Number theory

Number Theory is the study of the properties of integers, using the tools of modern mathematics to address many basic unanswered questions (e.g. concerning the distribution of prime numbers among integers, or solubility in integers of polynomial equations). Techniques used include many from analysis, algebra and geometry.

We aim to maintain the size of this research area as a proportion of the EPSRC portfolio. During the previous Delivery Plan period, we invested significantly in this area through programme grants. While this has contributed to a platform of excellent research ongoing in the UK, it has also caused localised research funding and created a significant critical mass of people working on specific topics in Number Theory. As a result, the balance of funding must be re-focused to prevent further localisation in a few geographical areas.

To achieve this, during the current Delivery Plan period, we will:

  • Support high-quality research on a smaller scale than during the previous Delivery Plan. Currently, there are no plans to continue investment through programme grants
  • Continue to support novel research emerging from the significant intradisciplinary overlap between Number Theory and research areas within mathematical sciences (e.g. by building on links to topics such as representation theory, algebraic geometry, additive combinatorics and probability theory)
  • Establish and strengthen interdisciplinary links through topics such as string theory, cryptology and theoretical computer science, as these are key mechanisms to accelerate the impact of Number Theory research. This will make a significant contribution, for instance, to nationally important fields such as cyber-security
  • Monitor the balance of researchers across career stages and ensure appropriate support. We will achieve this by funding standard-mode grants of the highest quality and where fellowship proposals align with the Mathematical Sciences Theme fellowship priority areas.
Highlights:

The UK continues to be world-leading in many aspects of Number Theory, with a particular focus on: L-functions; Diophantine geometry; and number theory in the Langlands programme. Number Theory researchers have received numerous European Research Council and Clay Institute awards. The partnerships between academic Number Theory research and the Heilbronn and Clay Institutes emphasise the potential opportunities for transformative research in this area. The number of speakers from UK-based research institutions invited to give plenary talks at the International Congress of Mathematics is a further reflection of success.

Number Theory contributes significantly to ongoing research in cryptography, an area of national importance as it plays a key role in information security (especially cyber-security), (Evidence source 1,2). Number Theory’s role can be underlined by the case studies highlighted in the Research Excellence Framework (REF) 2014 exercise which were provided by Government Communications Headquarters (GCHQ), (Evidence source 3). Number Theory has strong links to all areas of pure mathematics, especially algebra and geometry in relation to the Langlands programme. Other applications include communications technology and, more generally, engineering and physics (Evidence source 4,5).

The area has intradisciplinary links to all areas of pure mathematics, in particular through representation theory, algebraic geometry, additive combinatorics and probability theory. Sustaining these and strengthening interdisciplinary links (e.g. to string theory, cryptology and theoretical computer science) is key to maintaining this area’s health in the UK (Evidence source 4,5).

The research community remains relatively small, with researchers typically located in the south of England, due to the location of the main security services (e.g. GCHQ) and the Heilbronn and Clay Institutes. The Pure Mathematics Centre for Doctoral Training (CDT) is training the next generation of researchers in Geometry and Number Theory.

Over the previous Delivery Plan, we funded three programme grants in this research area. This has built a significant critical mass of researchers. However, this increase in funding has also imbalanced the mathematical sciences portfolio and this needs addressing (Evidence source 4,5,6).

This are will continue to underpin all of the Prosperity Outcomes, due to its support for data science and robotics and autonomous systems. It is especially relevant to the following Connected, Healthy, Productive and Resilient 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

Number Theory research plays a key role in encryption and security technologies.

H2: Improve prevention and public health

H3: Optimise diagnosis and treatment

Privacy and confidentiality in patient records, diagnostic outputs and treatment requires novel application of research related to data security and anonymisation.

P4: Drive business innovation through digital transformation

Number Theory research plays a key role in cryptography, digital signatures and data security.

R2: Ensuring a reliable infrastructure which underpins the UK economy

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

Fundamental research from Number Theory will play a role in the development of data storage, encryption technologies and cyber-security. 

  1. Department for Business, Innovation & Skills (BIS), Competitive Analysis of the UK Cyber Security Sector (PDF), (2013)
  2. Cabinet Office, The Cyber Security Strategy 2011-2016: Annual Report (PDF), (2016)
  3. Research Excellence Framework (REF) 2014, Impact Case Studies, (2014)
  4. EPSRC, EPSRC Pure Mathematics Evidence and Engagement Workshop Report, (2016)
  5. EPSRC, EPSRC Community Overview Document: Number Theory
  6. EPSRC Mathematical Sciences Regional Workshops 2016

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 number theory (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