Stacks in Poisson Geometry

TL;DR

This thesis explores the theory of stacks in Poisson geometry, establishing foundational equivalences, introducing new structures, and classifying special Poisson manifolds with a focus on symplectic groupoids and b-symplectic manifolds.

Contribution

It provides a rigorous proof of the equivalence between internal groupoids and geometric stacks, introduces a new site for Dirac structures, and classifies b-symplectic manifolds up to Morita equivalence.

Findings

Proved the equivalence between internal groupoids and geometric stacks.

Introduced a new site for Dirac structures.

Classified b-symplectic manifolds up to Morita equivalence.

Abstract

This thesis is divided into four chapters. The first chapter discusses the relationship between stacks on a site and groupoids internal to the site. It includes a rigorous proof of the folklore result that there is an equivalence between the bicategory of internal groupoids and the bicategory of geometric stacks. The second chapter discusses standard concepts in the theory of geometric stacks, including Morita equivalence, stack symmetries, and some Morita invariants. The third chapter introduces a new site of Dirac structures and provides a rigorous answer to the question: What is the stack associated to a symplectic groupoid? The last chapter discusses a remarkable class of Poisson manifolds, called b-symplectic manifolds, giving a classification of them up to Morita equivalence and computing their Picard group.