This project aims to investigate geon-type solutions in Einstein's theory of Gravitation. Geons are reminiscent of stars, consisting of self-trapped nonlinear gravitational waves, in some sense they are gravitational wave stars. A very important application of geons, provided they have good stability properties and sufficiently long lifetimes, is that they can provide a natural component for the mysterious dark matter content of the Universe without invoking unobserved new type of matter fields.I plan to develop a new approach to construct geon-type solutions, which circumvents the formidable difficulties posed by Einstein’s equations without any obvious symmetry. The approach will be based on combining analytical methods (suitable perturbation theory, asymptotic analysis) and numerical ones (spectral methods in space and time). The main challenges are the following: the mathematical and numerical construction of geons by solving the gravitational field equations, the study of their stability, the computation of their lifetime, and the investigation of their consequences on our Universe.The numerical construction of geon solutions will be made possible by the numerical code named Kadath, which has been developed recently by my proposed host researcher, P. Grandclément at Paris Observatory. The Marie Curie grant would make possible a two-year personalized training period, which would enable me to apply spectral methods professionally in General Relativity by using the state of the art program libraries on large computer clusters. The research collaboration in the multidisciplinary environment of Paris Observatory would considerably widen my research scope, and would present an important step in my research career in that I would be able to obtain a senior researcher position and be able to found a research group in my home country.