Geometric structures on Lie groupoids and differentiable stacks

Abstract: In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375], we have discussed the correspondence between two notions of a gerbe over a stack. In [arXiv:2012.08447], we have developed the notion of Chern-Weil map for principal bundles over a Lie groupoid when the Lie groupoid is equipped with a connection. In [arXiv:2012.08442], we discuss the notion of connection on the principal bundle over a Deligne Mumford stack using the notion of Atiyah sequence. The work in [arXiv:2103.04560] introduces the notion of a topological groupoid extension and discusses the correspondence between (Morita equivalent classes of) topological groupoid extensions and gerbes over topological stacks.