Case Studies and Films

The magic of numbers - finding structure in randomness

Mathematics is full of challenges that remain unanswered. The field of Number Theory is home to some of the most intense and fascinating work. Two Oxford mathematicians, Ben Green and Tom Sanders, have recently made an important breakthrough in an especially tantalising problem relating to arithmetic structure within the whole numbers.

The mathematics of violent plastic deformation

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?

The Mathematics of Shock Reflection-Diffraction and von Neumann’s Conjectures

As part of our series of research articles deliberately focusing on the rigour and complexity of mathematics and its problems, Oxford Mathematician Gui-Qiang G Chen discusses his work on the Mathematics of Shock Reflection-Diffraction.

Improving the quality and safety of x-rays

X-ray imaging is an important technique for a variety of applications including medical imaging, industrial inspection and airport security. An X-ray image shows a two-dimensional projection of a three-dimensional body. The original 3D information can be recovered if multiple images are given of the same object from different viewpoints. The process of recovering 3D information from a set of 2D X-ray projections is called Computed Tomography (CT).

Mitigating the impact of buy-to-let on the housing market

Much has been written about the buy-to-let sector and its role in encouraging both high levels of leverage and increases in house prices. Now Oxford Mathematician Doyne Farmer and colleagues from the Institute for New Economic Thinking at the Oxford Martin School and the Bank of England have modelled that impact.

Hummingbirds, umbrellas and hopper poppers do it. But why not as quickly as expected?

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 .

From Birds to Bacteria: Modelling Migration at Many Scales

The use of mathematical models to describe the motion of a variety of biological organisms has been the subject of much research interest for several decades. If we are able to predict the future locations of bacteria, cells or animals, and then we subsequently observe differences between the predictions and the experiments, we would have grounds to suggest that the local environment has changed, either on a chemical or protein scale, or on a larger scale, e.g.

Well behaved cities - what all cities have in common

How are people, infrastructure and economic activity organised and interrelated? It is an intractable problem with ever-changing infinite factors of history, geography, economy and culture playing their part. But a paper by Oxford Mathematician Hyejin Youn and colleagues suggests “a mathematical function common to all cities.”

Mathematical theories of consciousness

How a complex dynamic network such as the human brain gives rise to consciousness has yet to be established by science. A popular view among many neuroscientists is that, through a variety of learning paradigms, the brain builds relationships and in the context of these relationships a brain state acquires meaning in the form of the relational content of the corresponding experience.

How do networks shape the spread of disease and gossip?

A new approach to exploring the spread of contagious diseases or the latest celebrity gossip has been tested using London’s street and underground networks. Results from the new approach could help to predict when a contagion will spread through space as a simple wave (as in the Black Death) and when long-range connections, such as air travel, enable it to seemingly jump over long distances and emerge in locations far from an initial outbreak.

Did Value at Risk cause the crisis it was meant to avert?

What were the causes of the crisis of 2008? New research by Oxford Mathematicians Doyne Farmer, Christoph Aymanns, Vincent W.C. Tan and colleague Fabio Caccioli from University College London shows that managing risk using the procedure recommended by Basel II (the worldwide recommendations on banking regulation), which is called Value at Risk, may have played a central role.  

Mathematics enables faster computer simulations of biology

Numerous processes across both the physical and biological sciences are driven by diffusion, for example transport of proteins within living cells, and some drug delivery mechanisms. Diffusion is an unguided process which is of great importance at small spatial scales.

Constructing reaction systems - an inverse problem in mathematics

There is a wide class of problems in mathematics known as inverse problems. Rather than starting with a mathematical model and analysing its properties, mathematicians start with a set of properties and try to obtain mathematical models which display them. For example, in mathematical chemistry researchers try to construct chemical reaction systems that have certain predefined behaviours. From a mathematical point of view, this can be used to create simplified chemical systems that can be used as test problems for different mathematical fields.

The mathematics of species extinction

Correctly predicting extinction is critical to ecology. Claim extinction too late, and you may be taking resources away from a species that actually could be saved. Claim extinction too early, and you may cause the true extinction due to stopping resources, such as removing protection of its habitat. There is a balance to be sought, and it's clear that we're not quite there because every year several species that were thought to be extinct are rediscovered.

