Time:  3:30pm  4:30 pm 
Room: 
Wean Hall 8220

Speaker: 
Vahagn Aslanyan Department of Mathematical Sciences CMU 
Title: 
Geometry of strongly minimal sets in differentially closed fields

Abstract: 
I will discuss Zilber's Trichotomy conjecture and some structures (theories) where it holds (the conjecture in its general form was refuted by Hrushovski). In particular, by a result of Hrushovski and Sokolovic differentially closed fields satisfy Zilber's trichotomy. However, understanding whether a given definable set is strongly minimal or, given a strongly minimal set, understanding the nature of its geometry is not an easy task. I will show how one can use the AxSchanuel theorem for the jfunction to deduce strong minimality and geometric triviality of the differential equation of the jfunction (I will also explain why it is an important example). This result was first proven by Freitag and Scanlon using the analytic properties of the jfunction. My approach is completely abstract, I actually prove that once there is an AxSchanuel type statement of a certain form for a differential equation E(x,y) then some fibres of E are strongly minimal and geometrically trivial. 