Area of investment and support

Logic and combinatorics

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.

• Partners involved: Engineering and Physical Sciences Research Council (EPSRC)

The scope and what we're doing

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.

Logic

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.

Combinatorics

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 cybersecurity) 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.

Why we're doing it

Logic

The UK mathematical logic community is small but continues to deliver research of international quality. The UK has strong international expertise in three main areas of logic – proof, model and set theory – with model theory highlighted as a particular strength.

UK expertise includes those actively working in computer science and philosophy departments with close ties to mathematics. The EPSRC pure mathematics workshop in 2016 highlighted the strong intradisciplinary links with other areas of fundamental mathematics.

In particular, exploiting links to number theory, combinatorics, algebraic geometry, topology and geometric group theory were highlighted as potential opportunities for further strengthening of ties to areas of pure mathematics. Links beyond mathematics – for example to computer science and measurement theory – are of national importance due to the role research from logic plays in national security.

Combinatorics

This is a rapidly evolving field of mathematics with connections to many research areas (for example algebra, mathematical analysis, optimisation, number theory, statistics, theoretical computer science and statistical physics).

The UK has a world-leading reputation in this area, with particular strengths in topics such as extremal, additive, enumerative and algebraic combinatorics. The UK’s strength in these has been rewarded by high-profile awards and funding from the European Research Council.

Despite recent growth in the number of researchers working in this area, the interface between algorithms, combinatorial optimisation and combinatorics remains under-represented in the UK compared to communities working on this in the US and the EU.

Logic and Combinatorics

UK expertise in model theory continues to be world-leading, while strength remains in proof and set theory, but computability theory expertise has declined. In combinatorics, the UK’s standing has significantly increased, with extremal, probabilistic, algebraic and enumerative combinatorics being at the forefront of research in this area. Research in combinatorics is difficult to identify explicitly due to its underpinning role across the pure mathematics research areas.

Both Logic and Combinatorics are underpinning fundamental research areas and so play a key role in supporting ongoing research in other areas of the mathematical sciences and other disciplines such as information and communication technologies. They both have potential to play a key role in data science and through applications to coding and encryption theory.

Research stemming from these research areas could therefore also align with and complement future activities at the Alan Turing Institute.

The capacity of researchers working in logic is hard to ascertain due to research ongoing not just in mathematics departments but also in philosophy and computer science departments. Furthermore, the number of researchers focused solely on combinatorics is difficult to quantify due to the fluidity of research topics between pure mathematics research areas.

However, it has been identified that, while the capacity of people working in the field of combinatorics has generally increased, there is still a need to grow capacity with a focus on topics such as algorithms and combinatorial optimisation.

View evidence sources used to inform our research strategies.

Past projects, outcomes and impact

Visualising our portfolio (VoP) is a tool for users to visually interact with the EPSRC portfolio and data relationships. Find out more about research area connections and funding for logic and combinatorics.

Find previously funded projects on Grants on the Web.

Last updated: 16 March 2023