"The ultimate goal of this project is to explore, by theoretical considerations and numerical calculations, the mysterious and presently at large unknown interiors of compact astrophysical objects. Comparison of theoretical results with observations will allow for putting constraints on the properties of extremely dense matter, which is not likely to be accomplished in Earth-based experiments in near future. This intrinsically multidisciplinary project demands to combine knowledge from various fields of physics of different scales, especially the General Theory of Relativityand the modern theory of dense matter (quantum mechanics, thermodynamics, nuclear physics, theory of super-fluidity...), astrophysics as well as applied mathematics (numerical algorithms) and advanced programming in order to use and develop numerical methods for solving multi-dimensionalcomputational problems than cannot be solved analytically. The Applicant will address, through careful research and development of state-of-the art numerical libraries, a broad range of interesting unsolved astrophysical problems, such as the state of differentially rotating compact stars, oscillation and stability of non-linear modes for rapidlyrotating stars, dynamical re-configuration of the star structure induced by the appearance of exotic phase core (so-called mini-collapse), inclusion of realistic processes (i.e. viscosity, magnetic fields), elastic and thermal properties of the crust of spinning-down pulsars as well as direct computation of resulting gravitationalwave emission in various physical settings. The Applicant plans also to allocate a part of the time to explorecompletely new numerical approach, namely wavelet methods for solvingpartial differential equations: this approach unites the locality of finite-differences methods and the accuracy and theoretical foundations of spectral methods and therefore seems ideal for realistic astrophysical simulations."