Leeds

A theorem of R. Nevanlinna from 1922 characterizes the Cauchy

transforms of finite positive measures on the real line as the functions in

the Pick class that satisfy a certain growth condition on the real axis; this

result is important in the spectral theory of self-adjoint operators.

(The Pick class is the set of analytic functions in the upper half-plane

$\Pi$ with non-negative imaginary part). I will describe a higher-dimensional

analogue of Nevanlinna's theorem. The $n$-variable Pick class is defined to

be the set of analytic functions on the polyhalfplane $\Pi^n$ with non-negative

imaginary part; we obtain four different representation formulae for functions

in the $n$-variable Pick class in terms of the ``structured resolvent" of a

densely defined self-adjoint operator. Structured resolvents are analytic

operator-valued functions on the polyhalfplane with properties analogous to those of the

familiar resolvent of a self-adjoint operator. The types of representation that a

function admits are determined by the growth of the function on the imaginary polyaxis $(i\R)^n$.

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.