Our group's research focuses on exotic phases of quantum condensed matter. Microscopically, they are comprised of electrons but exhibit properties that belie these humble ingredients. Their excitations may carry fractional quantum numbers or obey exchange statistics that are neither bosonic nor fermionic. An essential question is how to 'get fractions by combining integers' and what conditions may prompt a microscopic system to do so. We address this challenge by deriving and applying duality relations. The best-known experimental system exhibiting such phenomena is the fractional quantum Hall effect, which is a central part of our work. Additionally, we use insights and techniques developed there to predict new phases of matter in other topological systems and quantum magnets.