Title: Lie Groupoids and Related Structures
Speaker: Matias del Hoyo
Institution: Universidade Federal Fluminense
Date & Time: February 18, 19, and 20, 2025, 10:00–11:00
Location: Seminar Room 5, Mathematics Building (3rd Floor), Jilin University
Abstract: Lie groupoids generalize Lie groups and can be viewed as structures on manifolds encoding both internal and external symmetries. Driven by connections to Poisson geometry, non-commutative geometry, and mathematical physics, their theory has evolved significantly over recent decades. This talk provides an overview of Lie groupoids, including basic definitions, key examples, and constructions. The Linearization Theorem and its applications—with a particular focus on Riemannian metric theory—will also be discussed.
Abstract: The infinitesimal geometry of a Lie groupoid is captured by its Lie algebroid, a structure generalizing regular foliations and Lie algebra bundles. Not all abstract Lie algebroids are integrable, and topological obstructions can be explicitly computed. The transverse geometry of a Lie groupoid is represented by its orbit stack, a structure generalizing manifolds and orbifolds. Two groupoids yield the same stack if and only if they are Morita equivalent, with many invariants and constructions being Morita invariant. This talk reviews these constructions, highlights key results, and discusses the speaker’s contributions to the field.
Abstract: Higher Lie groupoids arise from the simplicial approach to higher category theory. Notable examples include global counterparts of Courant algebroids in Poisson geometry. Analogous to how Lie groupoids relate to Lie algebroids and stacks, higher Lie groupoids connect with differential graded manifolds and higher stacks. However, these theories remain in their infancy, with foundational questions unresolved. This talk introduces the theory of higher Lie groupoids, presents the speaker’s recent work, and discusses open problems.
Prof. Matias del Hoyo has been a faculty member at the Universidade Federal Fluminense (Niterói, Brazil) since 2016. He earned his Ph.D. in Mathematics from the Universidad de Buenos Aires (Argentina) in 2010 and conducted postdoctoral research at the Instituto Superior Técnico (Lisbon, Portugal) and Utrecht University (Netherlands) from 2011 to 2013. He also served as a Visiting Professor at IMPA (Rio de Janeiro, Brazil) from 2014 to 2016 and at the Center de Recerca Matemàtica (Barcelona, Spain) in 2019. His research focuses on Lie groupoids, Lie algebroids, and differentiable stacks, with applications in Poisson geometry.
