In these talks I will present joint works with Angus Macintyre on model theory of adeles of number fields. The ring of adeles of a global field was introduced independently in different contexts by Weil, Chevalley, and Artin. It has revolutionized modern number theory in various aspects. In the first talk, I shall present results on quantifier elimination and decidability for adeles and for general restricted products of structures, and as a consequence, solution to a problem of Ax from 1968 on decidability of the class of all the residue rings of the integers. In the second talk, I will present results on elementary equivalence in adele rings in relation to arithmetical equivalence of number fields, and axioms for rings elementarily equivalent to restricted products. I shall also briefly discuss some recent works of myself on emerging applications of the ideas above to a model theory for automorphic representations, Langlands program, zeta and L-functions, and counting and equidistribution in algebra and number theory. I will present a number of open problems.

Please send an email to r.zoghi@gmail.com to join the webinar.