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. 

Several well-known formulas involving reflection groups of finite-dimensional algebraic systems break down in infinite dimensions, but there is often a predictable way to correct them. Oxford Mathematician Thomas Oliver talks about his research getting to grips with what structures underlie the mysterious correction process.

Oxford Mathematician Francis Bischoff talks about his recent work on generalized Kähler geometry and the problem of describing its underlying degrees of freedom.

Oxford Mathematician Alan Lauder works on elliptic curves and modular forms, and methods for constructing points on the former using the latter. Read more here.

Oxford Mathematician Nils Matthes talks about trying to understand old numbers using new techniques.

"The Riemann zeta function is arguably one of the most important objects in arithmetic. It encodes deep information about the whole numbers; for example the celebrated Riemann hypothesis, which gives a precise location of its zeros, predicts deep information about the prime numbers. In my research, I am mostly interested in the special values of the Riemann zeta function at integers $k\geq 2$,

Have you ever picked up a glass to find that the coaster it was resting on remains stuck to the bottom? If so, then you have experienced the ability of fluid to stick two surfaces together. When the bottom of the glass is wetted, for example by accidentally spilling a drink, then this fluid can fill the gap between the glass and coaster. The surface tension of the liquid then provides a pulling force on the coaster that keeps it attached to the glass.

Certain inflammatory and infectious diseases, including atherosclerosis and tuberculosis, are caused by the accumulation inside immune cells of harmful substances, such as lipids and bacteria. A multidisciplinary study published in Proceedings B of the Royal Society, by researchers from the Universities of Oxford and Sydney, has shown how cell cannibalism contributes to this process.

Oxford Mathematician Ian Griffiths talks about his work with colleagues Galina Printsypar and Maria Bruna on modelling the most efficient filters for uses as diverse as blood purification and domestic vacuum cleaners.

The formation of liquid drop patterns on solid surfaces is a fundamental process for both industry and nature. Now, a team of scientists including Oxford Mathematician Andreas Münch and colleagues from the Weierstrass Institute in Berlin, and the University of Saarbrücken can explain exactly how it happens.

In this collaboration with researchers from the Umeå University and the University of Zurich, Renaud Lambiotte from Oxford Mathematics explores the use of higher-order networks to analyse complex data.

Oxford Mathematician Ric Wade explains how right-angled Artin groups, once neglected, are now central figures in low-dimension topology. 

We’re all familiar with liquid droplets moving under gravity (especially if you live somewhere as rainy as Oxford). However, emerging applications such as lab-on-a-chip technologies require precise control of extremely small droplets; on these scales, the forces associated with surface tension become dominant over gravity, and it is therefore not practical to rely on the weight of the drops for motion.

Oxford Mathematician Yinan Wang talks about his and colleagues' work on classification of elliptic Calabi-Yau manifolds and geometric solutions of F-theory.

Statistical mechanics (or thermodynamics) is a way of understanding large systems of interacting objects, such as particles in fluids and gases, chemicals in solution, or people meandering through a crowded street. Large macroscopic systems require prohibitively large systems of equations, and so equilibrium thermodynamics gives us a way to average out all of these details and understand the typical behaviour of the large scale system.

If you type fundamental anagram of calculus into Google you will be led eventually to the string of symbols 6accdæ13eff7i3l9n4o4qrr4s8t12ux, probably accompanied by an explanation more or less as follows: this is a recipe for an anagram - take six copies of a, two of c, one of d, one of æ and so on, then rearrange these letters into a chunk of Latin.

As you settle into your seat for a flight to a holiday destination or as part of yet another business trip, it is very easy to become absorbed by the glossy magazines or the novel you've been waiting forever to start reading. Understandably, the phrase "safety features on board this aircraft" triggers a rather unenthusiastic response. But you may be surprised by some of the incredible technology just a few feet away that is there to make sure everything goes smoothly.

Free suspended liquid films or sheets are often formed during industrial production of sprays as well as in natural processes such as sea spray. Early experimental and theoretical investigations of them were done by French physicist Felix Savart, who observed liquid sheets forming by a jet impact on a solid surface, or by two jets impacting each other (1833), and British physicist Arthur Mason Worthington, a pioneer in investigation of the crown splash forming after impact of a drop onto a liquid surface.

Over the last few years, the study of the physiological mechanisms governing the movement of fluids in the brain (referred to as the brain waterscape) has gained prominence. The reason? Anomalies in the brain fluid dynamics are related to diseases such as Alzheimer's disease, other forms of dementia and hydrocephalus. Understanding how the brain waterscape works can help discover how these diseases develop. Unfortunately, experimenting with the human brain in vivo is extremely difficult and the subject is still poorly understood.

Oxford Mathematician Mehdi Yazdi talks about his study of tangled ropes in 3-dimensional space.

