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We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a quadratic number of different heterochromatic trees; and if the colouring is beautiful there is an exponential number of different such trees., Typos corrected, figures added

In 2003 Daniel K. Biss published, in the Annals of Mathematics, what he thought was a solution of a long standing problem culminating a discovery by Gelfand and MacPherson. Six years later he was encouraged to publish an "erratum" of his prove, observed by Nikolai Mnev; up to now, the homotopy type of the MacPhersonian had remained a mistery... The aim of this lecture is to convince the attendee of the fact that, using a compleatly different aproach to those used before, we can prove that the acyclic MacPhersonian has the homotopy type of the affine Grassmannian.

In the study of topological ?$(n_k)$? configurations in the projective plane we have ?$n$? pseudo-lines and ?$n$? points. Precisely ?$k$? of these points are incident with each pseudo-line and, vice versa, precisely ?$k$? pseudo-lines are incident with each point. We describe a non-orientable 2-manifold associated in a natural way to a topological ?$(n_k)$? configuration in the projective plane. Apart from being interesting in its own right, this manifold defines an equivalence class within topological configurations. It can be used to distinguish topological ?$(n_k)$? configurations.