Pfaffian groupoids

TL;DR

This thesis explores the geometry of PDEs through Lie groupoids equipped with multiplicative differential forms, introducing new results and dual perspectives using forms and distributions.

Contribution

It presents novel findings on Pfaffian groupoids, combining differential forms and distributions to advance the understanding of geometric structures related to PDEs.

Findings

Develops new theory of Pfaffian groupoids

Bridges differential forms and distributions in Lie groupoids

Provides original results on the geometry of PDEs

Abstract

This thesis is about the study of Lie groupoids endowed with a compatible (multiplicative) differential 1-form. The motivation and scope of the present work is to study the geometry of PDEs using the formalism of Lie groupoids and multiplicative forms; as such, ideas from the two theories have to be introduced and explained from our point of view (which may not be the same as in the literature!) before new results can be presented. Therefore the thesis can be naturally split in two halves: the first, consisting of chapters 1, 2 and 3, recall the ideas and methods which are used in the second half, where the majority of original results are presented. It is important to remark that when considering multiplicative structures on Lie groupoids we shall employ two (equivalent) points of view: the one using differential forms and the dual picture with distributions.