Title: Desingularizing Singular Symplectic Structures
Speaker: Eva Miranda
Institution: Polytechnic University of Catalonia
Date & Time: October 18, 2024, 20:00–22:00
Location: Zoom ID: 904 645 6677, Password: 2024
The study of symplectic structures on manifolds with boundaries has naturally led to the identification of a “simple” class of Poisson manifolds: those that are symplectic away from a critical hypersurface but degenerate along it. Known in the literature as
b-symplectic or
log-symplectic manifolds, they arise in contexts such as the space of geodesics of the Lorenz plane and serve as natural phase spaces for problems in celestial mechanics, including the restricted 3-body problem. Geometrically, these manifolds can be described as open symplectic manifolds equipped with a cosymplectic structure on their open ends.
The technique of
“deblogging” or desingularization associates families of smooth structures to singular symplectic structures: for even exponents (
b²ᵏ-symplectic structures), it yields standard symplectic structures, while for odd exponents (
b²ᵏ⁺¹-symplectic structures), it produces folded symplectic structures. This method exhibits strong convergence properties and generalizes to odd-dimensional contact geometry, unifying symplectic, folded-symplectic, contact, and Poisson geometries under a single framework.
The desingularization toolkit finds broad applications, including the construction of action-angle coordinates for integrable systems, KAM theory, quantization, and counting periodic orbits.
Eva Miranda is a Chair in Geometry and Topology at the Polytechnic University of Catalonia (UPC) and a member of the Centre de Recerca Matemàtica (CRM). She has held visiting professorships at the Paris Observatory, MIT, the University of Toulouse, and the University of Paris 7, and was an honorary professor at CSIC and an Affiliate Researcher at the Paris Observatory. She directs the Geometry and Dynamical Systems Laboratory at UPC and leads the Geometry of Manifolds and Applications research group.
Miranda’s honors include two consecutive ICREA Academia prizes (2016, 2021), a Chair of Excellence from the Paris Mathematical Sciences Foundation (2017–2018), a Humboldt Foundation Bessel Award (2022), and the François Deruyts Prize (2022). She was an invited speaker at the 8th European Congress of Mathematics (8ECM), a Hardy Lecturer (2023, London Mathematical Society), and a Nachdiplom Lecturer (2025, ETH Zurich). In 2024, she was appointed Gauss Professor by the University of Göttingen.
