I will sketch the proof that period maps associated to variations of pure Hodge structures on complex quasi-projective varieties are tame (a recent result of Bakker, Tsimerman and myself). As a corolloray (using also the results from lecture 2) one obtains a simpler proof of a famous result of Cattani-Deligne-Kaplan: the algebraicity of Hodge loci; as well as the algebraicity of images of period maps. Speaker(s): Bruno Klingler (Humboldt University)