Duistermaat-Heckman measures for Hamiltonian groupoid actions

Zwaan, Luka Marinus Simon

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

Description

Title

Duistermaat-Heckman measures for Hamiltonian groupoid actions

Author(s)

Zwaan, Luka Marinus Simon

Issue Date

2024-04-23

Director of Research (if dissertation) or Advisor (if thesis)

Loja Fernandes, Rui A

Doctoral Committee Chair(s)

Lerman, Eugene M

Committee Member(s)

• Junge, Marius
• Berwick Evans, Daniel

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Poisson manifolds
• Hamiltonian actions
• symplectic groupoids
• integral affine structures

Abstract

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.

Graduation Semester

2024-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2024 Luka Marinus Simon Zwaan

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