Singular symplectic manifolds

 23 September - 16 December 2025, HS 2025 ETHZ
Tuesdays, 13:15 - 15:00

Additionally some Mondays 10:15-12:00 will be devoted to problem sessions
Location: HG G 43, ETHZ

First lecture: 23 September

Notes will be taken by Isaac Ramos Reina with the kind assistance of Reto Kaufmann

Extra problem sessions will be held on:

1. October 27 at 10:15
   
   

2. November 24 at 10:15
   
   

All extra sessions will take place in the same room.

Please note that there will be no class on November 4. This session will instead be held on November 10 at 10:15 a.m., also in the same room.

This minicourse explores the rich geometry of symplectic structures with singularities—manifolds that are symplectic everywhere except along a critical hypersurface where the structure degenerates. Known as b-symplectic or log-symplectic manifolds, they naturally appear in problems of celestial mechanics (such as the restricted three-body problem) and in the study of the Lorenz plane.

A key tool in this field is desingularization, a unifying method that connects symplectic, folded-symplectic, contact, and Poisson geometries. This approach provides powerful models for understanding dynamics, quantization, and symmetries.

In the course, we will:

• Investigate toric actions and quantization on singular symplectic manifolds.

• Examine potential frameworks for defining Floer homology in the singular setting.

• Explore how Poisson manifolds can be studied through symplectic models built on Lie algebroids.

• Highlight applications to celestial mechanics and beyond.
  
  

By the end, participants will see how singular symplectic manifolds enrich the landscape of modern geometry, offering new pathways for applications in mechanics, quantization, and topology.