Duistermaat-Heckman measures for Hamiltonian groupoid actions

This thesis treats two problems related to Poisson manifolds of compact types: the existence of Poisson manifolds of strong compact type, and the generalisation of classical Duistermaat-Heckman results to the setting of Hamiltonian actions of symplectic groupoids. In Chapter 4 we prove that all strongly affine circles and 2-tori appear as the leaf space of a regular Poisson manifold of strong compact type. These Poisson manifolds are all fibrations over their leaf space with symplectic leaves diffeomorphic to the smooth manifold underlying a K3 surface. In Chapter 5 we show that for a Hamiltonian action of a regular, source proper symplectic groupoid with sufficiently nice properties there is an analogue of the Duistermaat-Heckman measure which is a polynomial measure with respect to the natural integral affine structure.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Zwaan, Luka Marinus Simon

Contributors dc:contributor

• Loja Fernandes, Rui A
• Lerman, Eugene M
• Junge, Marius
• Berwick Evans, Daniel

• Poisson Manifolds
• Hamiltonian Actions
• Symplectic Groupoids
• Integral Affine Structures

Statement dc:rights

• Copyright 2024 Luka Marinus Simon Zwaan

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/124365

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F124365