Abstracts
Sebastián Daza (Universidade de Coimbra)
Title: Deformations of Lie algebroid morphisms
Abstract: The theory of Lie algebroids allows us to study many different geometrical objects such as Lie algebras, foliations, Lie algebra actions between many other important examples.
In deformation theory we want to understand a structure by studying how it interacts with nearby structures. The idea is to obtain a cochain complex which codifies the model of the tangent space of the moduli space of a structure up to equivalences.
The deformation complex of a Lie algebroid structure is well-known.
In this talk I will introduce the deformation complex for morphisms of Lie algebroids. Moreover, we will talk about the Fréchet structures behind the deformation problem and how they relate with the cochain complex.
Ricardo Campos (CNRS - University of Toulouse)
Title: From Lie algebroids to curved Lie algebras
Abstract: A classical slogan from the 1970s says that infinitesimal deformation problems are controlled by differential graded Lie algebras. Recent work suggests that one can also formulate this principle for “unpointed” deformation problems.
Two different generalizations of dg Lie algebras capture this idea of unpointedness: On one hand, one can allow them to live over a more general manifold, leading to (dg) Lie algebroids. On the other hand, we can introduce a differential squaring to a non-zero “curvature,” giving curved Lie algebras. In this talk I will explain these notions and show that they are closely related.
The results are based on arXiv:2103.10728, joint work with Damien Calaque and Joost Nuiten.
Carlos Florentino (Universidade de Lisboa)
Title: Fano polygon spaces
Abstract: The moduli space of polygons in euclidean 3-dimensional space, with prescribed side lengths, are naturally symplectic and compact Kähler manifolds. In this talk, we classify all the smooth polygon spaces, with n sides, which are Fano varieties, in terms of the length vector. For all n>4, and all Picard numbers r between 1 and n-1, these Fano polygon spaces are toric varieties, and the corresponding toric polytopes can be explicitly determined. This is joint work with Leonor Godinho (IST)
Lennart Obster (Universidade de Coimbra)
Title: Representation theory for Lie groupoids revisited: VB-groupoids, representations up to homotopy, and fat extensions
Abstract: As for Lie groups, Lie groupoids come with an adjoint representation. Its definition is a bit more subtle: the adjoint representation can be “singular” in a precise sense. A derived viewpoint lead to the adjoint representation being introduced as a 2-term representation up to homotopy (ruth). Later, it was realised that vector bundles over groupoids (VB-groupoids) are a type of double structure that is equivalent to 2-term ruths.
In joint work with João Nuno Mestre and Luca Vitagliano, while studying multiplicative tensors on groupoids, we realised there is yet another point of view.
In this talk, a category of “fat extensions” will be introduced. This category is equivalent to the category of VB-groupoids (resp. 2-term ruths). The emphasis will be on explaining aspects of the various correspondences that we find enlightening for the study of multiplicative tensors. If time permits, we will also briefly introduce the infinitesimal analogue (for Lie algebroids) of the discussion.
