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401-5751-DRL 2 Credits DR D-MATH

Singular Symplectic Manifolds

Lecturers & Examiners: Dr. Eva Miranda
Doctoral students of I-Math (UZH) need to send an email to Jessica Bolsinger ( ) with the course number. The email should have the subject „Graduate course registration (ETH)“.
VVZ CR n/a

Last Updated: 2026-06-01 11:30:59

Abstract

Nachdiplom lecture

Content

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 Lorentz 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.

General Information

Language
English
Levels
DR

Examination

Type
ungraded semester performance

Course Components

Type Title Time & Place Hours
lecture Singular Symplectic Manifolds
If you plan to attend the lecture, please register by 21 September. For the registration form see Please note that there is no class on 4 November. This session will instead be held on 10 November (10:15 - 12:00 in HG G 43).
  • Tue 13:15-15:00 (HG G 43)
  • 10.11 Date 10:15-12:00 (HG G 43)
2 h weekly

Offered In