While physics is grappling with new problems dominated by strong interactions and (or) correlations, two main strategies are available to the theorists. The first consists in developing sophisticated approximations and numerical techniques for studying ab-initio models; the second in elaborating analytical methods to solve exactly the main models that capture the physics of interest. This latter approach is less general, but essential for our fundamental understanding, for developing controlled approximations, and to provide benchmarks for the numerical simulations. The purpose of this proposal is to obtain exact results for two types of problems. The first concerns phase transitions in 2+1 dimensional electrons gases in presence of disorder and mostly without interactions. A classical example is the transition between plateaux in the integer quantum Hall effect, where, despite of a wealth of numerical and experimental data, the values of the critical exponents are not known exactly. The study of the corresponding conformal field theories is made difficult by the non-unitarity and non-compactedness of the associated target space. In the last few years, steady progress has occurred on these questions. The understanding of sigma models on supergroups has improved thanks to developments around the AdS/CFT conjecture. Deep relations have been uncovered between the non-simplicity of lattice algebras and logarithmic properties of the associated conformal field theories. The formalism of Schramm-Loewner evolutions has transformed our understanding of fractal properties and geometrical phase transitions. We plan to build on this progress to further our understanding of critical properties in disordered electronic systems. In particular, we plan to understand better the role of continuous symmetries and indecomposability in logarithmic theories, the topology of renormalization group flows, and the probabilistic nature of wave functions and electronic trajectories. The second problem concerns out of equilibrium transport in nanosystems such as quantum dots. This topic, of crucial importance for applications and experiments, is even more challenging that the foregoing one. Indeed, the physical phenomena involved are not suited to the traditional methods of solid state physics; they often are non-perturbative (implying spin-charge separation, charge fractionalization and non Fermi liquid strong coupling fixed points), and are difficult to study numerically. Following works by some of us, and independently by the group of N. Andrei at Rutgers, it seems possible to find realistic models where transport can be studied exactly. We plan to obtain a complete solution for these models - from the I-V characteristic to the full counting statistics and the entanglement entropy. We plan moreover to use these solutions to obtain benchmarks for new numerical techniques using time-dependent density matrix renormalization group. Finally, we plan to explore more fundamental properties, such as renormalization and fixed points out of equilibrium, the integrability of the Keldysh formalism, and fluctuation-dissipation relations à la Gallavotti-Cohen. Finally, the two problems share common features, which we plan to study as well - from the role of non-compact target spaces in the description of transport to the reformulation of the replica method using contours `a la Keldysh'. This project brings together two groups (IPhT and LPTENS) whose expertises are complementary, and who have in the past accomplished important progress on related problems.