Final Activity Report Summary - CHAOSQCD (Search for the Anderson transition in quantum chaos and quantum chromodynamics) The main research results during the fellowship could be summarized as follows, with the publication number referring to relevant sections of the progress report: 1. Publications 4 and 7: The development of a theory, in collaboration with J. Wang, which predicted the magnitude of localisation effects in quantum chaos, namely in systems whose classical dynamics was chaotic. We specifically determined the type of classically chaotic motion leading to a quantum metal insulator transition and the conditions under which this transition could be studied experimentally by using ultra cold atoms' techniques. This was one of the main objectives of the research proposal. 2. Publication 3: The development of an analytical treatment of the Anderson, i.e. metal-insulator, transition in non-interacting systems of any dimension greater than two. This formalism predicted that the upper critical dimension for localisation was infinity. Moreover, it provided explicit analytical expressions for different quantities describing the transition, such as critical exponents, as a function of the space dimensionality. We also carried out, in collaboration with E. Cuevas, a numerical analysis of the dependence of localisation on the spatial dimension of the system. This was one of the main goals of the fellowship. 3. Publication 8: Based on numerical results he conjectured that the chiral phase transition in Quantum chromodynamics (QCD) was induced by a metal-insulator transition in the QCD Dirac operator. This work was noticed by the scientific community. Dr Garcia-Garcia was invited to different international conferences so as to present these results. 4. Publication 1: In collaboration with Prof. Boris Altshuler, we developed a theory for finite size effects in superconducting clean grains that described, within the Bardeen, Cooper and Schrieffer (BCS) formalism, how the superconducting gap depended on the shape, size and number of electrons inside the grain. For symmetric grains these effects might switch on and off superconductivity by just adding a small number of Cooper pairs. By the time of the project completion, I was working on possible practical, i.e. nanotechnological, applications of these findings. 5. Publication 2: I developed an explicit analytical expression of the dependence of the Planck and Stefan-Boltzmann radiation laws on the size and shape of the blackbody. These results were of relevance in cosmology, sonoluminiscence and the definition of radiance standards. 6. Publication 9: In collaboration with experimentalists I developed a theoretical description of shell effects in Sn nanograins. This paper was submitted to Nature. 7. Publication 10: The development of a theory of phase transitions in strongly interacting systems based on high energy ideas (Maldacena conjecture).