Analysis and Geometry in Several Complex Variables

In this project, the complex analysis groups of the University of Vienna, the Jagiellonian University in Krakow, and the University of Wuppertal join forces to address contemporary problems in complex analysis. The research project aims at exploiting synergies from the different expertise present in the three institutions to find innovative solutions to the problems, while at the same time strengthening the collaborative environment in the field in Europe and providing training opportunities for young researchers. The main strengths of the groups in the project are different subfields of complex analysis: Invariant distances and pluripotential theory for the group from Krakow, CR geometry and Bergman theory for the group from Vienna, and Complex Geometry and pluripotential theory for the group from Wuppertal. The project revolves around 11 problems each of which will be addressed by teams combining researchers from the participating institutions; roughly they fall into three main categories: Properties of invariant distances; geometric and pluripotential properties of special domains; and induced structures on the boundaries of domains. The work on the problems will not only significantly advance our current knowledge, but also lead to an interchange of ideas between the groups present in the project which will be fruitful for the future of the field. Concretely, the problems collected in the proposal are: A. Questions about the uniqueness of invariant distances on complex manifolds and applications to holomorphic maps. B. Zero sets of the Bergman kernel function and the Wiegerinck problem. C. Balogh-Bonk estimates on finite type domains. D. Steinness of ball quotients. E. Invariant metric(s) on the boundary of a strictly pseudoconvex domain. F. Hyperbolicity of (unbounded) model domains. G. Deformations of spheres. H. Characterizing the existence of real holomorphic vector fields on hermitian manifolds. I. Questions about the Brezis-Merle inequality. J. Pluripotential theory in worm domains and isoperimetric inequalities. K. Functional-analytic characterizations of properties of polynomially convex sets.