Have you ever forgotten to replace the lid of the blender before beginning to puree your mango and passion-fruit smoothie? If you have, you'll have witnessed the catastrophic explosion of fruit and yoghurt flung haphazardly around the kitchen in an unpredictable manner. This is a consequence of the complicated and turbulent fluid dynamics present within the machine, the exact behaviour of which is unknown.

Oxford Mathematician Paul Moore talks about his application of mathematical tools to identify who will be affected with Alzheimer's.

Cycling science is a lucrative and competitive industry in which small advantages are often the difference between winning and losing. For example, the 2017 Tour de France was won by a margin of less than one minute for a total race time of more than 86 hours. Such incremental improvements in performance come from a wide range of specialists, including sports scientists, engineers, and dieticians. How can mathematics assist us?

Oxford Mathematician Ricardo Ruiz Baier, in collaboration mainly with the biomedical engineer Alessio Gizzi from Campus Bio-Medico, Rome, have come up with a new class of models that couple diffusion and mechanical stress and which are specifically tailored to the study of cardiac electromechanics. 

The brain is the most complicated organ of any animal, formed and sculpted over 500 million years of evolution. And the cerebral cortex is a critical component. This folded grey matter forms the outside of the brain, and is the seat of higher cognitive functions such as language, episodic memory and voluntary movement.

Oxford Mathematician Yuuji Tanaka describes his part in the advances in our understanding of gauge theory.

Precise forecasting in the first few days of an infectious disease outbreak is challenging. However, Oxford Mathematical Biologist Robin Thompson and colleagues at Cambridge University have used mathematical modelling to show that for accurate epidemic prediction, it is necessary to develop and deploy diagnostic tests that can distinguish between hosts that are healthy and those that are infected but not yet showing symptoms.

As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Andrew Dancer discusses his work on Ricci Flow.

Taxation and death may be inevitable but what about crime? It is ubiquitous and seems to have been around for as long as human beings themselves. A disease we cannot shake. However, therein lies an idea, one that Oxford Mathematician Soumya Banerjee and colleagues have used as the basis for understanding and quantifying crime.

Oxford Mathematician Nick Trefethen was recently awarded the George Pólya Prize for Mathematical Exposition by the Society for Industrial and Applied Mathematics (SIAM) "for the exceptionally well-expressed accumulated insights found in his books, papers, essays, and talks." Here Nick refllects on the award, his approach to mathematics and the ever-expanding role of Numercial Analysis in the world.

We all know there is no guaranteed way of beating the bank in a casino or predicting the tossing of a coin. Well maybe. Perhaps a little more thought and a large dose of mathematics could help optimise our strategies.

Oxford Mathematician Sam Palmer tackles a crucial issue in our understanding of the risks of serious diseases such as cancer.

Consider a function $f: X \rightarrow X$, an initial value $a_0 \in X$, and an iterative sequence $(a_n)$ given by

Applied mathematics provides a collection of methods to allow scientists and engineers to make the most of experimental data, in order to answer their scientific questions and make predictions. The key link between experiments, understanding, and predictions is a mathematical model: you can find many examples in our case-studies. Experimental data can be used to calibrate a model by inferring the parameters of a real-world system from its observed behaviour.

Elementary particles in two dimensional systems are not constrained by the fermion-boson alternative. They are so-called "anyons''. Anyon systems are modelled by modular tensor categories, and form an active area of research. Oxford Mathematician André Henriques explains his interest in the question.

When mathematicians solve a differential equation, they are usually converting unbounded operators (such as differentiation) which are represented in the equation into bounded operators (such as integration) which represent the solutions.  It is rarely possible to give a solution explicitly, but general theory can often show whether a solution exists, whether it is unique, and what properties it has.  For this, one often needs to apply suitable (bounded) functions $f$ to unbounded operators $A$ and obtain bounded operators $f(A)$ with good properties.  Th

Knots are isotopy classes of smooth embeddings of $S^1$ in to $S^3$. Intuitively a knot can be thought of as an elastic closed curve in space, that can be deformed without tearing. Oxford Mathematician Daniele Celoria explains.

Diophantine approximation is about how well real numbers can be approximated by rationals. Say I give you a real number $\alpha$, and I ask you to approximate it by a rational number $a/q$, where $q$ is not too large. A naive strategy would be to first choose $q$ arbitrarily, and to then choose the nearest integer $a$ to $q \alpha$. This would give $| \alpha - a/q| \le 1/(2q)$, and $\pi \approx 3.14$.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies.

Oxford Mathematicians Álvaro Cartea and Leandro Sánchez-Betancourt talk about their work on employing stochastic optimal control techniques to mitigate the effects of the time delay when receiving information in the marketplace and the time delay when sending instructions to buy or sell financial instruments on electronic exchanges.

