Topological structures and the pointwise convergence of some multiple averages for commuting transformations

``Topological structures’’ associated to a topological dynamical system are recently developed tools in topological dynamics. They have several applications, including the characterization of topological dynamical systems, computing automorphisms groups and even the pointwise convergence of some averages. In this talk I will discuss some recent developments of this subject, emphasizing applications to the pointwise convergence of some averages.