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Javier de la Nuez Gonzlez (KIAS)
The question of whether two non-abelian free groups of different finite ranks satisfy the same collection of first order sentences was originally formulated by Tarski in the 1950s and remained open until 2001, when it was answered in the affirmative by Z. Sela in a celebrated series of papers relying heavily on geometric group theory techniques (an independent solution is due to Kharlampovich and Myasnikov). In this series of talks I will present some of the core ideas in Sela's proof. In particular, I will give an outline of the shortening argument and two of its applications: the construction of Makanin-Razborov diagrams for the analysis of solution sets of equations over free groups and the existence of formal solutions witnessing the validity of sentences. I will assume a certain degree of familiarity with Bass-Serre theory.