Groups definable in partial differential fields with an automorphism

This is a joint work with Ronald Bustamante Medina and Zoé Chatzidakis.
In this talk we are interested in differential and difference fields from the model-theoretic point of view. A differential field is a field with a set of commuting derivations and a difference-differential field is a differential field equipped with an automorphism which commutes with the derivations.
Cassidy studied definable groups in differentially closed fields, in particular she studied Zariski dense definable subgroups of simple algebraic groups and showed that they are isomorphic to the rational points of an algebraic group over some definable field. In this talk we study groups definable in existentially closed difference-differential fields. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an
analogue of Phyllis Cassidy’s result for partial differential fields.