Graduate Seminar on Higher Differential Geometry - 2021

What is the seminar about?

This weekly seminar is devoted to some of the research topics of Ph.D. students and Postdocs of our group. This semester we are going to have a working group on higher differential geometry. We plan to cover topics such as:

• Lie groupoids and differentiable stacks (Cristian and Fabricio). We will follow the survey by Matias del Hoyo available here.

• Higher Chern-Weil theory (Sebastián).

• Equivariant cohomology of differentiable stacks (Alejandro).

• Generalized complex structures and Lie theory (Carlos).

Next talk: June 18

Title: Actions on a differentiable stacks

Speaker: Alejandro Barbosa

Abstract: We will discuss the concept of action on a differentiable stack and we describe how this induces an action in the Lie groupoid associated to the differentiable stack.

Past talks

June 18

Title: Differentiable stacks

Speaker: Alejandro Barbosa

Abstract: We will discuss the concept of differentiable stacks and we make some computations with the classifying stack and the quotient stack.

June 11

Title: Restricting the group of equivariant cohomology and Integration of equivariant forms.

Speaker: Alejandro Barbosa

Abstract: We will discuss a restriction of the group in the equivariant cohomology for a compact connected Lie group acting on a smooth manifold M and we will introduce the notion of integration for equivariant forms.

June 04

Title: Equivariant cohomology (Slides)

Speaker: Alejandro Barbosa

May 21

Title: The Chern-Weil-Lecomte morphism

Speaker: Sebastián Herrera

Abstract: In this talk we will recall Lecomte's construction of the Chern-Weil morphism. Then, we will see how this extends to the setting of L_infty algebras extensions together with a representation up to homotopy of its base. We will also discuss some results on the naturality of the Chern-Weil-Lecomte map for L_infty algebras and applications to principal 2-bundles. The talk is based on an ongoing work joint with Cristian Ortiz.

May 14

Title: L_infty algebras

Speaker: Sebastián Herrera

April 30

Title: Invariant generalized complex structures on flag manifolds

Speaker: Carlos Varea

Abstract: The aim of this talk is to describe the invariant generalized complex structures on a maximal flag manifold in terms of a fixed root system associated to the complex semisimple Lie algebra determining such flag. This is a joint work with Luiz A. B. San Martin.

Title: Equivariant Cohomology Models for Differentiable Stacks

Speaker: Alejandro Barbosa

Abstract: We introduce the concept of equivariant cohomology in the smooth manifold case and the notion of differentiable stacks. Then we consider an action of a Lie group on a differentiable stack in the sense of Romagny and consider the stacky quotient associated to this action. Consequently, we construct an atlas that makes these stacky quotients a differentiable stacks. Then we provide a Borel model for equivariant cohomology in this context and in order to follow the classical approach for equivariant cohomology, we build a Cartan model for differentiable stacks and we prove that both models compute the same cohomology as the proposed by the Borel model.

April 23 (Friday - 17:00)

Title: LA-grupoides e PB-grupoides

Speaker: Sebastián Herrera

April 16 (Friday - 17:00)

Title: Fibrados principais e Teoria de Chern-Weil (Slides)

Speaker: Sebastián Herrera

Abstract: Nesta palestra vamos introduzir a noção de fibrado principal, conexões e curvatura, falaremos sobre algumas de suas equivalências em termos de levantamento horizontal. Também vamos construir o homomorfismo de Chern-Weil clássico e, por último, discutiremos sua importância na classificação de fibrados principais a menos de isomorfismo.

April 09 (Friday - 17:00)

Title: Principal bi-bundles (Slides)

Speaker: Fabricio Valencia

Abstract: We will introduce the notion of principal groupoid bundle with the aim of treating some features of the structure and properties of principal groupoid bi-bundles.

April 01 (Thursday - 17:00)

Title: Equivalences of Lie groupoids (Slides)

Speaker: Fabricio Valencia

Abstract: We will introduce the notions of both categorical and weak equivalences of Lie groupoids. We will exhibit a characterization of weak equivalences in terms of the transversal geometric data of the Lie groupoids in question and then we will define differentiable stacks using the notion of weak equivalence.

March 25 (Thursday - 17:00)

Title: Basics on Lie groupoids II (Slides)

Speaker: Cristian Ortiz

Abstract: We will review the definition of the normal representation of a Lie groupoid. Then, we will introduce principal groupoid bundles.

March 18 (Thursday - 17:00)

Title: Basics on Lie groupoids (Slides)

Speaker: Cristian Ortiz

Abstract: We will introduce Lie groupoids and discuss the main examples. Finally, the notion of Lie groupoid representation will be presented.