In this proposal we investigate different aspects of superconductivity with the ultimate goal of finding novel ways - that can be tested experimentally - to increase substantially the critical temperature (Tc) of a superconductor/superfluid. Motivated by recent experimental advances in cold atom, manipulation of nanostructures and theoretical advances in high energy physics, we propose to achieve this goal by studying: 1) finite size effects in different models of high Tc superconductivity both theoretically and experimentally, 2) superconductivity in systems that do not thermalize, 3) superconductivity induced in systems with Efimov states (three particles bound states that occur in situations in which the two body interaction does not lead to bound states). In relation to 1) we aim a description, mostly analytical, of finite size effects in different mean field descriptions of high Tc superconductor. Then, for the models leading to a highest Tc's we plan to carry out a more refined theoretical analysis whose results can be used to describe superconductivity in realistic systems. Finally, in collaboration with experimentalists,we aim to chose the materials and parameters (size, grain shape...) most suitable for experimental studies, show experimentally that the critical temperature can be substantially (>15%) increased and propose technological applications. In relation to 2) we first provide a quantitative description of the stability of the equivalent of a Cooper's trimer in many body systems described by Efimov physics. Then we explore the feasibility of ground states based on a collection of Efimov states by using Monte Carlo techniques. If successful, we aim to describe quantitatively the resulting superconducting state andits stability to thermal fluctuations.In relation to 3) we first address the role of Anderson-Mott localization effects in the route to thermalization in a closed many body system by using exact diagonalization techniques, random matrix theory and the finite size scaling method. Based on these results we put forward a characterization of thermalization in closed many body systems. Finally we investigate superconductivity in systems that do not thermalize. Specifically we aim to identify the non-thermal quasiparticle distribution that enhances Tc the most.A fully theoretical/analytical descritption of these systems is challenging since many of them are strongly interacting. In high energy physics the Anti de Sitter (AdS) - conformal field theory (CFT) correspondence, provides, in certain cases a theoretical framework to tackle these problems. In relation with this problem we explore to what extent this technique provides a really quantitative description of quantum critical points and certain aspects of high temperature superconductivity.