Previous Talks

Previous talks - 2021

November 22

Title: Higher vector bundles

Speaker: Matias del Hoyo - UFF

Abstract: Lie groupoids provide a unified framework for classic geometries, and they can be used to model stacks in differential geometry, giving finite-dimensional solutions to moduli problems. The use of simplicial manifolds allows us to study Lie groupoids via their nerve, and to develop a theory for higher groupoids and stacks, by exporting the methods of homological algebra. In this talk, I will discuss some recent works in the study of vector bundles over Lie groupoids, their classification, their relation with representation theory, and their interpretation at the level of stacks.

November 08

Title: Index theorems for transitive Lie algebroids and representations up to homotopy

Speaker: James Waldron - Newcastle University UK

Abstract: It is a fact that if G is a compact Lie group then the Euler characteristic of G is equal to zero. There are many different proofs, each using some amount of the structure theory of Lie groups, Hopf algebras, and differential or algebraic topology. One proof reduces the problem to linear algebra via an averaging technique of Cartan and the restatement of the problem in terms of Lie algebras. Motivated by this, I will explain how to generalise the 'Cartan' proof to the setting of principal G-bundles. This involves a general vanishing result for the Euler characteristic of a transitive Lie algebroid, the proof of which is an application of the Atiyah-Singer index theorem.

If there is time I will explain how these ideas can be generalised to the equivariant setting, and to 'representations up to homotopy' / 'cohesive modules'. In the case of the tangent bundle these formulas can be interpreted as index theorems for 'infinity local systems'.

The first part of the talk is based on parts of arXiv:1908.06861, and the second is work in progress.

October 25

Title: About the structure of local Lie groupoids

Speaker: Alejandro Cabrera - UFRJ

Abstract: In this talk, we will give an overview about local Lie groupoids. We begin by detailing their definition and some examples. We also review their motivation in the context of Poisson geometry and quantization. Finally, we present briefly some more recent constructive results about how to integrate a Lie algebroid into a local Lie groupoid and mention some applications.

October 11

Title: Invariant complex Dirac structures on flag manifolds

Speaker: Carlos Varea - USP

Abstract: The concept of complex Dirac structure extends the notion of generalized complex structure, which arises when the real index is zero. The aim of this talk is to describe the invariant Dirac structures with constant real index on a maximal flag manifold, pointing out the differences from the case of invariant generalized complex structures. This is a joint work with C. Ortiz.