Monday, 16 May 2016

What We Cannot Know - Marcus du Sautoy Public Lecture now online

The rolling of dice in a casino, Heisenberg's uncertainty, the meaning of consciousness. All are explored as Marcus takes us on a personal journey into the realms of the scientific unknown. Are we forever incapable of understanding all of the world around us or is it perhaps just a question of language, not having the right words to describe what we see?










Tuesday, 10 May 2016

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.  
The team made a very simple model for the banking system that captured the key elements of risk management under Value at Risk. Providing the banks only take modest risks, the financial system remains stable. But if they take higher risks, or if the banking sector gets larger, the market begins to spontaneously oscillate, in a way that resembles the period leading up to and including the Global Financial Crisis. For about 10 - 15 years prices and leverage slowly rise while volatility slowly falls, then prices and leverage suddenly crash and volatility spikes, as they did in the crisis.  
The key problem is that Value at Risk manages risk as if each bank existed in its own universe. But if all banks follow it, the buying and selling necessary to maintain individual risk targets can destabilise the market.  
The team then investigated alternative methods of managing risk and demonstrated that it is possible to do much better. The best policy depends on the size of the banking sector in relation to the rest of the market and how much risk the banks take. While the model does not show that the financial crisis and the period leading up to it were due to the use of Value at Risk, it does suggest that they could have been caused by it, and that the housing bubble may have just been the spark that triggered the crisis.
Wednesday, 4 May 2016

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. However, if you replace water with a material that does not expand upon freezing, similar damage occurs.

Oxford Mathematician Rob Style and colleagues have looked at this problem and come up with an alternative explanation. In their research, published in the Journal of Physical Chemistry, they carried out experiments on a model particle/water system designed to establish the dominant factors that cause frost heave, and then compared the results to existing theories. They found that the cooling rate and how well packed the particles are initially make big differences to the amount of heave that occurs. The latter in particular is normally overlooked when predicting heave rates. 

These findings will be important in tackling ice segregation occurring in a wide range of situations, not only in model laboratory experiments and theories but to wider geological and industrial processes that affect us all, such as frozen food production and, of course, those potholes in the road.

Wednesday, 4 May 2016

From social media to transportation systems - the interconnectedness of networks

What is a network and how can you use mathematics to unravel the relationships between a variety of different things? How can this understanding then be applied to a range of different settings?

In this Oxford Sparks podcast Oxford Mathematician Mason Porter studies how things are connected using mathematics. He builds up models of these connections to represent them as networks. But what are the basic components of a network? In the podcast Mason describes how from social networks to transport systems to locating a lost umbrella, the mathematics of networks can be used to address a range of apparently unconnected problems and how organisations around the world are using them to penetrate their ever-growing mass of data.

Friday, 29 April 2016

Three Oxford Mathematicians elected Fellows of the Royal Society

Congratulations to Oxford Mathematicians Martin Bridson, Marcus du Sautoy and Artur Ekert who have been elected Fellows of the Royal Society. Martin is Whitehead Professor of Pure Mathematics, a Fellow of Magdalen College and Head of the Mathematical Institute in Oxford. He has been elected for his many distinguished contributions to group theory and topology. Marcus is Charles Simonyi Professor for the Public Understanding of Science and a Fellow of New College and has been elected for his outstanding achievements in promoting the understanding of science and mathematics to a global audience and for eminent research that has completely transformed the study of zeta functions of groups. Artur is Professor of Quantum Physics at the Mathematical Institute and a Fellow of Merton College.  Artur has been elected FRS for his work on quantum physics, quantum computation and cryptography.

Thursday, 21 April 2016

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 researchers (working in collaboration with Tufts University in the US) derived a system of differential equations to capture the mechanics of the energy build-up and 'extreme acceleration' of the reptile's tongue. 

Derek Moulton, Associate Professor of Mathematical Biology at Oxford, said: 'if you are looking at the equations they might look complex, but at the heart of all of this is Newton's Second Law – the sort of thing that kids are learning in A-levels, which is simply that you're balancing forces with accelerations.

'In mathematical terms, what we've done is used the theory of non-linear elasticity to describe the energy in the various tongue layers and then passed that potential energy to a model of kinetic energy for the tongue dynamics.'

Special collagenous tissue within the chameleon's tongue is one of the secrets behind its effectiveness. This tissue surrounds a bone at the core of the tongue and is surrounded itself by a muscle.  Professor Moulton added: 'the muscle – the outermost layer – contracts to set the whole thing in motion.  We’ve modelled the mechanics of the whole process, the build up and release of energy.'

