Through a distributed research agenda, we are developing a new geometrical language unifying diagrammatic methods in algebraic topology and quantum representation theory with graphical techniques from quantum computation, computational linguistics, and software and hardware design. This research effort is organized into four cross-disciplinary strands (the language of geometry and the geometry of language, the computational structure of quantum spacetime, higher structures for computational systems, and applications in geometric computation), and the research projects of the Centre address such fundamental problems as the flow of information in quantum protocols, the design of natural language processing platforms, the automated analysis of large unstructured data sets, and the classification of quantum field theories.