Quantum Hall states are prototypical topological states of matter whose Hall conductance is topologically quantized to an integer or rational fraction multiple of the fundamental conductance quantum. A significant consequence of this quantization is that the Hall conductance value can be made independent of variations from device to device, within acceptable limits. Such topologically quantized properties are thus highly desirable for metrology or industrial purposes. Formulating a microscopic picture of fractional quantum Hall states and the characterization of all topological responses of quantum Hall states are frontier areas of condensed matter research, with far reaching technological consequences such as realizing anyonic topological quantum computation. In this dissertation, I will present my research on these topics.<br>

The discovery of topological quantum states of matter has required physicists to look beyond Landau’s theory of symmetry-breaking, previously the main paradigm for<br>studying states of matter. This has led also to the development of new topological theories for describing the novel properties. In this dissertation an investigation in this<br>frontier research area is presented, which looks at the interplay between the quantum geometry of these states, defects and disorder. After a brief introduction to the topological quantum states of matter considered herein, some aspects of my work in this area are described. First, the disorder-induced band structure engineering of topological insulator surface states is considered, which is possible due to their resilience from Anderson localization, and believed to be a consequence of their topological origin.<br>Next, the idiosyncratic behavior of these same surface states is considered, as observed in experiments on thin film topological insulators, in response to competition between<br>hybridization effects and an in-plane magnetic field. Then moving in a very different direction, the uncovering of topological ‘gravitational’ response is explained: the<br>topologically-protected charge response of two dimensional gapped electronic topological states to a special kind of 0-dimensional boundary – a disclination – that encodes spatial curvature. Finally, an intriguing relation between the gravitational response of quantum Hall states, and their response to an apparently unrelated perturbation – nonuniform electric fields is reported. <br>