry seminar on Tuesday 21 January 2020 at 11ham in the Salle des Profs. Marvin’s title is Deformation of Coisotropic Algebras and his abstract is below.

Symmetry reduction is one of the key concepts in classical as well as quantum physics. Geometrically, this can be understood as Marsden-Weinstein reduction on symplectic manifolds or, more generally, in the case of a Poisson manifold, as reduction of a coisotropic submanifold. In this talk I will present an algebraic generalization of coisotropic reduction, which also encompasses various quantum reduction schemes. Following ideas from deformation quantization we will discuss deformations of such coisotropic algebras. Obstructions to such deformations will then be understood as cohomology classes in a suitably defined coisotropic Hochschild cohomology.

Mario Garcia-Fernandez (ICMAT, Madrid) will speak in the geometry seminar on Tuesday 24 March 2020 at 11ham in the Salle des Profs. Mario’s title is Gravitating vortices with positive curvature and his abstract is below.

I will overview recent work with Chengjian Yao and Vamsi Pingali in arXiv:1911.09616, where we give a complete solution to the existence problem for gravitating vortices with positive topological constant c>0, as introduced in arXiv:1510.03810. Our main result establishes the existence of solutions provided that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. To this end, we use a continuity path starting from Yang’s solution with c=0, and deform the coupling constant α towards 0. A salient feature of our argument is a new bound S(g) \geq c for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.