Title: A caraterística de Euler Abstract: A topologia trata sobre o estudo dos espaços e as funções contínuas. Na topologia algébrica se utilizam invariantes algébricos para entender e classificar eles. Nesta palestra vamos discutir um dos invariantes fundamentais, a característica de Euler. Revisaremos sua história de mais de 250 anos, mostraremos algumas aplicações clássicas, e relacionaremos com a teoria de homologia, com intenção de mostrar o rol central que estas ideias ocupam na matemática moderna.
Title: On Hausdorff symplectic realizations Abstract: A symplectic realization is a way to present a given Poisson manifold as a quotient of a symplectic one. Such a desingularization, which always exists, may not reflect the original dynamics, it may fail to be complete. The existence of complete symplectic realizations is equivalent to the integrability of the associated Lie algebroids. This problem is ruled by topological obstructions and forces us to deal with non-Hausdorff manifolds. In this talk I will review symplectic realizations, the interplay between Poisson manifolds and Lie algebroids, and a recent collaboration with D. Lopez where we pay special attention to Hausdorff integrations.
Title: Higher vector bundles 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 allow 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 of my 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.
Title: Discrete dynamics and differentiable stacks Abstract: In a joint work with A. Cabrera (UFRJ) and E. Pujals (IMPA) we study actions of discrete groups over connected manifolds by means of their orbit stacks. Stacks are categorified spaces, they generalize manifolds and orbifolds, and they remember the isotropies of the actions that give rise to them. I will review the basics, show that for simply connected spaces the stacks recover the dynamics up to conjugacy, and discuss the general case. I will also describe several examples, involving irrational rotations of the circle, hyperbolic toral automorphisms, and the construction of lens spaces.
