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The Centre for Applied Compositionality

Exploring the interface between mathematics, computation and the foundations of science

Vision

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.

Projects

Some ongoing projects include:

  • Compositional graph rewriting (e.g. single-pushout, double-pushout, double-pushout with interfaces, etc.) as a computable model for discrete differential geometry, the fundamental structure of spacetime and the categorical foundations of quantum gravity.
  • Symmetric monoidal categories as models of multiway computation and their relationship with categorical quantum mechanics. Applications to quantum information theory (e.g. compact-closed dagger-symmetic monoidal categories, ZX-calculus, etc.)
  • Higher categorical models of causality in multiway systems and the relationship with axiomatic quantum field theory/quantum gravity. Fibration/foliation duality as a generalisation of the duality between functorial and algebraic quantum field theories. General models of causal semantics in multiway systems.
  • Higher homotopies in multiway systems and relationships with homotopy type theory, synthetic differential geometry and the emergence of spatiality in discrete/computational systems.

  • Symmetric monoidal categories as models of Petri nets, their generalisations and their causal semantics, with applications to chemical reaction networks, packet switching networks, interaction networks between biotic systems, resource-based quantum computing, etc.
  • Diagrammatic reasoning languages (especially those based on monoidal string diagrams) for quantum computation, abstract algebra, tensor and array manipulation, metamathematics, etc.
  • Decorated cospans as models for open networks and complex systems (e.g. ecological networks, trophic networks, systems biology, open quantum systems, etc.)
  • Higher category theory and dependent/homotopy type theory as a foundation for automated theorem-proving; formal compositional semantics for pure mathematics.
  • Speculative compositional models for natural language processing and cognition (e.g. Lambek calculus and categorial grammars).

Contact

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.).