Deep learning has become an important topic across many domains of science due to its recent success in image recognition, speech recognition, and drug discovery. Deep learning techniques are based on neural networks, which contain a certain number of layers to perform several mathematical transformations on the input.

Oxford Mathematician Ben Green on a tale of conjectures, mistaken assumptions and eventual solutions: a tale of mathematics.

"The famous discrete mathematician Ron Graham sadly passed away last year. I did not know him well, but I had the pleasure of meeting him a few times. On the first such occasion, in Vancouver in 2004, he mentioned one of his favourite open questions over lunch. This concerns the size of certain "van der Waerden numbers", a kind of arithmetic variant of graph Ramsey numbers.

During the early growth of the brain, an extraordinary process takes place where axons, neurons, and nerves extend, grow, and connect to form an intricate network that will be used for all brain activities and cognitive processes. A fundamental scientific question is to understand the laws that these growing cells follow to find their correct target.

By pooling resources between cells, colonies of bacteria can exhibit behaviours far beyond the capabilities of an individual bacterium. For example, bacterial populations can encase themselves in a self-generated polymer matrix that shelters cells in the core of the population from the external environment. Such communities are termed “bacterial biofilms”, and show increased tolerance to antimicrobial treatments such as antibiotics.

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.

Social distancing is integral to our lives these days, but distancing also underpins the ordered patterns and arrangements we see all around us in Nature. Oxford Mathematician Priya Subramanian studies the defects in such patterns and shows how they relate to the underlying pattern, i.e. to the distancing itself.

Tissue oxygenation plays a crucial role in the growth of cancerous tumours and their response to treatments. While it may seem intuitive that reducing oxygen delivery to a tumour would be a treatment therapy, low oxygen levels (hypoxia) can significantly reduce the effectiveness of treatments such as radiotherapy and some chemotherapies. Therefore, understanding the dynamics of a tumour's red blood cells - which carry oxygen through the vasculature - is of vital importance.

One of the great puzzles of the current COVID-19 crisis is the observation that older people have a much higher risk of becoming seriously ill. While it is usually commonly accepted that the immune system fails progressively with age, the actual mechanism leading to this effect was not fully understood. In a recent work, Sam Palmer from Oxford Mathematics and his colleagues in Cambridge have proposed a simple and elegant solution to this puzzle.

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.

A set of integers greater than 1 is primitive if no number in the set divides another. Erdős proved in 1935 that the series of $1/(n \log n)$ for $n$ running over a primitive set A is universally bounded over all choices of A. In 1988 he conjectured that the universal bound is attained for the set of prime numbers. In this research case study, Oxford's Jared Duker Lichtman describes recent progress towards this problem:

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.

Ben Green and collaborators discover that the well-known "birthday paradox" has its equivalent in the divisors of a typical integer.

"The well-known "birthday paradox'' states that if you have 23 or more people in a room - something difficult to achieve nowadays without a very large room - then the chances are better than 50:50 that some pair of them will share a birthday. If we could have a party of 70 or more people, the chance of this happening rises to 99.9 percent.

In modern Cryptography, the security of every cryptosystem is required to be formally proven. Most of the time, such formal proof is by contradiction: it shows that there cannot exist an adversary that breaks a specific cryptosystem, because otherwise the adversary would be able to solve a hard mathematical problem, i.e. a problem that needs an unfeasible amount of time (dozens of years) to be concretely solved, even with huge computational resources.

How do we design face masks that efficiently remove contaminants while ensuring that we can still breathe easily? One complicating factor with this question is the fact that the properties of the material that we start off with for our face mask can be very different when in use. A key example is seen when you stretch the mask around your face to put it on. In doing so, you also stretch the pores, i.e., the holes in the material that allow the air to pass through.

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

The Oberwolfach Institute is a famous venue for mathematical conferences and events. Its dining hall contains some circular tables of various sizes; I don't know how big they are but I know they seat $n$ people in total. I'm inviting $n$ people to a conference, and to foster good communication, I want to write some seating plans so that each pair of people sit next to each other exactly once during dinner. Is this possible?

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.