Interests

My main contributions are in singular foliations and Lie groupoids. My starting point was symplectic and Poisson geometry, together with Lie theory. My research has since expanded towards higher structures, diffeology, and applications to mathematical physics.

I am now working on a special class of diffeological groupoids related to singular foliations and on higher differential and geometric structures.

Publications

1. Garmendia, A. and Miranda, E.: “E-symplectic vs almost regular Poisson”.
   Journal of the Institute of Mathematics of Jussieu.
   DOI 10.1017/S1474748026101595.
   arXiv 2410.11641. March 2026.

   This paper examines almost regular Poisson manifolds, as studied by Androulidakis and Zambon, and E-symplectic manifolds, as studied by Miranda and Scott. They reveal a natural bi Lie algebroid 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.

2. Garmendia, A. and Cattafi, F.: “PB-groupoids vs VB-groupoids”.
   Revista Matemática Iberoamericana.
   DOI 10.4171/RMI/1580.
   arXiv 2406.06259. Aug 2025.

   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.

3. Garmendia, A. and Paycha, S.: “Principal bundle groupoids, their gauge group and their nerve”.
   Journal of Geometry and Physics.
   DOI 10.1016/j.geomphys.2023.104865.
   arXiv 2108.07352. Sept 2023.

   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.

4. Garmendia, A. and Villatoro, J.: “Integration of singular foliations via paths”.
   International Mathematics Research Notices.
   DOI 10.1093/imrn/rnab177.
   arXiv 1912.02148. Sept 2021.

   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.

5. Garmendia, A. and Zambon, M.: “Quotients of singular foliations and Lie 2-group actions”.
   Journal of Noncommutative Geometry.
   DOI 10.4171/JNCG/434.
   arXiv 1904.08890. Dec 2021.

   In this note we exhibit that 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.

6. Garmendia, A. and Yudilevich, O.: “On the inner automorphisms of a singular foliation”.
   Mathematische Zeitschrift.
   DOI 10.1007/s00209-018-2212-0.
   arXiv 1804.06103. Oct 2019.

   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.

7. Garmendia, A. and Zambon, M.: “Hausdorff Morita equivalence of singular foliations”.
   Annals of Global Analysis and Geometry.
   DOI 10.1007/s10455-018-9620-6.
   arXiv 1803.00896. Feb 2019.

   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.

Preprints

1. Garmendia, A. ; Miyamoto, D. and Ryvkin, L.: “A singular Serre-Swan theorem via tepui fibrations”.
   arXiv 2510.20936. Oct 2025.

   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.

2. Garmendia, A.: “Groupoids and singular foliations”. PhD Thesis, KU Leuven.
   arXiv 2107.10502. Jul 2019.

   PhD Thesis where I explain the relation between singular foliations and groupoids. I also give a definition of Morita equivalence for singular foliations, explain its consequences on the groupoid level and with quotients.

Other

1. Garmendia, A and Ryvkin, L in Young mathematicians’ column EMS.
   DOI 10.4171/MAG/7. 2021

   Leonid Ryvkin and I contributed a short article on how we organise an international online seminar during the Pandemic in the EMS Magazine.