Wednesday, 12 April 2017

Doireann O'Kiely wins the Lighthill-Thwaites Prize

Oxford Mathematician Doireann O'Kiely has been awarded the biennial Lighthill-Thwaites Prize for her work on the production of thin glass sheets. The prize is awarded by the Institute of Mathematics and its Applications to researchers who have spent no more than five years in full-time study or work since completing their undergraduate degrees. Oxford Mathematicians Nabil Fadai and Zachary Wilmott were also among the five finalists. The prize was presented at the British Applied Mathematics Colloquium on Tuesday 11 April.

Doireann's work focuses on mathematical modelling of real-world systems, primarily in fluid mechanics.  She conducted her study of the production of thin glass sheets via the redraw process in collaboration with Schott AG.

Monday, 3 April 2017

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.

Suppose $p$ is prime and consider the map $x\mapsto x^2+1$ on the ring $\mathbb{Z}_p$. For example, if $p=5$ then $0\to 0^2+1=1$, $1\to 2$, $2\to 5=0$, $3\to 10=0$ and $4\to 17=2$. If we iterate the map we get sequences, such as $0\to 1\to 2\to 0\to 1\ldots$ and $4\to 2\to 0\to 1\to 2\to 0\ldots$. Clearly any such sequence must eventually repeat and enter a cycle. The process produces a directed graph, with vertices labeled $0,1,\ldots, p-1$.

The questions I am interested in include - How far must one go before entering a cycle? How long are the cycles? And how many distinct cycles are there? For example, for $p=31$, if one starts at 13 there are 6 steps before one gets a repetition; and there are 3 cycles, of lengths 1, 1 and 3.

What I have shown is the following. Firstly, any path repeats after $O(p/\log\log p)$ steps; and secondly the sum of the lengths of all the cycles is $O(p/\log\log p)$. (The notation means that there is a numerical constant $C$, which I've not bothered to compute, so that the number of steps is at most $Cp/\log\log p$.) Of course $\log\log p$ tends to infinity extremely slowly, but the results do show that eventually $p$ is sufficiently large that one always gets a repetition within $p/1000$ steps, for example.

Other people (including Fernando Shao, at the Mathematical Institute here in Oxford) have looked at this and related questions, but mine are the first quantitative results of this type. There is some interesting Galois theory of relevance, but my approach is more geometric.

You can read my paper which will be published in due course in a memorial volume for Klaus Roth, the first British Fields Medalist."

Friday, 24 March 2017

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.

During intercourse, more than 50 million sperm set out to fertilise an egg, but only 10 make it to the final destination, before a single sperm wins the race and makes contact. The journey involved is treacherous and little known, and key to understanding fertility.

The findings, newly published in the journal Physical Review Letters, showed that a sperm’s tail creates a characteristic rhythm that pushes the sperm forward, but also pulls the head backwards and sideways in a coordinated fashion. The team now aim to use this research to understand how larger groups of sperm behave and interact, a task that would be impossible using observational techniques.

By analysing these movements, researchers noticed that a swimming sperm moves the fluid in a coordinated rhythmic way, which can be captured to form a relatively simple mathematical formula. Using this formula in practical medicine could mean that the complex and expensive computer simulations currently used in infertility screening, would no longer be needed.

Dr Hermes Gadêlha, from the University of York’s Department of Mathematics and formerly from Oxford Mathematics, said: "‘Numerical simulations are used to identify the flow around the sperm, but as the structures of the fluid are so complex, the data is particularly challenging to understand and use. Around 55 million spermatozoa are found in a given sample, so it is understandably very difficult to model how they move simultaneously. ‘We wanted to create a mathematical formula that would simplify how we address this problem and make it easier to predict how large numbers of sperm swim. This would help us understand why some sperm succeed and others fail."

The research demonstrated that the sperm has to make multiple contradictory movements, such as moving backwards, in order to propel it forward towards the egg.

The journey to fertilisation is not easy, says Dr Gadelha: "Every time someone tells me they are having a baby, I think it is one of the greatest miracles ever – no-one realises the complexities involved, but the human body has a very sophisticated system of making sure the right cells come together."

Speaking on the value and future uses of the research, Oxford Mathematician, Eamonn Gaffney, co-author,said: "mathematically analysing slow motion video of human sperm swimming reveals a graceful choreography with a surprisingly simple and elegant fluid flow around the cell as it moves. This will make studying the dynamics of sperm populations simpler, which can find numerous applications such as developing a predictive understanding of sperm control in prospective microdevices for sperm handling and isolation in sperm research and assisted reproductive technologies."

Now that the team has a mathematical formula that can predict the fluid movement of one sperm, the next step is to use the model for predictions on larger numbers of cells. The team also believe that it will have implications for new innovations in infertility treatment.

Monday, 13 March 2017

James Grogan wins Gold at STEM

Oxford Mathematician James Grogan has won Gold for Mathematics at STEM for Britain, a poster competition and exhibition for early career researchers held at the Houses of Parliament on 13 March 2017. James's poster and work is focused on understanding tumour development and treatment.

Around 150 researchers presented at STEM (Science, Technology, Engineering and Mathematics) for Britain this year, including 5 Oxford Mathematicians – James (of course), Lucy Hutchinson, Christoph Siebenbrunner, Edward Rolls and Ben Sloman. 

Monday, 13 March 2017

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. 

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, transport, housing, and education. Fuelled by rapid urbanisation and pressing policy concerns, an unparalleled inter-disciplinary research agenda has emerged that spans the humanities, social and physical sciences. From a quantitative perspective, this agenda embraces the new wave of data emerging from both the private and public sector, and its promise to deliver new insights and transformative detail on how society functions today. The novel application of tools from mathematics, combined with high resolution data, to study social, economic and physical systems transcends traditional domain boundaries and provides opportunities for a uniquely multi-disciplinary and high impact research agenda.

