Abstract: The classical Picard theorem from complex analysis states that a holomorphic map from the complex plane to itself omits at most one point. Of course, the exponential mapping shows that Picard's theorem is sharp. In higher dimensional quasiconformal geometry (i.e. when we allow bounded distortion of the conformal structure), Picard's theorem appears in the form of Rickman's theorem (1980): A quasiregular mapping from the Euclidean space to itself the cardinality of the omitted set is bounded by a constant depending only on the dimension and distortion of the mapping. Also this result is sharp. In this talk, I will discuss these Picard type results and their cohomological counterparts for quasiregular mappings from Euclidean spaces to closed Riemannian manifolds, especially the following result: If a closed Riemannian n-manifold admits a quasiregular mapping from the n-Euclidean space, then its de Rham cohomology embeds into the exterior algebra of R^n. 

Bio-Sketch: Pekka Pankka received his PhD from the University of Helsinki in 2005, and is now a professor at the University of Helsinki and the Vice President of Finnish Mathematical Society. His research interests include analysis of complex variables, metric geometry, geometric topology, and focus on quasiconformal mappings in particular. In these areas, he has produced more than 40 research articles, publishing on renowed journals including Acta Mathematica and Duke Mathematical Journal.