|Frank Wagner is a professor at the Institut Camille Jordan of Lyon University, and a member of the Institut universitaire de France. His research interests are in mathematical logic, more specifically in model theory and its interactions with algebra, group theory and multiplicative combinatorics. Gold medalist at the 1983 International Mathematical Olympiad, he is keenly interested in mathematics education and participates regularly in science outreach activities such as Math@Lyon.|
"Ultraproducts, asymptotics, and model theory:" Ultraproducts are a means to construct, given some infinite class of structures, some sort of logical limit incorporating the asymptotic behaviour of the structures in the class. Contrary to the more common limit constructions, it not only is applicable to an arbitrary collection of structures using the same language, but at the same time forces the existence of the limit. In particular, it can be (and has been) applied to characterise the asymptotic behaviour of certain classes of finite structures (such as finite fields), giving rise to what are called pseudo-finite structures.
In the lectures I aim to present the logical notions used (language, structure), give the ultraproduct construction and prove its main properties.I shall also derive the fundamental theorem of model theorem, the compactness theorem, and apply ultraproducts to field and group theory.