"The discovery potential of the Large Hadron Collider and, in particular, its heavy-ion program, will be largely conditioned by the existence of strong theoretical groups able to interpret the experimental findings. We present a project on QCD under extreme conditions with the clear aim to provide the necessary tools to understand the LHC data.The kinematical reach of the LHC increases by more than one order of magnitude in virtuality and more than two in fraction of momentum x present ones at the Relativistic Heavy Ion Collider. In these conditions, new regimes of QCD will be probed for the first time, providing unprecedented possibilities for the characterization of the produced medium. We intend an extensive study of jets in heavy-ion collisions, including the essential improvements on the description of the in-medium shower evolution as well as the hydrodynamical description of the medium. One of the priorities of the project is the study of nuclear parton distribution functions, both with linear and non-linear evolution equations, with special emphasis on the development of a global fit for practical applications. An important part of the project is devoted to the relation between the findings in heavy-ion collisions and string theory. Although still speculative, several string theory computations related with heavy-ion observables have been recently proposed."