Scientists discover how a common garden weed expels its seeds at record speeds

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.

Are big-city transportation systems too complex for human minds?

Many of us know the feeling of standing in front of a subway map in a strange city, baffled by the multi-coloured web staring back at us and seemingly unable to plot a route from point A to point B. Now, a team of physicists and mathematicians has attempted to quantify this confusion and find out whether there is a point at which navigating a route through a complex urban transport system exceeds our cognitive limits.

When a droplet hits a surface

Understanding how droplets impact surfaces is important for a huge range of different applications. These range from spray painting, inkjet printing, fertiliser application and rainfall to crime-scene blood-splatter analysis and hygiene situations (men’s urinals being a familiar example). High speed movies show that when droplets hit surfaces fast enough, they often splash, emitting a corona of new, tiny droplets on impact.

Comparing the social structure of different cities

People make a city. Each city is as unique as the combination of its inhabitants. Currently, cities are generally categorised by size, but research by Oxford Mathematicians Peter Grindrod and Tamsin Lee on the social networks of different cities shows that City A, which is twice the size of City B, may not necessarily be accurately represented as an amalgamation of two City Bs.

Predicting and managing energy use in a low-carbon future

If effectively harnessed, increased uptake of renewable generation, and the electrification of heating and transport, will form the bedrock of a low carbon future. Unfortunately, these technologies may have undesirable consequences for the electricity networks supplying our homes and businesses. The possible plethora of low carbon technologies, like electric vehicles, heat pumps and photovoltaics, will lead to increased pressure on the local electricity networks from larger and less predictable demands.

How weights and pulleys might explain the hunting techniques of toads

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.

Mathematics enables a better understanding of damage during brain surgery

For over a hundred years, when confronted by swelling in the brain, surgeons more often than not have resorted to decompressive craniectomy, the traditional route to reducing swelling by removing a large part of the skull. However, while this might be the standard procedure, its failure rate has been worryingly high, primarily because the consequences on the rest of the brain have been poorly understood. 
 

Using geometry to choose the best mathematical model

Across the physical and biological sciences, mathematical models are formulated to capture experimental observations. Often, multiple models are developed to explore alternate hypotheses.  It then becomes necessary to choose between different models.

Unleashing the mathematics of the chameleon's tongue

The chameleon's tongue is said to unravel at the sort of speed that would see a car go from 0-60 mph in one hundredth of a second – and it can extend up to 2.5 body lengths when catching insects. Oxford Mathematicans Derek Moulton and Alain Goriely have built a mathematical model to explain its secrets. 

The mathematics of poaching and gamekeeping

How do we stop poaching? You may think the answer lies in finding a way of giving gamekeepers an advantage over poachers. Oxford Mathematician Tamsin Lee and David Roberts from the University of Kent decided to look at the interaction between rhino poachers and a gamekeeper to predict the outcome of the battle. Their conclusions suggest alternative ways of tackling the problem.

The universal structure of language

Semantics is the study of meaning as expressed through language, and it provides indirect access to an underlying level of conceptual structure. However, to what degree this conceptual structure is universal or is due to cultural histories, or to the environment inhabited by a speech community, is still controversial. Meaning is notoriously difficult to measure, let alone parameterise, for quantitative comparative studies.

How predictable is technological progress?

Everyone knows that Moore’s law says that computers get cheaper at an exponential rate.  What is not as well known is that many other technologies that have nothing to do with computers obey a similar law. Costs for DNA sequencing, some forms of renewable energy, chemical processes and consumer goods have also dropped at an exponential rate, even if the rates vary and are typically slower than for computers.

Mitigating the impact of frost heave

Frost heave is a common problem in any country where the temperature drops below 0 degrees Celsius. It’s most commonly known as the cause of potholes that form in roads during winter, costing billions of dollars worth of damage worldwide each year. However, despite this, it is still not well understood. For example, the commonly accepted explanation of how it occurs is that water expands as it freezes, and this expansion tears open the surrounding material.

The Mathematics of Tumour Growth

The Mathematics of Sea Shells

The Mathematics of Poking

International Brain Mechanics and Trauma Lab

Maths for Industry