We group our doctoral programme under six key areas of Complexity Science:agent-based modelling;networks and emergent behaviour;self-organisation and assembly;nonlinear dynamics and modelling in the presence of noise;spatio-temporal complexity, quantification and modelling;management and bounding of complexity.The development of these themes and applications proposed within them addresses societal, financial and technical performance at the system level, key for national competitiveness. Particular key current societal problems are also addressed, such as crime, terrorism, epidemics, computer viruses, and understanding how to control their spread.We propose medically related projects of clear importance to the NHS (diabetes, back pain, consultation effectiveness), financial applications address the effective management of UK markets and financial intervention, and we include a diverse range of environmental interests for the UK, from habitats to our skies and space weather. Our neuro-related network projects and intelligent agent and trusted agent researches relate to close UK priorities identified by DTI Foresight programme, and we also look to longer term possibilities such as self-assembling molecular-scale technology.