Wednesday, 8 March 2017

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. Why is this wrinkled or crumpled state preferred to the ‘free lunch’ offered by mirror buckling? In a series of papers Oxford Mathematician Dominic Vella and colleagues address this question for the case of the beach ball: an elastic shell with an internal pressure.

Wrinkling is caused by compressive forces within the shell (just as a piece of paper buckles when you compress it at its edges). The key insight is that wrinkling allows the shell to relax the compressive stress so that there is essentially no compression in the direction perpendicular to the wrinkles, and a very high tension along the wrinkles. This change in the stress causes the shell to adopt a new kind of shape,  that is qualitatively different to mirror buckling. To determine the energetic cost of this new shape requires a detailed calculation of how wrinkles behave - we find that the wrinkle pattern is intricate, changing spatially (see picture) and also evolving as the degree of poking changes. However, we also show that despite this, the energetic cost of the wrinkling is relatively small, and so this wrinkly shape is an approximate isometry (a ‘ wrinkly isometry’).

The shell now has two choices of cheap deformations to adopt: the wrinkly isometry or mirror buckling. The final piece of the puzzle is to realize that elastic energy is not the only energy that matters in this system - since the shell has an internal pressure, the gas within it must also be compressed. In this case, the wrinkly isometry displaces less gas and hence costs less energy; this is why it is the preferred state.

For more information about the research funded by the European Research Council (ERC):

Monday, 6 March 2017

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.

Angiogenesis, the formation of new blood vessels from existing ones, is a key characteristic of tumour progression. The purpose of anti-angiogenic (AA) cancer therapies is to disrupt the tumour’s blood supply, inhibiting delivery of oxygen and nutrients. However, such therapies have demonstrated limited benefit to patients; they do not consistently improve survival. It has been suggested that normalisation, the process by which vessels transition from the leaky state that is typical of tumour vasculature (the arrangement of the vessels) to a state where blood perfusion is increased, is the reason why some AA therapies lack efficacy.

Lucy and her colleagues have developed a mathematical model that accounts for the biochemistry of angiogenesis and vessel normalisation. They have shown that the model exhibits four possible long term behaviour regimes that could correspond to vascular phenotypes (an organism's observable traits) in patients. By identifying parameters in the mathematical model that lead to transitions between these behaviour regimes, the model can be used to determine which biological perturbations would theoretically lead to a transition between a given pre-treatment phenotype and the goal post-treatment phenotype. The implication of these mathematical models is that it enables 'personalised' therapy, where the normalisation level of a tumour's vasculature is taken into account when selecting the most beneficial type of anti-angiogenic treatment for a given tumour.

The research can be examined in more detail in the Journal of Theoretical Biology.


Friday, 3 March 2017

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.

Oxygen distributions in living tissues depend on the positioning of the smallest blood vessels, the capillaries. In healthy tissue the capillary spacing is tightly regulated and vessels are evenly spaced. In diseased tissue, such as tumours, the capillary spacing can be heterogeneous, with large regions left with no vessels. In the clinic, we can estimate the capillary distribution using imaging techniques such as functional MRI, which is then used to target radiation. However, MRI can’t resolve small-scale variations in capillary spacing, potentially leaving pockets of low oxygen, radiotherapy resistant cells undetected.

Mathematical modelling can be used to study how important small-scale, undetectable, variations in capillary spacing are to overall radiotherapy response. If they are relatively unimportant it may be possible to use lower radiation doses and reduce damage to nearby tissues. Thanks to the recent development of high-resolution imaging techniques by colleagues Bostjan Markelc and Ruth Muschel in the CRUK/MRC Oxford Institute for Radiation Oncology, Oxford Mathematicians James Grogan, Anthony Connor (former), Philip Maini and Helen Byrne and Joe Pitt-Francis in the Department of Computer Science have been able to use real, 3D, vessel data to model tumour oxygen distributions and radiotherapy responses for the first time.

