The investigator of the project: "On locality and randomness in non-relativistic systems" will focus on two major topics of interest. The first explores a quasi-local structure of the dynamics corresponding to non-relativistic systems. Due to the fact that such systems have no equivalent to a finite speed of light, the dynamics associated with, for example, a nearest neighbor Hamiltonian does not generally preserve locality in the strict sense. An approximate form of locality, or quasi-locality, has been demonstrated for many systems governed by Hamiltonians with short-range interactions. Recent developments have shown that these estimates, known as Lieb-Robinson bounds, are valid in a variety of contexts and useful in a number of important applications. The investigator proposes the study of several systems for which there are no known quasi-locality results. Given these estimates, certain intriguing questions may be within reach of rigorous and detailed analysis. The next area of focus concerns random perturbations of specific quantum spin systems. For single particle systems, it is well known that disorder leads to localization: a metal with sufficiently many impurities loses its conductance properties. This phenomenon should persist in many-particle interacting systems, and the investigator proposes to analyze the emergence of localization in random quantum spin systems. In fact, a dynamical form of localization, in terms of explicit Lieb-Robinson bounds, is proposed as a possible means of quantifying and thus further investigating this predicted behavior. In condensed matter physics, a wealth of intriguing, physically relevant phenomena is described in the context of quantum many-body systems. For example, the oscillations of atoms in a crystal and the alignment of spins in a magnet can both be quite effectively described by quantum lattice models. These quantum lattice systems also provide a framework for more theoretical considerations. In fact, discrete versions of quantum field theory and the basic models in the theory of quantum information and computation can be expressed in this setting. The main goal of this proposal is to further develop the theory of quantum lattice systems by analyzing a number of specific models. A crucial ingredient in this proposal will be the investigation of locality estimates which demonstrate that disturbances do not propagate arbitrarily fast through the systems under consideration. Several important applications of these locality bounds have already appeared in the mathematics and physics literature. As a new example, the investigator proposes a detailed analysis of locality properties for random systems. Random systems are particularly interesting because they are predicted to exhibit a metal-insulator phase transition with increasing disorder. Sufficiently detailed analysis may suggest design specifications for materials, e.g. fiber-optic cables, that necessarily have impurities dispersed throughout. In this case, the potential impact of this proposal, beyond the scope of mathematics and physics, may be substantial. Moreover, interested graduate students working on the project will learn the language and techniques used by scientists in a number of fields. After completing their thesis, they should be well prepared for either an academic position in the sciences or a more technical role in industry.