Alex Torzewski - 15h45 - Studying rational points via p-adic period mappings.

We outline a recent method of Lawrence-Venkatesh to study rational points on varieties via $p$-adic period mappings. More specifically, for varieties which come equipped with a family for which the fibres "vary sufficiently", by considering the variation of the cohomology of the fibres, one may show that the rational points must live in a proper subspace. In the case of smooth projective curves of genus $> 1$, such families exist and this yields a proof of the finiteness of their rational points, i.e. an alternative proof of the Mordell Conjecture.