This project is concerned with the study of geometrical and analytical aspects of the space of discrete representations of a surface group into the group of isometries of complex hyperbolic space, that is the complex hyperbolic quasi-Fuchsian space QC. It is the important dvelopment of the project the researcher I. Platis is carrying out as a Marie Curie fellow at the University of Durham under the supervision of J.R. Parker. Both projects are parts of a wider programme which aims to understand fully the ge ometry of QC. The main scientific objective of this project is to make further significant contributions to the wider research plan. There is a variety of deep geometrical results concerning the spaces of real hyperbolic isometries such us Teichmueller sp ace, many of them proved with tha aim of strong analytical tools such as quasiconformal mapping theory. An analogous theory already exists and it is fair to ask how it can be implemented in the complex hyperbolic setting. this question is considered one of the main problems in Complex Hyperbolic Geometry. Platis has already proved important and useful preliminary results in this area. By continuing his work on the subject in the Department of Mathematics of the University of Thessaloniki and by collaborati ng with N. Mandouvalos who is considered the top expert of Analysis and Hyperbolic Geometry in Greece, Platis will be able to give answers to the problem and simultaneously, his scientific and professional career will be immensely benefited so that after t he termination of this project he will be established as an independent researcher of Complex Hyperbolic Geometry in Greece.The Department is initially offering him a two-year contract with the view to being able to make this permanent subsequently. The fu nding of this proposal would be a major step on the path to reintegrating the researcher into a stable career in his home country.
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