Exploring the interface between mathematics, computation and the foundations of science
The central goal of the Centre for Applied Compositionality is to study the purely structural aspects of elementary computations, and the algebra of how they compose, in the service of constructing more explicit algorithmic models for fundamental physics, quantum information theory, complex systems theory, chemical reaction networks, ecological networks, mathematical reasoning, cognitive neuroscience, linguistics and many other fields.
Our principal defining methodology is to employ powerful abstract mathematical and computational methods from areas such as (higher) category theory, (higher) topos theory and homotopy type theory in order to study complex scientific and technological systems in terms of the properties of their constituent components and the formal syntactic rules for how those components may be "glued" together to form composite structures.
Some of our core philosophical themes include an emphasis on the multiway nature of computations, the emergence of spatial structure (in the form of pretopology or pregeometry) in abstract systems, the relationship between different models of observation and different notions of mathematical equivalence, the nature of causality and the structure of causal relations in arbitrary compositional models, the interplay between causal structure and multiway structure, etc.
Many of these themes may be thought of as being unified by a single broad focus on functoriality (as well as the deeply related concepts of universality, naturality, adjunction and their various generalisations) as a robust and formal methodology for describing the precise relationship between the abstract syntax of a computational model and the concrete semantics of the system that it is attempting to describe.
Some ongoing projects include:
The Centre for Applied Compositionality is principally based at the University of Cardiff (joint between the School of Mathematics, the School of Physics and Astronomy and the School of Computer Science and Informatics), with a small offshoot based residually at the University of Cambridge.
We are always interested in hearing from potential collaborators, master's students, PhD students, postdocs or visiting researchers with overlapping interests.
If you wish to make an inquiry, please contact the Centre's director (Jonathan Gorard) at GorardJ@cardiff.ac.uk (Cardiff University), jg865@cantab.ac.uk (Cambridge University) or JonathanG@wolfram.com (Wolfram Research, Inc.).