UK Biobank is an intensively characterised prospective cohort of 500,000 adults aged 40–69 years when recruited between 2006 and 2010. The study was established to enable researchers worldwide to undertake health-related research in the public interest. The existence of such a large, detailed prospective cohort with a high degree of participant engagement enabled its rapid repurposing for coronavirus disease-2019 (COVID-19) research. In response to the pandemic, the frequency of updates on hospitalisations and deaths among participants was immediately increased, and new data linkages were established to national severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) testing and primary care health records to facilitate research into the determinants of severe COVID-19. UK Biobank also instigated several sub-studies on COVID-19. In 2020, monthly blood samples were collected from approximately 20,000 individuals to investigate the distribution and determinants of SARS-CoV-2 infection, and to assess the persistence of antibodies following infection with another blood sample collected after 12 months. UK Biobank also performed repeat imaging of approximately 2,000 participants (half of whom had evidence of previous SARS-CoV-2 infection and half did not) to investigate the impact of the virus on changes in measures of internal organ structure and function. In addition, approximately 200,000 UK Biobank participants took part in a self-test SARS-CoV-2 antibody sub-study (between February and November 2021) to collect objective data on previous SARS-CoV-2 infection. These studies are enabling unique research into the genetic, lifestyle and environmental determinants of SARS-CoV-2 infection and severe COVID-19, as well as their long-term health effects. UK Biobank’s contribution to the national and international response to the pandemic represents a case study for its broader value, now and in the future, to precision medicine research.

Certain ideas recur in many areas of mathematics. One example is groups of symmetries, which appear in the Galois theory of equations and in Lie groups. Lie groups are of great value in physics, where Noether’s theorem enables us to derive a conservation law for every case in which a function known as the Lagrangian is invariant under a one-parameter Lie group. The importance of this approach can be seen from the fact that the laws of the conservation of energy, linear momentum and angular momentum are all outcomes of Noether’s theorem, though they can of course be derived by simpler methods. The full power of Noether’s approach is shown in its applications to quantum field theory, where it can be used to find conserved currents and charges.

Calculating Fisher's F-ratio is a key step in a number of statistical procedures involving null hypothesis significance testing. This is particularly so in the case of ANOVA (analysis of variance) in its several forms, but even multiple regression includes a test of significance of the overall model which employs an F-ratio. The present paper aims at making the basic ideas behind this common statistic more comprehensible by providing a visual counterpart to, and justification for, its algebraic definition.

As an example of how the definition works, consider the following very simple set of data comprising an independent variable consisting of three groups, where the values of the dependent variable are 1, 2, 3 for the first group, 4, 5, 6 for the second group and 7, 8, 9 for the third group. The groups could represent three drug treatments, and the numbers, a measure of clinical outcome for each of nine participants. One might represent this set of data as a row vector thus: (1, 2, 3, 4, 5, 6, 7, 8, 9).