Our understanding of wave phenomena underpins many technologies upon which our society depends, for example, radar, sonar, mobile phones, ultrasound, optical fibres, and crack-detection in structures.Many wave phenomena can be described mathematically by Partial Differential Equations'' (PDEs); and information about the physical processes can be obtained by studying these mathematical models. Mathematicians have a toolkit of techniques to study PDEs and extract useful information. This project seeks both to sharpen'' two tools, which the investigator has played a key role in developing, and to combine them, not only with each other, but also with other cutting-edge techniques from different areas of mathematics. This combination will increase the power of these techniques and allow mathematicians to apply them in new situations, further increasing our knowledge of waves.Some examples of problems this project will investigate are:- The scattering of sound and electromagnetic waves from obstacles with sharp corners and edges. These problems are of fundamental importance to many engineering applications, and hence have been extensively studied for many years. However, the current mathematical tools are still not powerful enough to solve many important practical problems.- The propagation of waves through so-called meta-materials'', artificial materials engineered to produce properties not found in nature, and periodic media'', which have applications in photonic crystals used in optical communication.- The detection of cracks in the surface of materials; this is obviously important for testing the integrity of many engineering structures, but in particular nuclear and chemical reactors.