In this case study I report on a collaboration with Mo Dick Wong (Durham University), in the area of analytic and probabilistic number theory. We studied questions on a *random model* for the Möbius $\mu$ function. This function is one of the most elusive functions in number theory, and encodes deep questions on primes. Let's define it properly.

Globally kidney disease is forecast to be the 5^{th} leading cause of death by 2040, and in the UK more than 3 million people are living with the most severe stages of chronic kidney disease. Chronic kidney disease is often due to autoimmune damage to the filtration units of the kidney, known as the glomeruli, which can occur in lupus, a disease which disproportionally affects women and people of non-white ethnicities, groups often underrepresented in research.

One of the greatest challenges in cancer treatment is maximising the response to a given drug - how can doctors get the greatest impact for the patient from the drug?

Traditionally, the answer to this has been ‘maximum tolerated dose’ (MTD) therapy, where the patient continually receives a high drug dose, with no breaks in treatment.

Systems of differential equations have a key role in biological and chemical models. These models come with parameters that show the model’s dependency on the environmental effects and often have unknown values. Model simulations from observation are desired not to be affected by the values of the parameters. In other words, we would like the parameters to be identifiable from the input-output behaviour of the system.

In mathematics, as in footwear, order matters. Putting on your socks before your shoes produces a different result to putting on your shoes before your socks. This means that the two operations 'putting on socks' and 'putting on shoes' are *noncommutative*. Noncommutative structures are widespread in mathematics, appearing in subjects ranging from group theory to analysis and differential equations.

We all know that mathematical activity goes on nowadays in a great variety of settings – not just in academia, but across the whole range of industry, education, and beyond. This diversity in mathematics is by no means new, and yet the study of the history of mathematics has often failed to capture it.

Holography is one of a set of powerful tools which theoretical physicists use to understand the fundamental aspects of nature. The holographic principle states that the entire information content of a theory of quantum gravity in some volume is equivalent (or dual) to a theory living at the boundary of the volume without gravity. The boundary degrees of freedom encode all the bulk degrees of freedom and their dynamics and vice versa.

Systems of chemical reactions figure prominently in models throughout the natural sciences. Biology in particular gives rise to hugely complicated reaction networks which, somehow, perform intricate and delicate functions. How do we make sense of these complex networks, and try to understand the relationships between their structure and function?

Mathematical modelling played a key role in describing the spread of the COVID-19 pandemic; now a different kind of maths is helping us understand how immune cells interact in the lungs of patients with severe COVID-19.

In a damaged lung with a massive immune cell infiltrate, as seen with severe COVID-19 infection, it can be difficult to figure out which cells are involved in causing lung injury.

For the first several centuries of its existence, an education at the University of Oxford entailed a basic grounding in a range of different subjects, rather than the specialised study of a single discipline. The goal was to turn out well-rounded individuals rather than narrow experts. Nevertheless, the university often tried, wherever possible, to provide advanced instruction in specific areas for those students who were interested.

In this case study we'll highlight new world records, going 23% beyond the Riemann Hypothesis. To explain, we start with the (last digit of) *prime numbers*: \[ {\color{blue}{\bf 2}}, {\color{green}{\bf3}}, {\color{orange}{\bf5}}, {\color{red}{\bf7}}, 1{\bf1}, 1{\color{green}{\bf3}}, 1{\color{red}{\bf7}}, 1{\color{purple}{\bf9}}, 2{\color{green}{\bf3}},\ldots \] After some thought, we may realize that no such last digit may be even (after ${\color{blue}{\bf 2}}$ itself), else the whole number is even; nor may ${\color{orange}{\bf5}}$ appear again.

During the COVID-19 pandemic, mathematical modelling played a major role in informing public health policy responses. A key question for public health policy makers is whether the introduction of a virus into a population is likely to lead to sustained transmission. This is critical for understanding the epidemic and/or pandemic potential of a novel virus – notably, for example, following the first detected COVID-19 cases in Wuhan, China.

Many modern challenges facing industry and government relate to cleaning: removing pollution from water or industrial waste gases, or decontaminating the environment after chemical spills. Mathematicians in Oxford are collaborating with various industrial and government partners to model specific cleaning challenges, deepening understanding of these processes and working towards optimal cleaning solutions.

Researchers at the University of Oxford’s Botanic Garden and in Oxford Mathematics have shown that the shape, size and geometry of carnivorous pitcher plants determines the type of prey they trap. The results have been published today in the *Proceedings of the National Academy of Sciences (PNAS)*.

In many modern applications, a key bottleneck is the solution of a matrix problem of the form Ax=b where A is a large matrix. In numerical weather prediction, such systems arise as a sub-problem within data assimilation algorithms. In this setting, finding the most likely initial condition with which to initialise a forecast is equivalent to finding the (approximate) solution x.

Symmetry underpins all physics research. We look for fundamental and beautiful patterns to describe and explain the laws of nature. One way of explaining symmetry is to ask: "what is the full set of operations I can do to my real-world experiment or abstract theory written on paper that doesn't change any physical measurements or predictions?'' There are simple symmetries we are perhaps already familiar with. For example, lab-based physics experiments usually don't care if you wait an hour to do the experiment or if you rotate your apparatus by 90 degrees.

Inspired by jumping insects, Oxford Mathematicians have helped develop a miniature robot capable of leaping more than 40 times its body length - equivalent to a human jumping up to the 20th floor of a building. The innovation could be a major step forward in developing miniature robots for a wide range of applications.

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