Exploring Steiner Chains with Möbius Transformations
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.
Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials
Oxford Mathematician Dmitry Belyaev is interested in the interface between analysis and probability. Here he discusses his latest work.
"There are two areas of mathematics that clearly have nothing to do with each other: projective geometry and conformally invariant critical models of statistical physics. It turns out that the situation is not as simple as it looks and these two areas might be connected.
Oxford Mathematics Research looks at Ricci Flow
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.
A continuum of expanders
As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician David Hume discusses his work on networks and expanders.
Heterogeneity in cell populations - a cautionary tale
Researchers from Oxford Mathematics and Imperial College London have provided a “'mathematical thought experiment' to inspire caution in biologists measuring heterogeneity in cell populations.
How our immune systems could help us understand crime
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.
Knots and the nature of 3-dimensional space
It is an intriguing fact that the 3-dimensional world in which we live is, from a mathematical point of view, rather special. Dimension 3 is very different from dimension 4 and these both have very different theories from that of dimensions 5 and above. The study of space in dimensions 2, 3 and 4 is the field of low-dimensional topology, the research area of Oxford Mathematician Marc Lackenby.
Numerical Analyst Nick Trefethen on the pleasures and significance of his subject
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.
How do biomembranes form micro-structures in our cells?
The human body comprises an incredibly large number of cells. Estimates place the number somewhere in the region of 70 trillion, and that’s even before taking into account the microbes and bacteria that live in and around the body. Yet inside each cell, a myriad of complex processes occur to conceive and sustain these micro-organisms.
Some advice for gamblers from Oxford Mathematics
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.
Iteration of Quadratic Polynomials Over Finite Fields - new research from Professor Roger Heath-Brown
As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Eminent Oxford Mathematician and number theorist Roger Heath-Brown discusses his latest work.
"Since retiring last September I've had plenty of time for research. Here is something I've been looking into.
The physics of the frog and the lily pad
A resting frog can deform the lily pad on which it sits. The weight of the frog applies a localised load to the lily pad (which is supported by the buoyancy of the liquid below), thus deforming the pad. Whether or not the frog knows it, the physical scenario of a floating elastic sheet subject to an applied load is present in a diverse range of situations spanning a spectrum of length scales. At global scales the gravitational loading of the lithosphere by mountain ranges and volcanic sea mounts involve much the same physical ingredients.
Creating successful cities - how mathematical modelling can help
Oxford Mathematician Neave O’Clery recently moved to Oxford from the Center for International Development at Harvard University where she worked on the development of mathematical models to describe the processes behind industrial diversification and economic growth. Here she discusses how network science can help us understand the success of cities, and provide practical tools for policy-makers.
Mathematics and Politics: The International Congresses of Mathematicians
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.
Oxford Mathematics Research - Rates of convergence in the method of alternating projections
As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician David Seifert discusses his and his collaborator Catalin Badea's work.
Improving the performance of solar cells
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 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?
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.
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.
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.
Hair today, gone tomorrow. But have scientists found a new way to stimulate hair growth?
How does the skin develop follicles and eventually sprout hair? Research from a team including Oxford Mathematicians Ruth Baker and Linus Schumacher addresses this question using insights gleaned from organoids, 3D assemblies of cells possessing rudimentary skin structure and function, including the ability to grow hair.
Oxford Mathematics Research: Nikolay Nikolov on his latest research into Sofic Groups
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.
The mathematics of abnormal skull growth
Mathematics is delving in to ever-wider aspects of the physical world. Here Oxford Mathematician Alain Goriely describes how mathematicians and engineers are working with medics to better understand the workings of the human brain and in particular the issue of abnormal skull growth.
The mathematics of glass sheets - how to make their thickness uniform
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.
Mathematical physicist James Sparks talks about his research into exact results in the AdS/CFT correspondence
As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician James Sparks discusses his latest work.
"Two great successes of 20th century theoretical physics are Quantum Field Theory and General Relativity.
Modelling the architecture of the brain
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.
Using mathematical modelling to improve our understanding of radiotherapy
New methods for localising radiation treatment of tumours depend on estimating the spatial distribution of oxygen in the tissue. Oxford Mathematicians hope to improve such estimates by predicting tumour oxygen distributions and radiotherapy response using high resolution images of real blood vessel networks.
