How maths changes the world: mathematical science impact case studies drawn from REF 2021

Abstract

The mathematical sciences are an essential—albeit often invisible—part of the UK's national infrastructure, woven through almost every sector of the economy and most of the systems and services on which society depends. They are fundamental to our resilience, quality of life, and prosperity: recent analysis by AcadMathSci found that people working in mathematical science occupations directly contributed nearly half a trillion pounds to the UK economy in 2024, equivalent to about 19% of total GDP. But what does this contribution mean in practice? One does not have to look far for examples. The Research Excellence Framework (REF) is the UK's system for assessing the quality and impact of research carried out in higher education institutions. Its most recent exercise, REF 2021, included many detailed impact case studies showing how university research has delivered tangible benefits far beyond academia: from improving public services and protecting lives to supporting new technologies, strengthening businesses, and informing public policy. This report distils a subset of the 176 publicly available REF 2021 impact case studies associated with the mathematical sciences$^1$ into concise, accessible summaries. The case studies reach across the economy and society, showing how mathematical ideas developed in one setting can provide the methods, evidence, and insight needed to solve important problems elsewhere. Together, they illustrate both the breadth of the UK's mathematical research and the variety of ways in which it is changing the world. This collection is necessarily a selective and time-bound snapshot, rather than a comprehensive account, of the mathematical sciences' leading-edge impact. For example, REF 2021 captured research and impact developed before the recent rapid advances in artificial intelligence (AI) capabilities: AI-related case studies were thus far less prominent than they are likely to be in future exercises. Work connected to national security is also substantially under-represented: a REF 2021 overview report noted that many highly confidential case studies related to national security (i.e., not among the 176 mathematical science case studies$^1$ in the public domain) were based on "mathematical research of the very highest calibre." Despite such constraints, these case studies point consistently to the same conclusion. When we invest in the mathematical sciences—in education, research, and people—we strengthen the UK’s capacity to innovate, respond to uncertainty, and improve lives. That investment helps build a more productive, resilient, and prosperous future, while generating knowledge and tools with benefits reaching far beyond the problems they were first developed to solve. Maximising the UK-wide impact of the mathematical sciences, and sustaining that impact into the future, requires national capacity to identify, connect, and utilise mathematical expertise. AcadMathSci is working to build that national capacity by bringing together education, academia, business, industry and government; translating evidence into forms useful for decision-makers, policymakers, and industry; and helping ensure that the mathematical sciences are recognised and supported as a strategic national asset. With sustained support, AcadMathSci will be able to advance its wider work to strengthen the mathematical sciences across the UK. Repeating and extending this exercise following REF 2029 would form one part of that work: capturing a new generation of impact case studies, strengthening the evidence base for support for the mathematical sciences, and revealing many further ways in which the mathematical sciences are transforming lives, society, and the economy.