Wednesday, 25 May 2016

Zubin Siganporia wins Outstanding Tutor award

Congratulations to Oxford Mathematics' Zubin Siganporia who has won the award for Outstanding Tutor for the Mathematical, Physical and Life Sciences Division in the 2016 Oxford University Student Union Student Led Teaching Awards.


Wednesday, 25 May 2016

Andrew Wiles presented with the Abel Prize in Oslo

The work of Oxford University Professor Sir Andrew Wiles was celebrated as having 'heralded a new era in number theory' as he received the top international prize for mathematics. 

Sir Andrew received the 2016 Abel Prize from Crown Prince Hakon of Norway at the prize ceremony in Oslo on 24 May. He was awarded the prize 'for his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory'.

The ceremony at the University Aula was attended by more than 400 guests, from members of the international mathematics community to local residents. 

Professor Ole Sejersted, President of the Norwegian Academy of Science and Letters, which presents the Abel Prize, said: 'Mathematicians have tried to prove Fermat's Last Theorem for 350 years, without success, indicating that mathematicians regard this as one of the great mathematical puzzles.

'The Abel Committee says that Sir Andrew's work has heralded a new era in number theory. To me, this indicates that the work on the theorem required the development of an entirely new mathematical foundation, the significance of which goes far beyond the actual proving of the theorem.'

Accepting the prize, Sir Andrew Wiles said: 'As a ten-year-old eager to explore mathematics I rummaged in the popular mathematics section of my local public library and found a copy of a book called The Last Problem by E.T. Bell. I did not even have to open the book. On the bright yellow front cover it told the story of the 1907 Wolfskehl prize offered for the solution of a famous mathematical problem. The problem itself was on the back cover. I was hooked.

'It was a wonderful find for me. Apparently inside mathematics there was hidden treasure! A little over 300 years previously a Frenchman by the name of Pierre de Fermat had solved a beautiful sounding problem, but he had buried the proof and now there was a prize for finding it!

'Fermat did not leave any clues because he did not have a solution, but nature itself leaves clues. I just had to find them. There was never going to be a one-line proof. Nor do proofs come just because one has been born with mathematical perfect pitch. There is no such thing. One has to spend years mastering the problem so that it becomes second nature. Then, and only then, after years of preparation is one's intuition so strong that the answer can come in a flash.

'These eureka moments are what a mathematician lives for; the bursts of creativity that are all the more precious for the years of hard work that go into them. The moment in the morning of September 1994 when I resolved my last problem is a moment I will never forget.'

Fermat's Last Theorem had been widely regarded by many mathematicians as seemingly intractable. First formulated by the French mathematician Pierre de Fermat in 1637, it states:

There are no whole number solutions to the equation xn + yn = zn  when n is greater than 2, unless xyz=0.

Fermat himself claimed to have found a proof for the theorem but said that the margin of the text he was making notes on was not wide enough to contain it. After seven years of intense study in private at Princeton University, Sir Andrew announced he had found a proof in 1993, combining three complex mathematical fields – modular forms, elliptic curves and Galois representations.

Sir Andrew not only solved the long-standing puzzle of the theorem, but in doing so he created entirely new directions in mathematics, which have proved invaluable to other scientists in the years since his discovery. The Norwegian Academy of Science and Letters said in its citation: 'Few results have as rich a mathematical history and as dramatic a proof as Fermat's Last Theorem.'

The Abel Prize is named after the Norwegian mathematician Niels Henrik Abel (1802-29) and was established in 2001 to recognize pioneering scientific achievements in mathematics. Abel himself did some of the early work on the properties of elliptical functions. Previous winners of the Prize include Britain's Sir Michael Atiyah and the late US mathematician John Nash.

Accompanying the prize-giving ceremony is a series of 'Abel week' activities aimed particularly at young people, including the awarding of the Holcombe Memorial Prize for an outstanding teacher of mathematics and the UngeAbel contest for teams of secondary pupils. This year's winning teacher and young winners were in the audience for the Abel Prize ceremony. 

Friday, 20 May 2016

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. Unfortunately, even with our most sensitive biomedical imaging techniques, we are unable to see exactly where the tumour has spread. Thus, even though a surgeon may cut out the worst affected areas there may still be infected areas that are left to grow, allowing the tumour to re-emerge. Surgeons are, thus, turning to the predictive power of mathematics, which allows them to anticipate where the cells will be, even if they cannot be seen.

Simulations capturing the main effects of tumour invasion have been produced since the early 2000s. However, more recently, it has been experimentally observed that the tumour cells appear to congregate within the interfaces of the brain’s white and grey matter (see Figure 1), which is not accounted for by the previously developed models.

Oxford Mathematical biologists Thomas Woolley, Philip Maini and Eamonn Gaffney, in collaboration with José Belmonte-Beitia from the Universidad de Castilla-La Mancha and Jake Scott from the Moffitt Cancer Center have revisited this problem and found that they can reproduce these new observations by altering how a cell senses its surroundings.

The original model assumed that the cells could sense their environment far from their current locations. This leads to the simulations showing that the cells spread out evenly over the entire space. By assuming that the cells can only sense a local region the simulations show that the cells tend to form high density peaks along the white-grey matter interfaces (see Figure 2).

Critically, this new formulation changes our understanding of how the cells invade our brain tissue. Not only could this lead to better predictions of how tumours spread, but, consequently, more lives could be saved using mathematical modelling.

Figure 1. A high resolution image of a brain tumour inside a mouse. Green represents the core of the tumour, yellow represents dispersing cells and the grey cloudy region represents the interface between the white and grey matter. Reprinted from The American Association for Cancer Research: Burden-Gulley et al. (2011).


Figure 2. Three time points illustrating the spread of the tumour cells. Each time point illustrates the same domain, but with the cells using the two different sensing rules. Initially, at day 0 (top image), a small number of tumour cells are initiated at the top left corner of the domain, after 300 days (middle image) and 600 days (bottom image) we see the cells spread out over the domain. Each domain is split in half along the horizontal mid-axis. The top of the domain is taken to be white matter, the bottom is taken to be grey matter. In each pair of simulations the top simulation shows the current standard and, thus, the cells spread out uniformly leaving the domain all one colour by the end. The bottom simulation illustrates the new sensing rule and we see that a high density of cells (shown as a red colour) forms along the interface.


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).