o-Minimal structures in Algebraic Geometry
Monday, 9.9.19, 00:00-01:00, Hörsaal II, Albertstr. 23b
Originating in model theory, o-minimality is a tameness property of real sets which enjoys important finiteness properties. O-minimal structures have recently found a number of important applications to algebraic and arithmetic geometry, including functional transcendence, the Andre-Oort conjecture, and Hodge theory. The aim of this summer school is to provide an introduction to these ideas for an audience of non-experts.\n\nWe plan for three lecture series on o-minimal Structures, delivered by Benjamin Bakker, Yohan Brunebarbe and Bruno Klingler.
Stably embedded pairs and applications
Wednesday, 11.9.19, 16:00-17:00, Raum 318, Ernst-Zermelo-Str. 1
A structure is called stably embedded if the trace of every externally definable is definable\nwith parameters from the structure. We will show different examples of theories for which the class of pairs of\nelementary substructures, where the smaller one is stably embedded in the bigger one, forms an elementary class\nin the language of pairs. When, in addition, the model-theoretic algebraic closure of a set is a model of the\ntheory, we show that definable types are uniformly definable. As an application, we obtain uniform definability\nof types in various NIP theories including the theory of algebraically closed valued fields, real closed valued\nfields, p-adically closed fields and Presburger arithmetic. This implies in return that the spaces of definable\ntypes in such theories are pro-definable.