At the beginning of the 20th century, Jacques Hadamard gave the definition of well-posed problems, with a view to classifying “correct” mathematical models of physical phenomena. Three criteria should be fulfilled: a solution exists, that solution is unique, and it should depend continuously on the parameters.

Knots are widespread, universal physical structures, from shoelaces to Celtic decoration to the many variants familiar to sailors. They are often simple to construct and aesthetically appealing, yet remain topologically and mechanically quite complex.

Knots are also common in biopolymers such as DNA and proteins, with significant and often detrimental effects, and biological mechanisms also exist for 'unknotting'.

Oxford Mathematician Katherine Staden provides a fascinating snapshot of the field of combinatorics, and in particular extremal combinatorics, and the progress that she and her collaborators are making in answering one of its central questions posed by Paul Erdős over sixty years ago. 

As recent breaches have demonstrated, security will be one of the major concerns of our digital futures. The collective intelligence of the mathematical community is critical to finding these flaws. A group of Oxford Mathematicians, both researchers and undergraduates, have done just that.

How can solar panels become cheaper? Part of the cost is in the production of silicon, which is manufactured in electrode-heated furnaces through a reaction between carbon and naturally occurring quartz rock. Making these furnaces more efficient could lead to a reduction in the financial cost of silicon and everything made from it, including computer chips, textiles, and solar panels. Greater efficiency also means reduced pollution.

Oxford Mathematician Kristian Kiradjiev has been awarded the Institute of Mathematics and its Applications (IMA) Early Career Mathematicians Catherine Richards Prize 2017 for his article on 'Exploring Steiner Chains with Möbius Transformations.' Here he explains his work.

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Nikolay Nikolov discusses his research in to Sofic Groups.

Oxford Mathematician Doireann O'Kiely was recently awarded the IMA's biennial Lighthill-Thwaites Prize for her work on the production of thin glass sheets. Here Doireann describes her work which was conducted in collaboration with Schott AG.

Oxford Mathematicians Tamsin Lee and Peter Grindrod discuss their latest research on the brain, part of our series focusing on the complexities and applications of mathematical research and modelling.

Systemic risk, loosely defined, describes the risk that large parts of the financial system will collapse, leading to potentially far-reaching consequences both within and beyond the financial system. Such risks can materialize following shocks to relatively small parts of the financial system and then spread through various contagion channels. Assessing the systemic risk a bank poses to the system has thus become a central part of regulating its capital requirements.

Social media for health promotion is a fast-moving, complex environment, teeming with messages and interactions among a diversity of users. In order to better understand this landscape a team of mathematicians and medical anthropologists from Oxford, Imperial College and Sinnia led by Oxford Mathematician Mariano Beguerisse studied a collection of 2.5 million tweets that contain the term "diabetes".

Many elastic structures have two possible equilibrium states. For example umbrellas that become inverted in a sudden gust of wind, nanoelectromechanical switches, origami patterns and even the hopper popper, which jumps after being turned inside-out. These systems typically move from one state to the other via a rapid ‘snap-through’. Snap-through allows plants to gradually store elastic energy, before releasing it suddenly to generate rapid motions, as in the Venus flytrap .

Plants use many strategies to disperse their seeds, but among the most fascinating are exploding seed pods. Scientists had assumed that the energy to power these explosions was generated through the seed pods deforming as they dried out, but in the case of ‘popping cress’ (Cardamine hirsuta) this turns out not to be so. These seed pods don’t wait to dry before they explode.

Glioblastoma is an aggressive form of brain tumour, which is characterised by life expectancies of less than 2 years from diagnosis and currently has no cure. The only intervention available to a patient is having the infected area of their brain cut away as soon as the tumour cells are observed.

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. In their recently published paper Oxford Mathematicians Dominic Vella and Alain Goriely and colleagues looked at the dynamics of the chain when one of the weights is removed and thus one end is pulled with constant acceleration.

This picture shows the "Z" machine at Sandia Labs in New Mexico producing, for a tiny fraction of a second, 290 TW of power - about 100 times the average electricity consumption of the entire planet. This astonishing power is used to subject metal samples to enormous pressures up to 10 million atmospheres, causing them to undergo violent plastic deformation at velocities up to 10 km/s. How should such extreme behaviour be described mathematically?

Organometal halide perovskite (OMHP) is hardly a household name, but this new material is the source of much interest, not least for Oxford Applied Mathematicians Victor Burlakov and Alain Goriely as they model the fabrication and operation of solar cells.

The International Congresses of Mathematicians (ICMs) take place every four years at different locations around the globe, and are the largest regular gatherings of mathematicians from all nations.  However, as much as the assembled mathematicians may like to pretend that these gatherings transcend politics, they have always been coloured by world events: the congresses prior to the Second World War saw friction between French and German mathematicians, for example, whilst Cold War political tensions likewise shaped the conduct of later congresses.