Their study, recently published in the Transactions on Biomedical Engineering, involved high resolution multi-photon imaging of well-vascularized murine colon adenocarcinoma (MC38) tumours in mice. The images were used to extract 3D blood vessel structures, which were used as inputs to multi-scale, multi-physics mathematical models of oxygen transport, tumour growth and radiation therapy. The models predicted that radiotherapy outcome was relatively insensitive to small-scale (order of 1 mm) variations in capillary spacing for the studied, well-vascularized tumour. They also demonstrated that previous 2D models, which used artificial tumour vessel data, overestimated the sensitivity of radiotherapy response to heterogeneities in capillary spacing. The newly predicted lack of sensitivity to variations in capillary spacing should help inform the on-going design of localized radiotherapy treatments.  However, further work is required to determine whether predictions extend to other tumour types, notably in humans which can have different vessel network structures to the studied mice.

Some of the main challenges in the study were the processing of 3D imaging datasets for integration with the mathematical models and the development of software for multi-scale, multi-physics simulation of tumour growth in large 3D domains. The resulting software is based on the open-source Chaste library, developed in the Mathematical Institute and Department of Computer Science, University of Oxford. The authors are now applying their integrated imaging and modelling approach to several related problems in the area of vascularized tumour treatment.

For further demos see the software page.

Wednesday, 1 March 2017

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.

As with conventional risk types, systemic risks need to be quantified. Currently global regulators propose a range of bank-specific indicators that measure size and interconnectedness to proxy systemic risk. Oxford Mathematician Christoph Siebenbrunner and colleagues tested to what extent such indicators are able to act as a proxy for different types of contagion effects in the financial system. They developed a model that allowed them to integrate domino-type network contagion effects with a market model for calculating the price impact of asset fire sales, and were able to demonstrate the existence of solutions to the resulting equations as well as providing algorithms to compute these solutions.

Testing the model empirically using real-world data, they compared the regulatory indicator set to the best-fitting alternative indicator set selected from a large universe of possible sets. The results showed that the regulatory indicator set represents a good selection of bank-specific indicators. However, bank-specific indicators alone are not able to capture the full extent of contagion effects, in particular for contagion channels that have far-reaching systemic impacts such as the effects of mark-to-market accounting in the presence of asset fire sales.


Thursday, 23 February 2017

Nick Trefethen wins the George Pólya Prize

Oxford Mathematician Nick Trefethen FRS has been awarded the prestigious George Pólya Prize by the Society for Industrial and Applied Mathematics (SIAM). The Prize for Mathematical Exposition, established in 2013, is awarded every two years to an outstanding expositor of the mathematical sciences.

Nick Trefethen is Professor of Numerical Analysis, University of Oxford, Fellow of Balliol College & Global Distinguished Professor, New York University. He is Head of Oxford Mathematics' Numerical Analysis Group. He is known for a succession of influential textbooks and monographs related to numerical mathematics, most recently 'Approximation Theory and Approximation Practice' which appeared in 2013. His next book will explore Ordinary Differential Equations.

Wednesday, 22 February 2017

Philip Maini awarded the Arthur T. Winfree Prize by the Society of Mathematical Biology

Oxford Mathematician Philip Maini has been awarded the Arthur T. Winfree Prize by the Society of Mathematical Biology for his work on mathematical modelling of spatiotemporal processes in biology and medicine. In the words of the citation Philip's work "has led to significant scientific advances not only in mathematics, but also in biology and the biomedical sciences. His mathematical oncology research has provided detailed insight into the design of combination cancer therapies."

Philip will receive his award at the 2017 Annual Meeting of the Society, to be held at the University of Utah in Salt Lake City from July 17-20, 2017.

Tuesday, 21 February 2017

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.

The first ICM, held in Zurich in 1897, emerged from the great expansion in international scientific activities (where 'international' usually meant just Europe and North America) that resulted from the improved communications and transport connections of the late nineteenth century.  The second ICM was held in Paris in 1900, alongside the many other conferences and exhibitions that were being staged there to mark the new century.  A noteworthy feature of the Paris ICM was a lecture given by the prominent German mathematician David Hilbert (1862-1943), in which he outlined a series of problems that he thought ought to be tackled by mathematicians in the coming decades.  Hilbert's problems went on to shape a great deal of twentieth-century mathematical research; just three remain entirely unresolved.

After Paris, a four-yearly pattern was established for the ICMs, and further meetings took place elsewhere in Europe (Heidelberg, 1904; Rome, 1908; Cambridge, 1912).  It was proposed that the 1916 congress would be held in Stockholm, but in the face of the war raging on the continent, it did not take place.

