Singular symplectic manifolds

23 September - 16 December 2025

Tuesdays, 13:15 - 15:00

Location: HG G 43

First lecture: 23 September

external page Course website

Abstract

The exploration of symplectic structures on manifolds with boundary has naturally led to the identification of a particularly tractable class of Poisson manifolds. These manifolds are symplectic away from a critical hypersurface, where the structure degenerates. In the literature, they are referred to as \(b\)-symplectic or log-symplectic manifolds. Such structures arise, for instance, in the study of the space of geodesics of the Lorenz plane, and they provide natural phase spaces for problems in celestial mechanics, such as the restricted three-body problem.

From a geometric perspective, these manifolds can be described as open symplectic manifolds endowed with a cosymplectic structure at their open ends. The technique of desingularization (or "deblogging") associates to a singular symplectic structure with even exponent (the so-called \(b^{2k}\)-symplectic structures) a family of smooth symplectic structures, while in the odd exponent case (the \(b^{2k+1}\)-symplectic structures) it produces a family of folded symplectic structures. This method has good convergence properties and extends naturally to the odd-dimensional setting. In this way, the desingularization procedure unifies, under a single framework, several geometries including symplectic, folded-symplectic, contact, and Poisson geometry.

In this minicourse we explore several aspects of the geometry of singular symplectic manifolds, including toric actions and quantization, with particular emphasis on applications to celestial mechanics. For \(b\)-symplectic manifolds we will also discuss several potential approaches to defining Floer homology, opening a new direction at the interface of symplectic topology and Poisson geometry. We use \(b\)-symplectic manifolds as a proof of concept for viewing Poisson manifolds through symplectic models built on Lie algebroids, and conclude by situating singular symplectic manifolds within the broader landscape of Poisson geometry. This perspective not only provides a practical pathway for applications in mechanics and quantization, but also opens the door to exporting tools from symplectic topology — such as Floer homology — to the Poisson realm.