Projectdetails

This project focuses on dualities which are 'dually' relevant, both in mathematics and informatics. Within mathematics, dualities underlie fundamental connections between algebra and geometry, and between logical syntax and semantics, e.g. in the various dualities extending Gelfand and Stone. Such dualities also appear in informatics where they relate (program) logics and computations. Dualities are best described and understood in the language of category theory as arising from certain (dual) adjunctions. Additional (mathematical/computational) structure then appears as endofunctors whose algebras and coalgebras capture the compound systems in the duality setting.

The aim of this project is to significantly advance interdisciplinary interaction between topological methods in algebra and coalgebraic methods in informatics by engaging two Ph.D. students and two senior researchers in addressing cutting-edge problems pertinent to both disciplines, and to seek shared solutions and shared understanding. Dualities based on discrete spaces are central in mathematics where profinite structures are concerned, in algebraic logic, and in many applications to informatics. Dualities based on continuous topological algebras such as the unit interval, the real or complex numbers play an important role in mathematical physics, algebraic geometry, and infinitary-valued logic. The introduction of stochastics in computing systems also makes this setting timely and relevant for informatics. Our main goal is to unify dualities for enriched discrete and continuous spaces. For mathematics, this project aims to unify topological methods in logic and in geometry and algebra. In computer science terms, this goal may be stated as aiming for a generalization of Abramsky's 'Domains in Logical Form' encompasing probabilistic computing systems.