Logic includes model theory, recursion theory, proof theory and set theory. Combinatorics is concerned with the study of discrete structures such as graphs and hypergraphs.
Logic and combinatorics are separate research areas but combined here due to their relatively small size.
Mathematical logic is divided broadly into four areas – model theory, recursion theory (also known as computability theory), proof theory and set theory – that have common origins in the foundations of mathematics, but now have very different perspectives. There is also a strong interface between logic and computer science, including topics such as automated reasoning and program extraction.
In its most basic form, combinatorics is concerned with the arrangement of discrete objects according to constraints. Combinatorics studies discrete structures such as graphs (also known as networks) and hypergraphs. This research area includes, for instance, algebraic and probabilistic combinatorics, combinatorial optimisation and Ramsey theory.
Although both areas are of a relatively small size, they continue to produce research of an international standard.
We aim to:
- continue to support and enhance the current UK research strengths in proof, model and set theory – the UK has significant influence in the applied aspects of model theory, and support of interactions with aspects of algebraic, geometric and number theoretic research are key to preserving this
- encourage and enable novel research to continue at the interface between mathematical logic and the application of logical research in computer science, through closer interactions with the Theoretical Computer Science research area
- ensure that the community has the appropriate people and skills balance for the UK to remain at the forefront of mathematical advances in this field, by funding through the standard funding opportunity mechanisms and strategic activities where possible.
We aim to:
- continue support of sub-fields of key UK strength (for example, extremal, additive, enumerative and algebraic combinatorics)
- encourage and foster relevant links with other research areas, including those within mathematical sciences and beyond (for example, theoretical computer science)
- encourage fundamental research in areas that could contribute to fields of national importance (for example data science and cyber-security) in the short to long term
- continue to support a strong training and skills base in this area, with interventions made where necessary
- work with the community to identify the most appropriate routes to maximise and highlight the impact of ongoing research to the wider scientific community.