[Todos CMAT] Seminario Sistemas Dinámicos

[Matias Carrasco]

Estimados,

este viernes 15/7 a las 14:30 en el salón de seminarios del IMERL tenemos el placer de recibir a Sébastien Alvarez (IMPA).

Abajo título y resumen.

Saludos

Title: Commuting vector fields in dimension 3

Abstract: One of the most fundamental problem in dynamical systems is the existence of fixed points. For actions of R or Z the basic tools are the index theories of Poincaré-Hopf and Lefschetz. We are interested in ations of R², or, this is equivalent, to pairs of commuting vector fields.

A famous theorem of Lima states that two commuting vector fields of a closed surface with non-zero Euler characteristic have a common zero. If we want to generalize this theorem to 3-dimensional manifolds, we immediatly face the following difficulty: these manifolds have zero Euler characteristic number. In this talk we will discuss the following conjecture of Bonatti, which is a semi-local version of Lima's theorem in dimension 3: let X and Y be two commuting vector fields of a 3-manifold, and let U be a relatively compact open set such that X does not vanish on the boundary of U. Then if the Poincaré-Hopf index of X does not vanish in U, X and Y have a common zero in U.

I will discuss this conjecture as well as the progress we have made in this direction together with C.Bonatti and B.Santiago.