Thursday, 2 June 2016

Oxford's Victorian Savilian Professors of Geometry - the latest in our history series

Our latest Oxford Mathematicians are the three Savilian Professors of Geometry who dominated Oxford’s mathematical scene during the Victorian era: Baden Powell (1796–1860), Henry John Stephen Smith (1826–83) and James Joseph Sylvester (1814–97). None was primarily a geometer, but each brought a different contribution to the role. Find out more.

The Savilians are the fourth in our series exploring Oxford's mathematical heritage.


Wednesday, 1 June 2016

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. A recent paper in the scientific journal Cell offers new insights into the biology and mechanics behind this process.

Several teams of scientists from different disciplines and countries including Oxford Mathematicians Alain Goriely and Derek Moulton and colleagues from Oxford's departments of Plant Sciences, Zoology and Engineering and led by Angela Hay, a plant geneticist in the Department of Comparative Development and Genetics at the Max Planck Institute for Plant Breeding Research (MPIPZ), worked together to discover how the seed pods of popping cress explode. A rapid movement like this is rare among plants; since plants do not have muscles, most movements in the plant kingdom are extremely slow.

But the explosive shatter of popping cress pods is so fast that advanced high-speed cameras are needed to even see the explosion. Richard Bomphrey, of the Royal Veterinary College at the University of London, explains: “Because the seeds are so small, aerodynamic drag slows them down immediately.” To compensate, the seeds are accelerated away from the fruit and get up-to-speed extremely quickly. In fact, they accelerate from 0 to 10 metres per second in about half a millisecond, “which is super fast!” says Bomphrey.

Hay’s teams of scientists discovered that the secret to explosive acceleration in popping cress is the evolutionary innovation of a fruit wall that can store elastic energy through growth and expansion, and can rapidly release this energy at the right stage of development.

Previously, scientists had claimed that tension was generated by differential contraction of the inner and outer layers of the seed pod as it dried. So what puzzled the authors of the Cell paper was how popping cress pods exploded while green and hydrated, rather than brown and dry. Their surprising discovery was that hydrated cells in the outer layer of the seed pod actually used their internal pressure in order to contract and generate tension. The authors used a computational model of three-dimensional plant cells, to show that when these cells were pressurised, they expanded in depth while contracting in length, “like the way an air mattress expands in depth, when inflated, but contracts in width,” explains Richard Smith, a computer scientist at MPIPZ.



Another unexpected finding was how this energy was released. The authors found that the fruit wall wanted to coil along its length to release tension, but it had a curved cross-section preventing this. “This geometric constraint is also found in a toy called a slap bracelet,” explains Oxford Mathematics's Derek Moulton. In both the toy and the seed pod, the cross-section first has to flatten before the tension is suddenly released by coiling. Unexpectedly, this mechanism relies on a unique cell wall geometry in the seed pod. As Moulton explains, “This wall is shaped like a hinge, which can open,” causing the fruit wall to flatten in cross-section and explosively coil.

According to Hay, their most exciting discovery was the evolutionary novelty of this hinged cell wall. They had evidence from genetics and mathematical modelling that this hinge was needed for explosive pod shatter, “but finding the hinge only in plants with explosive seed dispersal was the smoking gun,” says Hay.

These findings reinforce the description of evolution as a "tinkerer, not an engineer", made by the scientist Francois Jacob. It appears that the sophisticated mechanism of explosive seed dispersal in popping cress evolved via tweaking the shape of already-existing cellular components.

When asked what implications their results will have for other researchers, Smith answered: “It is likely that other processes in plants that were previously attributed to passive shrinkage by drying are in fact active processes, especially in green, hydrated tissues.”

This study is a good example of how the recent trend towards interdisciplinary, collaborative science can lead to a global understanding of the biological and physical mechanisms at play in a complex process. The authors of this Cell paper built up a comprehensive picture of explosive seed dispersal by relating observations at the plant scale all the way down to the cellular and genetic scales, and systematically linking each scale. As Oxford Mathematics's Alain Goriely says, “this approach was only made possible by combining state-of-the-art modelling techniques with biophysical measurements and biological experiments.”

The image above is of a mathematical model explaining the explosive dispersal of seeds from a common garden weed.

Tuesday, 31 May 2016

Heather Harrington awarded a Royal Society University Research Fellowship

Oxford Mathematician Heather Harrington has been awarded a Royal Society University Research Fellowship. The fellowships recognise outstanding scientists in the UK who are in the early stages of their research career and have the potential to become leaders in their field. Heather's work covers a range of topics in applied mathematics, including algebraic systems biology, inverse problems, computational biology, and information processing in biological and chemical systems.


Tuesday, 31 May 2016

F is for Fourier Transform - the Oxford Mathematics Alphabet (part six)

The Fourier transform is that rarest of things: a mathematical method from over 200 years ago which not only remains an active area of research in its own right, but is also an invaluable tool in nearly every branch of mathematics. Though originally developed by Fourier in 1807 to help solve certain partial differential equations, the transform is a living example of a remarkable feature of mathematics, that a tool created in one sub-discipline can break through these artificial classifications and become vital in another. Find out more about a method that has attracted the attention of mathematicians from Hardy and Littlewood to John Nash.

The Fourier Trasform is the latest in our Oxford Mathematics Alphabet, a sequence of 26 letters explaining key concepts and our latest research. 

Tuesday, 31 May 2016

Nigel Hitchin wins the Shaw Prize

Professor Nigel Hitchin FRSSavilian Professor of Geometry in the Mathematical Institute, University of Oxford has won the prestigious Shaw Prize in Mathematical Sciences for, in the words of the Prize Foundation "his far-reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance."

Professor Frances Kirwan FRS, a colleague in Oxford, paid tribute: "Nigel Hitchin has made fundamental contributions to the fields of differential and algebraic geometry and richly deserves the award of the Shaw Prize. His work has influenced a wide range of areas in geometry and mathematical physics, including symplectic and hyperkähler geometry, the theory of instanton and monopole equations, twistor theory, integrable systems, Higgs bundles, Einstein metrics and mirror symmetry."

Professor Martin Bridson FRS, Head of the Mathematical Institute in Oxford, said: "'it is a real joy to see Nigel Hitchin's profound and influential work recognised by the award of the 2016 Shaw Prize. His inspiring intellectual leadership in geometry has been matched throughout his career by many services to the mathematical community in the UK and across the world, for which we are all deeply grateful. Oxford has been extremely fortunate to have Nigel with us for so much of his career, and we are very proud of him."

Nigel said on news of the award: "I am delighted and honoured to be awarded this prize. Since most of my working life has been spent in Oxford, it is also a recognition of the support I have received here. I was pleased to note that my “twin” in New College, the Savilian Professor of Astronomy, won the Shaw prize a few years ago.”

The Shaw Prize is an annual award first presented by the Shaw Prize Foundation in 2004. Established in 2002 in Hong Kong it honours living individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The 2016 prize is worth US$1.2m to each winner.

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.