Area of investment and support

Area of investment and support: Number theory

The study of the properties of integers, using the tools of modern mathematics to address many basic unanswered questions.

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

The scope and what we're doing

Number theory is the study of the properties of integers, using the tools of modern mathematics to address many basic unanswered questions – for example, concerning the distribution of prime numbers among integers, or solubility in integers of polynomial equations.

Techniques used include many from analysis, algebra and geometry.

Supporting novel research

We will continue to support novel research emerging from the significant intradisciplinary overlap between Number Theory and research areas within mathematical sciences – for example, by building on links to topics such as:

representation theory
algebraic geometry
additive combinatorics
probability theory.

Strengthening interdisciplinary links

We will continue to 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.

Supporting our researchers

We will monitor the balance of researchers across career stages and ensure appropriate support. We will achieve this by funding new investigator awards, standard grants and programme grants of the highest quality and where fellowship proposals align with the mathematical sciences fellowship priority areas.

Why we're doing it

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 Institute 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 Mathematicians 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). 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).

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.

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 – for example, to string theory, cryptology and theoretical computer science – is key to maintaining this area’s health in the UK.

The research community remains relatively small, with researchers typically located in the south of England due to the location of the main security services, such as:

The London School of Geometry and Number Theory is training the next generation of researchers in geometry and number theory.

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 number theory.

Find previously funded projects on Grants on the Web.

Last updated: 20 February 2023