Leeds

After Sela and Kharlampovich-Myasnikov independently proved that non abelian free groups share the same common theory model theoretic interest for the subject arose.

 In this talk I will present a survey of results around this theory starting with basic model theoretic properties mostly coming from the connectedness of the free group (Poizat).

Then I will sketch our proof with C.Perin for the homogeneity of non abelian free groups and I will give several applications, the most important being the description of forking independence.

 In the last part I will discuss a list of open problems, that fit in the context of geometric stability theory, together with some ideas/partial answers to them.

Let T be a (one-sorted first order) geometric theory (so T

has infinite models, T eliminates "there exist infinitely many" and

algebraic closure gives a pregeometry). I shall present some results

about T_P, the theory of lovely pairs of models of T as defined by

Berenstein and Vassiliev following earlier work of Ben-Yaacov, Pillay

and Vassiliev, of van den Dries and of Poizat. I shall present

results concerning superrosiness, the independence property and

imaginaries. As far as the independence property is concerned, I

shall discuss the relationship with recent work of Gunaydin and

Hieronymi and of Berenstein, Dolich and Onshuus. I shall also discuss

an application to Belegradek and Zilber's theory of the real field

with a subgroup of the unit circle. As far as imaginaries are

concerned, I shall discuss an application of one of the general

results to imaginaries in pairs of algebraically closed fields,

adding to Pillay's work on that subject.