The proposed project belongs to the domain of Complex Analysis in Several Variables. It contains two lines of research, both having to do with the behavior of holomorphic functions at the boundary of their domain of definition.
The first topic is an analysis of the algebra of CR functions defined near a point p of the boundary bM of a domain M, in particular regarding the properties of the Taylor development. The known results suggest that some of these properties depend on the existence of a CR peak function at the point p.
The second one, instead, is of a global nature: it deals with the study of the Bergman space of certain non-compact complex manifolds M. Since such spaces, in general, can be trivial, we recently introduced a method to construct L^2 holomorphic functions in the case when M admits a large group of holomorphic automorphisms. This method depends, again, on the existence of appropriate holomorphic peak functions.