Understanding the risks banks pose to the financial system
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.
Modelling the impact of scientific collaboration
If nations are to grow, both economically and intellectually, they must foster scientific creativity. To do that they must create scientific environments that stimulate collaboration. This is especially true of developing countries as they seek to prosper in a global economy.
Mathematics and health promotion - discussing diabetes on Twitter
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".
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.
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 .
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.
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 the spread of brain tumours
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.
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.
Exploding the myths of Ada Lovelace’s mathematics
Ada Lovelace (1815–1852) is celebrated as “the first programmer” for her remarkable 1843 paper which explained Charles Babbage’s designs for a mechanical computer. New research reinforces the view that she was a gifted, perceptive and knowledgeable mathematician.
Shapes and Numbers - Oxford Mathematics Research considers number theory and topology
As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics, we look at Oxford Mathematician Minyhong Kim's research in to the relationship between number theory and topology. Minhyong Kim is Professor of Number Theory here in Oxford and Fellow of Merton College.
It is probably well-known that number theory is the source of some of the oldest and most accessible questions in mathematics:
How fast does the Greenland Ice Sheet move?
Governments around the world are seeking to address the economic and humanitarian consequences of climate change. One of the most graphic indications of warming temperatures is the melting of the large ice caps in Greenland and Antarctica. This is a litmus test for climate change, since ice loss may contribute more than a metre to sea-level rise over the next century, and the fresh water that is dumped into the ocean will most likely affect the ocean circulation that regulates our temperature.
Can Big Data root out corruption in Africa?
Many anticorruption advocates are excited about the prospects that “big data” will help detect and deter graft and other forms of malfeasance. But good data alone isn’t enough. To be useful, there must be a group of interested and informed users, who have both the tools and the skills to analyse the data to uncover misconduct, and then lobby governments and donors to listen to and act on the findings.
The mathematics of why our grandmothers love us
Michael Bonsall, Professor of Mathematical Biology at Oxford University's Department of Zoology, discusses his research in population biology, what it tells us about species evolution and, in particular, why grandmothering is important to humans. His research was done in conjunction with Oxford Mathematician Jared Field.
"What is mathematical biology?
The mathematics of sperm control
From studying the rhythmic movements, researchers at the Universities of York, Birmingham, Oxford and Kyoto University, Japan, have developed a mathematical formula which makes it easier to understand and predict how sperm make the journey to fertilise an egg. This knowledge will help scientists to gauge why some sperm are successful in fertilisation and others are not.
Why shells behave unexpectedly when poked - Oxford Mathematics Research
The classic picture of how spheres deform (e.g. when poked) is that they adopt something called 'mirror buckling' - this is a special deformation (an isometry) that is geometrically very elegant. This deformation is also very cheap (in terms of the elastic energy) and so it has long been assumed that this is what a physical shell (e.g. a ping pong ball or beach ball) will do when poked. However, experience shows that actually many shells don’t adopt this state - instead, beach balls wrinkle and ping pong balls crumple.
Modelling the growth of blood vessels in tumours
Cancer is a complex and resilient set of diseases and the search for a cure requires a multi-strategic approach. Oxford Mathematicians Lucy Hutchinson, Eamonn Gaffney, Philip Maini and Helen Byrne and Jonathan Wagg and Alex Phipps from Roche have addressed this challenge by focusing on the mathematical modelling of blood vessel growth in cancer tumours.
Applied mathematics: don’t think twice, it’s all right
In an interview with Rolling Stone Magazine in 1965, Bob Dylan was pushed to define himself: Do you think of yourself primarily as a singer or a poet? To which, Dylan famously replied: Oh, I think of myself more as a song and dance man, y’know. Dylan’s attitude to pigeonholing resonates with many applied mathematicians. I lack the coolness factor of Dylan, but if pushed about defining what kind of mathematician I am, I would say: Oh, I think myself more as an equation and matrix guy, y’know.
Oxford Mathematics Research - On the null origin of the ambitwistor string
As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Eduardo Casali discusses his work.
The impact of mathematics – human interactions!
Think of a mathematician and you might imagine an isolated individual fueled by coffee whose immaculate if incomprehensible papers may, in the fullness of time, via a decades-long dry chain of citations, be made use of by an industrialist (via one or two other dedicated mathematicians).
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).
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.
