Seminar

Joint Seminar on Mathematical Structure in Computer Science

Math Structures CS seminar is a weekly seminar involving several groups organised by the STyLo team. It provides a space to present ongoing research, share preliminary results, and discuss papers that are centred around mathematical structures in computer science. In particular, we facilitate in-depth discussions and presentations that may last through several sessions.

For questions or if you wish to participate, please contact Oisín Flynn-Connolly.

  • When: Fridays at 10am
  • Where: Location announced via the mailing list, but generally BE0.08 in the Gorleaus building.

The page of the previous academic years are here: 2025-2026.

30 October 2026 – Chase Ford

A Glimpse at Quantitative Algebra

Universal algebra is the model theory of equationally definable classes of structures for an algebraic signature consisting of finitary operation symbols, which makes it a uniform framework for the study of specific algebraic structures such as lattices or monoids. Equational theories are equivalently described by the category-theoretic concept of a finitary monad on the category Set of sets and maps: every finitary monad on Set is the free-algebra monad of an equational theory. More recently, the syntactic characterization of monads on base categories beyond Set has gained considerable attention. The aim of this talk is to provide a glimpse into the theory of quantitative algebra, which aims to extend the methodologies of universal algebra to algebraic structures whose carrier structure is given by equipping a metric space with nonexpansive operations and their relation to metric monads.

For some friendly introductions to the topic, see the following references below.

  1. Wechler, W., Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science, Springer (1992). https://doi.org/10.1007/978-3-642-76771-5

  2. Hyland, M., and A. J. Power, The category theoretic understanding of universal algebra: Lawvere theories and monads, Electronic Notes in Theoretical Computer Science 172 (2007) 437–458. https://doi.org/10.1016/j.entcs.2007.02.019

  3. Radu Mardare, Prakash Panangaden, and Gordon Plotkin. 2016. Quantitative Algebraic Reasoning. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS ‘16). ACM, 700–709. https://doi.org/10.1145/2933575.2934518

23 October 2026 – Henning Basold

Henning Basold will finish his set of lectures on coalgebras.

16 October 2026 – Oisín Flynn-Connolly

Final coalgebras in R-modules

Coinduction in R-modules, for R a commutative ring, is useful in e.g., describing the behaviour of linear weighted automata. In this talk, we describe the construction and properties of final coalgebras with a focus on R-modules. First we sketch the relevant background information on convergence ordinals for finitary functors in locally presentable categories. We then give an example of a binary intersection preserving, finitary functor on the category of abelian groups that fails to converge in ω+ω steps, thus settling a question of Adámek, Milius, and Moss. We relate this to the failure of the Artinian property for Z and show that all finitary functors on RMod preserving countable intersections and a generalised notion of monomorphism converge in ω (n + 2) steps, where n is the Krull dimension of R. Time permitting, we may also explain how conditions for behavioural equivalence to imply bisimulation may be given in terms of the Ext^1 functor from homological algebra. This is joint work with Henning Basold and Chase Ford.

9 October 2026 – Workshop

No seminar, but the Workshop on Reliable Software by Construction.

2 October 2026 – Henning Basold

Henning Basold will continue his set of lectures on coalgebras.

25 September 2026 – Henning Basold

Theory and Applications of Coalgebra as Theory of Observable Systems

Coalgebras provide a natural setting for describing (potentially) infinite behaviour of systems and recurrences via local observations. The theory of coalgebra comprises abstract reasoning principles for the equivalence of system behaviour, abstract composition and transformation operations, coalgebraic logic as generalisation of modal logic, and tools for analysing long-term behaviour.

In this talk, I will provide an introduction to the field of coalgebra and its techniques, and I will show that it provides a fundamental understanding of many recursive structures in mathematics and computer science. We will see in depth the usefulness of coalgebraic proof techniques to reason about smooth functions, and sketch out a broad range of examples from probability theory, logic, dynamical systems and programming languages. Moreover, we will illustrate the power of abstraction through a generalisation of the existence theorem of stationary distributions for Markov chains to general coalgebras. Finally, I will highlight a few ongoing research directions.