My research is in quantum information science, an interdisciplinary subject using ideas and methods from physics, mathematics and computer science to study information and computation in light of quantum mechanics. A central goal of my research is to determine the ultimate capabilities and limitations of computers, networks and other physical systems for computing and for processing information. Through my research, I want to gain a better understanding of the true physical nature of information and to make significant theoretical advances toward realizing a fault-tolerant quantum computer. My approach is two-fold: On one hand, I use mathematical tools, such as algebraic number theory, to study explicit constructions and algorithms for fault-tolerant quantum gates in finite dimensions. On the other hand, I use traditional information-theoretic methods, like random coding, to answer fundamental questions about entanglement, correlations and error correction in the asymptotic limit of arbitrarily many quantum systems. I also work at the interface between these complementary yet interrelated approaches, such as on arithmetical aspects of the quantum Hall effect and other topological phases in 2+1 dimensions. Besides helping to classify such theories, this may lead to new ways of thinking about the complexity of physical systems, to new quantum algorithms, or to new perspectives on quantum field theory.