The format is hybrid, with the possibility to attend in person either in Madrid (at ICMAT) or online. If you wish to attend online, please
contact us.
The recorded sessions are available in our YouTube channel. You can also download the latest version of the lecture
notes.
Season 2026-27
We meet on Tuesdays at 15:30 CEST.
We are currently focused on differentiabls stacks higher geometric structures, starting from the basics such as Morita equivalence.
Season 2025-26
Lecture 1 (7 October 2025, 15:30 CEST), Oscar Carballal (UCM, Madrid, Spain).
Abstract: We illustrate the need for groupoids through a series of examples, primarily
following
R. Loja (2021) and Weinstein (1996), and discuss some of their recent applications in
Mathematical
Physics. We also outline potential directions for future topics of our Reading Groupoid.
Lecture 2 (14 October 2025, 15:30 CEST), Bartosz Maciej Zawora (KMMF-UW, Warsaw, Poland).
Abstract: We will talk about Lie groupoids, give some examples, and prove some fundamental
results.
Abstract: We will introduce bisections and illustrate them with several examples.
Lecture 5 (18 November 2025, 15:30 CEST), Juan Manuel López Medel (ICMAT, Madrid, Spain).
Abstract: In this session we first record basic facts about local bisections and give an
explicit
formula for multiplication in the tangent groupoid. We continue with Section 1.5. Components and
Transitivity. We show that the identity–component subgroupoid C is generated by symmetric
neighbourhoods
of the unit section, is open, and yields connected reductions of principal bundles by choosing
components of the total space. Finally, orbits are shown to be submanifolds and transitive Lie
groupoids
are locally trivial.
Lecture 6 (25 November 2025, 15:30 CEST), Paula Alba San Miguel (UNED, Madrid, Spain).
Abstract: In this section we discuss Lie groupoid actions on smooth manifolds, the morphisms
between such actions, and their intrinsic definition. We also apply results from previous sections
to
the study of actions.
Lecture 7 (2 December 2025, 15:30 CEST), Tomasz Sobczak (KMMF-UW, Warsaw, Poland).
Lecture 8 (9 December 2025, 15:30 CEST), Juan Manuel López Medel (ICMAT, Madrid, Spain).
Abstract: We will mainly talk about linear actions, and the relation with representation. We
study a proposition that gives us a lot of examples as Riemannian frame groupoid, complex frame
groupoid... Lastly, we show how are the linear actions on the fundamental groupoid.
Lecture 9 (16 December 2025, 15:30 CEST), Oscar
Carballal (UCM, Madrid, Spain).
Abstract: We will present the exponential map and the adjoint formulas.
Lecture 14 (24 February 2026, 15:30 CEST), Oscar
Carballal (UCM, Madrid, Spain).
Abstract: We will introduce Lie algebroid connections and representations.
Lecture 15 (3 March 2026, 15:30 CEST), Arnau Mas (ICMAT, Madrid, Spain).
Lecture 16 (10 March 2026, 15:30 CEST), Juan Manuel López Medel (ICMAT, Madrid, Spain).
Abstract: In this lecture we study the problem of integration a Poisson manifold. Given a manifold (M, π), can we find a surjective submersion µ : (S, ω) → (M, π) being (S, ω). This is called a symplectic realization of the Poisson manifold.Moreover, we are going to see that there is a natural structure of groupoid on the symplectic manifold (S, ω) related by the multiplicative form. This is called a symplectic groupoid.
Abstract: Unlike Lie algebras, not all Lie algebroids can be integrated into a Lie groupoid. However, given a Lie algebroid it is always possible to construct an associated topological groupoid. In this talk, I shall explain the topological aspects of Lie's third theorem for Lie algebroids.
Abstract: Following my previous lecture, I shall explain and prove the Lie's third theorem for Lie algebroids, which states that a Lie algebroid is integrable iff its monodromy groups are locally uniformly discrete.