One particular strand of research concerns the fundamental question: how do cities move into new economic activities, providing opportunities for citizens and generating inclusive growth? Cities are naturally constrained by their current resources, and the proximity of their current capabilities to new opportunities. This simple fact gives rise to a notion of path dependence: cities move into new activities that are similar to what they currently produce. In order to describe the similarities between industries, we construct a network model where nodes represent industries and edges represent capability overlap. The capability overlap for industry pairs may be empirically estimated by counting worker transitions between industries. Intuitively, if many workers switch jobs between a pair of industries, then it is likely that these industries share a high degree of knowhow. 

This network can be seen as modelling the opportunity landscape of cities: where a particular city is located in this network (i.e., its industries) will determine its future diversification potential. In other words, a city has the skills and knowhow to move into neighbouring nodes. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities. 

Such models aid policy-makers, planners and investors by providing detailed predictions of what types of new activities are likely to be successful in a particular place - information that typically cannot be gleaned from standard economic models. Metrics derived from such networks are informative about a range of associated questions concerning the overall growth of formal employment and the optimal size of urban commuting zones. 

You can explore diversification opportunities for cities and states in Colombia using network mapping tools (as shown in the figure below) by visiting 

This research was conducted by Neave and colleagues primarily at the Center for International Development at Harvard University, in collaboration with Prof. Ricardo Hausmann, Eduardo Lora and Dr Andres Gomez. To see the working papers click the links:

The Path to Labour Formality: Urban Agglomeration and the Emergence of Complex Industry

City Size, Distance and Formal Employment Creation 


Figure caption (click on it to enlarge): a network of labour flows between industries for Colombia. Nodes represent industries, and are colored by sector. It is observed that closely related industries tend to cluster, driven by workers transitioning between similar economic activities. This network models the flow of know-how within the Colombian economy, and can be used to model the path dependent process of industrial diversification for urban centres.

Thursday, 9 March 2017

The importance of the STEM subjects - Oxford Mathematicians in the Houses of Parliament

STEM is an acronym that means a lot to those in the know and probably nothing to the vast majority of the population. However, STEM or Science, Technology, Engineering and Mathematics are where it is at, at least in so far as any nation wanting to improve human wealth and welfare has to have a rich talent in those subjects.

To encourage that talent every year parliament hosts STEM for Britain, a poster competition and exhibition for early career researchers Around 150 researchers are presenting this year and they include 5 Oxford Mathematicians – James Grogan, Lucy Hutchinson, Christoph Siebenbrunner, Edward Rolls and Ben Sloman. James and Lucy's work is focussed on understanding tumour development and treatment while Edward also works in mathematical biology. Christoph studies risks to the stability of the financial system and Ben works on modelling thermal and chemical effects in silicon production. All five will present posters at the Event on Monday 13 March at the House of Commons where over 100 parliamentarians are expected. STEM may be a curious acronym, but a lot rests on it.


Thursday, 9 March 2017

Oxford Mathematician and Computer Scientist Andreea Marzoca wins top spot in national student awards

Oxford Mathematics and Computer Science Undergraduate Andreea Marzoca has become joint winner of the The WCIT University IT Awards 2017. The awards recognise outstanding undergraduate and postgraduate IT students within the UK, and were created in 2015 by The Worshipful Company of Information Technologists Charity (WCIT Charity). Criteria for the award included academic excellence, overcoming adversity, entrepreneurial skills and contribution to charity or community. Andreea, and joint winner Joanna Joss (of Brunel University, London) are the first female winners of this award. 

Andreea is a 3rd year undergraduate studying Maths and Computer Science. She is also Vice President of OxWoCS (Oxford Women in Computer Science). Andreea received her award along with the other finalists at the WCIT 95th Business Lunch, held at the Saddlers’ Hall in the City of London, where the students also had the opportunity to network with high-ranking IT professionals from all around the UK. Each finalist was presented with a cheque and certificate by the City of London Alderman Sheriff, Peter Estlin.

Wednesday, 8 March 2017

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. At the other end of the spectrum is the use of Atomic Force Microscopes (AFM) to measure the properties of graphene and biological membranes, such as skin.

Information concerning the material properties of the floating elastic layer, and the physical properties of the fluid substrate, can be gleaned from the shape that the layer takes when sat upon.

Taking inspiration from the frog, Oxford Mathematicians Finn Box and Dominic Vella poked thin elastic sheets floating on a liquid bath and studied their resultant deformation. They found that for small loads, the resultant deformation remains axi-symmetric about the point of poking. In this case, the deformation is controlled either by the bending of the sheet or its stretching – the difference depends on the thickness of the sheet.  For sufficiently-high loads, a radial wrinkle pattern forms as the sheet buckles out-of-the-plane. Such wrinkle patterns are of interest not merely for their aesthetic appeal, but also as a means of generating patterned surfaces with tunable characteristics that can be used as photonic structures in photovoltaics, amongst other things.

And where does the frog sit in all of this? Well, the frog rests at the smaller end of the length scale spectrum and although the large and small scale situations contain the same physics, the latter are additionally affected by the surface tension of the liquid. Perhaps fortunately for the frog, it isn’t heavy enough to cause the lily pad to wrinkle. The researchers believe that they may be, however, and are looking forward to testing their findings next time they encounter a lily leaf large enough to support them.

Finn and Dominic's research will be published shortly.

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.