After the end of the First World War, the mathematicians of Western Europe realised that something ought to be done to help to rebuild their discipline and its international networks.  To this end, a group of mathematicians, many of whom hailed from the Western European countries that had been particularly devastated by the war, proposed to re-establish the ICMs with a meeting in 1920.  But in doing this, they made two bold statements.  The first was that the ICM would take place in Strasbourg: a French city that had been incorporated into Germany following the Franco-Prussian War of 1870-1, and that had only recently been returned to France by the Treaty of Versailles.  The second was that all mathematicians from Germany and her wartime allies would be barred from attending the congress.  The exclusion of German mathematicians extended also to the next congress (Toronto, 1924), but by the time of the 1928 ICM in Bologna, the more moderate voices had become louder, and Germans delegates were admitted.

Despite (or because of) the re-opening of relations with German mathematicians, tensions remained in the international mathematical community.  There were those who believed that the Germans should not have been re-admitted to the ICMs.  Moreover, some German mathematicians felt resentment at their earlier exclusion and so boycotted the 1928 congress.  In a bid to bring people back together and re-establish ties, the ICM returned in 1932 to Zurich – a deliberately neutral choice.  Similar reasoning resulted in Oslo being chosen as the venue for the 1936 ICM.

From the start, the Oslo congress was a political melting pot of different agendas; the effects of the wider European political situation were clearly visible.  The goal of the Nazi-led German contingent, for example, was clear: to showcase the best of 'Aryan mathematics'.  The expected Soviet delegation, on the other hand, was conspicuous by its absence.  Like the Germans, Russian mathematicians had had a difficult relationship with the ICMs.  Prior to the First World War, they had regularly attended in significant numbers, but had been rather less visible during the 1920s, following the October Revolution (1917) and subsequent Russian Civil War (1917-1922).  As the decade progressed, they began to reappear, but as Stalin increased his grip on power during the early 1930s, and sought to exercise greater control over the USSR’s scientific community, the ability of Soviet academics to travel to foreign conferences was gradually curtailed.

At the Oslo congress, around eleven Soviets were expected to attend, including two plenary speakers, one of whom, A. O. Gel’fond (1906-1968), was due to lecture on his solution to Hilbert's 7th problem.  However, when the congress convened in July 1936, it was announced that none of the Soviet delegates had appeared: all had been denied permission to travel.

Just like the proposed Stockholm ICM, the congress planned for 1940 did not take place.  It was not until 1950 that the ICMs resumed with a congress in Cambridge, Massachusetts.  No Soviet delegates attended, although several had been invited.  Shortly before the congress, the organisers had received a telegram from the president of the Soviet Academy of Sciences, making the rather transparent excuse that Soviet mathematicians were unable to attend due to pressure of work.

Following Stalin’s death in 1953, international relations thawed somewhat, and the numbers of Soviet delegates at the ICMs gradually increased.  The USSR's involvement in the international mathematical community expanded further in 1957 when it joined the International Mathematical Union.  The signal that the USSR was now fully engaged in world mathematics came in 1966 when it hosted the ICM in Moscow that year.  In the decades that followed, the ICMs provided a forum for mathematicians from East and West to establish personal contacts – but their organisation was certainly not free of difficulties arising from the Cold War political climate.

The International Congresses of Mathematicians provide an excellent means of studying the development of mathematics in the twentieth century: not only can we trace its technical developments and its trends by looking at the choices of plenary speakers, but we can also investigate the ways in which its conduct was affected by events in the wider world, and thereby see that mathematics is indeed a part of global culture.

Christopher Hollings

Oxford Mathematician Christopher Hollings is Departmental Lecturer in Mathematics and its History, and Clifford Norton Senior Research Fellow in the History of Mathematics at The Queen's College. More about the relations between the mathematicians of the East and the West can be found here. Christopher's podcast on the subject will feature shortly as part of the Oxford Sparks series.

Thursday, 16 February 2017

Ursula Martin elected Fellow of the Royal Society of Edinburgh

Oxford Mathematician and Computer Scientist Ursula Martin has been elected a Fellow of the Royal Society of Edinburgh, joining over 1600 current fellows drawn from a wide range of disciplines – science & technology, arts, humanities, social science, business and public service.

