Rosario Mennuni (Università di Milano) will give a talk on
Ordered groups, valued fields, and generic automorphisms
Abstract:
In the first part of this talk, I will present results from a joint work with Jan Dobrowoslki, where we prove that the category of ordered abelian groups with an automorphism has the Amalgamation Property, and study the class of existentially closed objects of this category. Since this class is not elementary, we study it in the framework of positive logic, that I will briefly introduce, where we show them to be NIP.
In the second part, I will talk about work in progress with Jan Dobrowolski and Francesco Gallinaro, where we apply the aforementioned result to the study valued difference fields, that is, fields equipped with a valuation and a compatible automorphism. Here too, we focus on existentially closed objects and, by using techniques from tropical geometry, prove that they satisfy several desirable properties, obtain an amalgamation result after adding angular components, and show that their are NTP2 in the sense of positive logic.