Logic and combinatorics

Note: 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 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 therefore intend to maintain their current size as a proportion of the EPSRC portfolio.

Logic:

By the end of the current Delivery Plan, we aim to have:

  • Continued 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. Support of interactions with aspects of algebraic, geometric and number theoretic research are key to preserving this
  • Encouraged and enabled novel research to continue at the interface between mathematical Logic and the application of logical research in computer science. The goal is to achieve this through closer interactions with the Theoretical Computer Science research area, which has established a significant overlap with Logic
  • Ensured that the community has the appropriate people and skills balance for the UK to remain at the forefront of mathematical advances in this field. This will be achieved through funding via standard-mode grant mechanisms and through strategic activities, where possible

Combinatorics:

By the end of the current Delivery Plan, we aim to have:

  • Continued support of sub-fields of key UK strength (e.g. extremal, additive, enumerative and algebraic Combinatorics) and encouraged and fostered relevant links with other research areas, including those within mathematical sciences and beyond (e.g. Theoretical Computer Science)
  • Encouraged fundamental research in areas that could contribute to fields of national importance (e.g. data science and cyber-security) in the short to long term
  • Continued to support a strong training and skills base in this area, monitored over the Delivery Plan and with interventions made where necessary
  • Worked with the community to identify the most appropriate routes to maximise and highlight impact of ongoing research to the wider scientific community
Highlights:

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 (e.g. to computer science and measurement theory) are of national importance due to the role research from Logic plays in national security. (Evidence source 1-4)

Combinatorics:

This is a rapidly evolving field of mathematics with connections to many research areas (e.g. Algebra, Mathematical Analysis, Optimisation, Number Theory, Statistics, Theoretical Computer Science and Statistical Physics). (Evidence source 1-4)

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. (Evidence source 1-4)

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 European Union.

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 over the previous Delivery Plan. In Combinatorics, the UK's standing has significantly increased over the last decade 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. (Evidence source 1-4)

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 (ICT). 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. (Evidence source 1-5)

This research area will continue to underpin the Connected, Healthy, Productive and Resilient Outcomes, in particular the following Ambitions:

C1: Enable a competitive, data-driven economy

C3: Deliver intelligent technologies and systems

C4: Ensure a safe and trusted cyber society

C5: Design for an inclusive, innovative and confident digital society

Logic and Combinatorics research is highly relevant as discrete algorithms are fundamental to data science and their theoretical foundation is in combinatorics and discrete mathematics.

H2: Improve prevention and public health

H3: Optimise diagnosis and treatment

Due to the large volumes of data that will be generated through research relevant to these fields, Logic and Combinatorics research will have an important role to play as they underpin discrete algorithms which could be key to data science.

P4: Drive business innovation through digital transformation

Research in areas such as Logic and Combinatorics is considered blue-sky thinking and so has an underpinning role to play.

R3: Develop better solutions to acute threats: cyber, defence, financial and health

Research in Logic and Combinatorics has been shown to have a significant impact in the application areas of data science and cyber-security. 

Research area connections

This diagram shows the top 10 connections between Research Areas within the EPSRC research portfolio. The depth of the segment relates to value of grants and the width of the segment relates to the number of grants shared by those two Research Areas. Please click to see the related Research Area rationale.

Maintain

We aim to maintain this area as a proportion of the EPSRC portfolio.

Visualising our Portfolio (VoP)
Visualising our portfolio (VoP) is a tool for users to visually interact with the EPSRC portfolio and data relationships.

EPSRC support by research area in Logic and combinatorics (GoW)
Search EPSRC's research and training grants.

Contact Details

In the following table, contact information relevant to the page. The first column is for visual reference only. Data is in the right column.

Name: Tom Robinson
Job title: Portfolio Manager
Section / Team: Pure Mathematics
Department: Mathematical Sciences
Organisation: EPSRC
Telephone: 01793 442892