The classical Bergman kernel method is a well-known orthonormalization
technique for approximating the conformal mapping of simply-connected
domains in the complex plane, in terms of orthonormal polynomials,
the so-called Bergman polynomials.
The main purpose of the talk is to present the convergence and stability
properties of a variant of the classical method, designed to reflect the
singularities of the mapping function on and outside the boundary of the domain.
This will bring us to the fascinating theory of the distribution of
the zeros of the Bergman polynomials.