A singular Serre-Swan theorem via tepui fibrations  AG,D Miyamoto,L Ryvkin

The classical Serre-Swan theorem asserts that any finitely generated projective module over the algebra C^\infty(M) of smooth functions of a manifold M can be realized as the sections of a vector bundle over M. In this article, we extend this theorem beyond the projective case by introducing a notion of singular vector bundle whose sections can realize all finitely generated C^\infty(M)-modules, up to invisible elements. We introduce tepui fibrations as the underlying geometric objects of these singular vector bundles, and show how these tepui fibrations can model singular foliations, their holonomy groupoids, and singular subalgebroids.
(Preprint 2025)

E-structures and almost regular Poisson manifolds AG, Eva Miranda

This paper examines almost regular Poisson manifolds, as studied by Androulidakis and Zambon, and E-symplectic manifolds, as studied by Miranda ans Scott. They reveal a natural bi-Liealgebroid and a Poisson groupoid. We also write the local formulas for the Poisson structure on the groupoid level and its relation to Poisson integration and symplectic realization.

(To appear in JIMJ, Preprint from 2024)

PB-groupoids vs VB-groupoids  Francesco Cattafi, AG

In this paper we introduce the notion of PB-groupoid with a structural Lie 2-groupoid, and extend the classical correspondence between vector bundles and principal bundles to VB-groupoids and PB-groupoids. 

DOI Revista Matemática Iberoamericana 2025

PB groupoids, their gauge group and their nerve AG, Sylvie Paycha

We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. 

DOI Journal of Geometry and Physics 2023

Integration of singular foliations via paths AG, Joel Villatoro

We give a new construction of the holonomy and fundamental groupoids of a singular foliation. In contrast with the existing construction of Androulidakis and Skandalis. This strategy is a direct extension of the classical construction for regular foliations and mirrors the integration of Lie algebroids via paths per Crainic and Fernandes. 

DOI IMRN 2021

Quotients of singular foliations and Lie 2-group actions AG, Marco Zambon

In this note we exhibit that it the assignment from singular foliations to its holonomy groupoid works well with quotients. Moreover, for quotients by a Lie group action, under suitable assumptions, yields a Lie 2-group action on the holonomy groupoid. 

DOI JNCG 2020

On the Inner Automorphisms of a Singular Foliation AG, Ori Yudilevich

We give an alternative proof of the (surprisingly non-trivial) fundamental fact that the time-one flow of an element of a singular foliation is an automorphism of the singular foliation. 

DOI Mathematische Zeitschrift 2019

Hausdorff Morita Equivalence of singular foliations AG, Marco Zambon

We introduce a notion of equivalence for singular foliations. We show that our notion of equivalence is compatible with ME of its holonomy groupoids. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's.  

DOI  Annals of Global Analysis and Geometry 2018

Work in Progress

Tepui fibrations, groupoids and foliations AG, David Miyamoto and Leonid Ryvkin

Follow the work on tepui fibrations and groupoids.

On PB Algebroids AG, Francesco Cattafi

We study the Lie functor of PB-groupoids and relate it with the one of VB-groupoids.

Other