How to deal with resistance? This is the headline question these days with regards to COVID vaccines. But it is an important question also in cancer therapy. Over the past century, oncology has come a long way, but all too often cancers still recur due to the emergence of drug-resistant tumour cells. How to tackle these cells is one of the key questions in cancer research. The main strategy so far has been the development of new drugs to which the resistant cells are still sensitive.

Take a piece of rope and knot it as you wish. When you are done, glue the two extremities together and you will obtain a physical realisation of what mathematicians also call a knot: a simple closed curve in 3-dimensional space. Now, put the knotted rope on a table and take a picture of it from above. It is now a planar projection of your knot. The mathematical equivalent of it is a knot diagram with multiple crossings as shown in the figure.

Oxford Mathematician Daniel Gulotta talks about his work on $p$-adic geometry and the Langlands program.

"Geometry is one of the more visceral areas of mathematics. Concepts like distance and curvature are things that we can actually see and feel.

For thirty years Oxford Mathematician Roger Penrose has challenged one of the key planks of Cosmology, namely the concept of Inflation, now over 40 years old, according to which our universe expanded at an enormous rate immediately after the Big Bang. Instead, fifteen years ago, Penrose proposed a counter-concept of Conformal Cyclic Cosmology by which Inflation is moved to before the Big Bang and which introduces the idea of preceding aeons.

Oxford Mathematician Ben Green on how and why he has been pondering footballs in high dimensions.

"A 3-dimensional football is usually a truncated icosahedron. This solid has the virtue of being pleasingly round, hence its widespread use as a football. It is also symmetric in the sense that there is no way to tell two different vertices apart: more mathematically, there is a group of isometries of $\mathbf{R}^3$ acting transitively on the vertices.

Social distancing measures to reduce the spread of the novel coronavirus are in place worldwide. These guideline are for everyone. We are all expected to reduce our contact with others, and this will have some negative impacts in terms of mental health and loneliness, particularly for the elderly and other vulnerable groups. So why should we follow measures that seem so extreme? The answer is simple. Social distancing works. It reduces transmission of the virus effectively and lessens the impact on already stretched healthcare services.

When Oxford Mathematician Alain Goriely was approached by his collaborator Ellen Kuhl from Stanford University to work on a travel restriction issue in Newfoundland he started a Coronavirus journey that ended up in the Canadian Supreme Court.

Mathematical models have been used throughout the COVID-19 pandemic to help plan public health measures. Attention is now turning to how interventions can be removed while continuing to restrict transmission. Predicting the effects of different possible COVID-19 exit strategies is an important current challenge requiring mathematical modelling, but many uncertainties remain.

Oxford Mathematician Katherine Staden talks about the mathematics of seating plans.

The Coronavirus disease pandemic (COVID-19) poses unprecedented challenges for governments and societies around the world. In addition to medical measures, non-pharmaceutical measures have proven to be critical for delaying and containing the spread of the virus. However, effective and rapid decision-making during all stages of the pandemic requires reliable and timely data not only about infections, but also about human behaviour, especially on mobility and physical co-presence of people.