Ursula's career has taken in Cambridge and Warwick and included spells across the Atlantic as well as recently at Queen Mary, University of London. From 1992 to 2002, she was Professor of Computer Science at the University of St Andrews in Scotland, the first female professor at the University since its foundation in 1411. Her work around theoretical Computer Science is accompanied by a passionate commitment to advancing the cause of women in science. She has also been a leading light in the recent study and promotion of the life and work of Victorian Mathematician Ada Lovelace and has been instrumental in examining and explaining Ada's mathematics as well as promoting her achievements as a woman.



Tuesday, 14 February 2017

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. 

One of the greatest strengths of applied mathematics is that it has established itself by defying simple categorisation. Applied mathematics, be it an art, a craft, or a discipline, is not bound to a particular scientific application, a particular mathematical universe, or a well-defined university department. The drawback is that applied mathematics usually gets no mega-funding or the limelight associated with big scientific breakthroughs. But its biggest advantage is that it can insert itself into all scientific disciplines and easily reinvent itself by moving fluidly from one field to the next, guided only by methods, theory, and applications: it is all equations and matrices. Many applied mathematicians see new challenges as an opportunity to expand their mathematical horizons, and in our rapidly changing modern new society such new challenges abound. Here are three of these.

Major scientific efforts are required for major society challenges. These include fighting climate change, optimising new renewable energy sources, developing new medical treatments, and understanding the brain. Traditionally, applied mathematicians involved with these collaborative efforts were considered a useful but small cog in a huge scientific machine, but it is now appreciated that quality science requires clever modelling, state-of-the-art numerical methods, and fundamental theoretical insights from simplified models. This is the realm of applied mathematics, and accordingly our role in these endeavours is bound to increase. By the end of the day, we may not get the fame, but we’ll certainly have the fun.

A second relatively recent development of applied mathematics is the theory of networks. Networks represent connections between multiple physical or virtual entities. They are found in information theory (web links, social connections), biological systems (gene regulatory networks, metabolic networks, evolutionary trees), and physical systems (axon connections, electric grid). Regardless of their origin, these networks share common mathematical features. Their analyses span many different fields of study, and network theory has now established tentacular connections to various parts of pure and applied mathematics, a network of its own.

For about five years there has been much excitement about BIG DATA. The initial hope was that one could go straight into data and use empirical methods to unravel the mysteries of the universe. Quite the opposite is happening. The success of many methods has shed a bright light on the need to understand the underlying mathematical structure of both data and methods. The subject now presents a rich field of study that brings all mathematical sciences together, including statistics and computer science.

These examples share a common thread that highlights a new trend in mathematical and scientific discoveries: beyond inter-, multi-, and supra-disciplinarity, we live in a post-disciplinary world. Things have changed, and Oxford University with its collegiate system, and the Mathematical Institute with its collegial atmosphere, are particularly well equipped to thrive in this new scientific world. But despite all the hype, we’re also fully aware that there’s nothing wrong with the old world, the old problems, or the old conjectures. We have an intellectual responsibility to promote and cherish these areas of knowledge defined by the great thinkers, past and present, especially if they are believed to be useless or irrelevant.

Bob Dylan in the same interview, foresaw yet another possible application of mathematics: What would you call your music? His reply: I like to think of it more in terms of vision music – it's mathematical music.

Alain Goriely, Professor of Mathematical Modelling, Oxford Mathematics, University of Oxford.

The caption next to Bob above is a scan of Alain’s brain. Bob sang: “My feet are so tired, my brain is so wired”. But will collaborative applied mathematics untangle the mystery of the author’s brain, mathematical or otherwise?

Alain's 'Applied Mathematics: A Very Short Introduction' will be published by OUP later in the Year. You can also watch his lecture via the Oxford Mathematics YouTube page.

Friday, 10 February 2017

The Truth is not enough - Tim Harford Oxford Mathematics Public Lecture now online

From the tobacco companies in the fifties to the arguments of the Brexit campaign, Economist and BBC Radio 4 Presenter Tim Harford takes us on a tour of truths, facts and the weapon that is doubt. Surely fact-checking websites and rational thinking are the best weapons to convince people of the truth? Or is in fact the truth simply not good enough. Do we have time or any inclination to hear it? Maybe we need to start with something simpler. Perhaps arousing people's curiosity might be just as important.

Watch Tim make his case in the latest of the successful Oxford Mathematics Public Lecture series.