Mário Edmundo (Universidade Aberta, Portugal)

Model theory (analytic part): from Grothendieck to André-Oort

Abstract: O-minimality is the analytic part of model theory and deals with theories of ordered, hence topological, structures satisfying certain tameness properties. O-minimality was isolated by van den Dries as the requisite condition to prove the basic structural results of semialgebraic geometry and then by Steinhorn and Pillay in its logical form. The definition of o-minimality is rather simple and innocent looking, surprisingly however this notion turned out to be very deep with somewhat unexpected applications: it generalizes semi-algebraic geometry and globally sub-analytic geometry and it is claimed to be the formalization of Grothendieck's notion of tame topology (topologie modérée). In this short course we will: (i) introduce the audience to the subject; (ii) mention the recent development of the formalism of the Grothendieck six operations on o-minimal sheaves (a generalization and a new approach to similar formalisms for semi-algebraic sheaves (Delfs) and sub-analytic sheaves (Kashiwara-Schapira)); (iii) mention the application of o-minimality to a recent unconditional proof of the Andre-Oort conjecture for arbitrary products of modular curves by J. Pila (previous proofs were known only in some special cases and some under the Generalized Riemann Hypothesis).