The researchers say the insights will be useful in biomimetics – copying from nature in engineering and design - for example in developing soft, elastic materials for robotics. They add that they also did the research because it was interesting and fun. Both pretty good reasons to study mathematics.

Thursday, 21 April 2016

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. weather patterns or changing distributions of predators/prey.

Early approaches were predominantly centred on the position jump model of motion, where agents undergo instantaneous changes of position according to a distribution kernel interspersed with waiting periods of stochastic length. To clarify, after a random period of time, the organism in question disappears in one location, and reappears in another nearby location. Equations for the probability that a particle is located in a position in space are called drift-diffusion equations - which are usually easy to solve numerically.
However, the position jump framework suffers from the limitation that correlations in the direction of successive runs are difficult to capture; this directional persistence is present in many types of movement. Furthermore, the diffusive nature of the position jump framework results in an unbounded distribution of movement speeds between successive steps – so theoretically an animal could be moving at any speed! Consequently Oxford Mathematician Jake P. Taylor-King and colleagues have been looking at other ways to address the issue.
Some organisms, whose sizes can differ by many orders of magnitude, have been observed to switch between different modes of operation. For instance, the bacterium Escherichia coli changes the orientation of one or more of its flagella between clockwise and anticlockwise to achieve a run-and-tumble like motion. As a result, during the runs, we see migration-like movement and during the tumbles, we see resting or local diffusion behaviour. To add to this complexity, it should be noted that the direction of successive runs are correlated. On a larger scale let's compare the migratory movements of vertebrates where individuals often travel large distances with intermittent stop-overs to rest or forage. An example is the lesser black-backed gull (Larus fuscus). Individuals of this species that breed in the Netherlands migrate southwards during Autumn. Even though the scales involved in these two processes differ by many orders of magnitude, one can use the same mathematical framework to model the observed motion.
When considering the movement of a `particle’ as a series of straight-line trajectories, the corresponding mathematical description is known as a velocity jump process [Othmer 1988]. Organisms travel with a randomly-distributed speed and angle for a finite duration, before undergoing a stochastic reorientation event. A big hurdle when using this approach is that the underlying differential equation involves the use of mesoscopic transport equations that need to be solved in a higher dimensional space than traditional drift-diffusion equations. Until recently [Friedrich 2006], the length of jumps has been modelled as exponentially distributed for mathematical ease. Therefore, it is assumed there is a constant rate at which animals reorientate.

The researchers' new approach allows the specification of any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are challenging to solve both analytically and numerically, and therefore it is necessary to both simplify and derive summary statistics.
For comparison between theory and experimental data, the researchers derived expressions for the mean squared displacement which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. A large time diffusive approximation is also considered via a Cattaneo approximation [Hillen 2004]. This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution. Therefore, two processes with the same means but different variances for how long an animal moves in the same direction can have different large scale observed behaviour. 

Finally, this method then enables us to switch between straight-line trajectory GPS (or tracking) data and some of the commonly studied differential equation models used within mathematical ecology. The main benefit of this approach is that velocity jump models can often be parameterised using smaller quantities of data than what may be required when using a position jump process. All of which enables us to better predict the future locations of animals and, in turn, to better understand the reasons for the choice of those locations.

(the image above shows the pattern of seagulls above the UK).


Wednesday, 20 April 2016

E is for Elliptic Curves

Appearing everywhere from state-of-the-art cryptosystems to the proof of Fermat's Last Theorem, elliptic curves play an important role in modern society and are the subject of much research in number theory today. Jennifer Balakrishnan, a researcher working in number theory, explains more in the latest in our Oxford Mathematics Alphabet.

Thursday, 14 April 2016

Rob Style wins 2016 Adhesion Society Young Scientist Award

Oxford Mathematician Rob Style has been awarded the 2016 Adhesion Society Young Scientist Award, sponsored by the Adhesion and Sealant Council, for his fundamental contributions to our understanding of the coupling of surfaces tension to elastic deformation.  Rob researches the mechanics of very soft solids like gels and rubber, in particular investigating why they don’t obey the same rules as hard materials that are more traditionally used by engineers.

Thursday, 14 April 2016

Jake Taylor King wins Lee Segel Prize

Oxford Mathematician Jake Taylor King has won the Lee Segel Prize for Best Student Paper for his paper 'From birds to bacteria: Generalised velocity jump processes with resting states.' Jake worked on his research with Professor Jon Chapman. The prize is awarded annually by the Society for Mathematical Biology. One of Jake's co-authors on the paper, Gabs Rosser, previously also studied Mathematics at Oxford in the Wolfson Centre for Mathematical Biology.