Stefan Baranczuk Title: Algebraic sets over finite fields. Reduction maps of Mordell-Weil groups. Abstract: In the first part of the talk I will discuss an elimination theorem for polynomials defining algebraic sets over finite fields. In the second, I will present results of a joint work with Grzegorz Banaszak on reduction maps of Mordell-Weil groups of abelian varieties over number fields, and two new local-global principles.