Electrons moving in a solid experience strong Coulomb repulsive interactions. These interactions can lead to extraordinarily fascinating and complex behavior at low temperatures, such as superconductivity and the quantum Hall effect. This proposal deals with the properties of many-electron systems in which the electron-electron interactions play an important role, changing the low-energy physics qualitatively, and leading to novel quantum states of matter which cannot be described by models of weakly interacting particles. In the first part of this proposal, I intend to pursue a new theoretical method of studying the properties of quantum critical points in metals. Such transitions are ubiquitous in many correlated materials, such as the high-temperature superconductors, and may hold the key to understanding some of the properties of these materials. The new technique relies on formulating a model which can be simulated numerically in an efficient manner, avoiding the well-known “fermion sign problem” which often hinders the study of models involvingfermions. Extensions of this technique to other fermionic models will also be explored.In the second part of the proposal, I will study properties of interaction-driven electronic phases that have an underlying topological structure. Such systems have drawn much attention in condensed matter physics, especially over the last few years, following the discovery of the topological insulators. I will propose new ways of realizing time-reversal invariant topological superconducting phases in composite structures of superconductors and semiconductor materials. Finally, I will study how interfaces between superconductors and quantum Hall systems, either in the integer or fractional regime, can lead to new types of topological phases which may be interesting from a quantum information processing perspective.