Friday, 21 June 2019

Sticking with droplets: How having a soft foot can improve capillary adhesion

Have you ever picked up a glass to find that the coaster it was resting on remains stuck to the bottom? If so, then you have experienced the ability of fluid to stick two surfaces together. When the bottom of the glass is wetted, for example by accidentally spilling a drink, then this fluid can fill the gap between the glass and coaster. The surface tension of the liquid then provides a pulling force on the coaster that keeps it attached to the glass.

It is believed that the climbing ability of insects is aided by this effect, albeit at a smaller scale. Insects are able to walk up walls and across ceilings, and have been observed holding over 100 times their own body weight while remaining adhered. Looking closely beneath their feet, an oily fluid is found that is believed to help stick their feet to a substrate. It has also been noted that the footpads of some insects are soft and deformable.

Inspired by these observations of insects, Oxford Mathematicians Matthew Butler and Dominic Vella have developed a mathematical model to investigate the combination of deformability and adhesion. They consider a system where the footpad is a thin, deformable sheet under tension (like a drum skin) in which the deformability can be varied by changing the tension in the sheet, and the separation of the solid surfaces can be controlled.

Calculations of the equilibria reveal two distinct stable states, distinguished by whether the deformable sheet is in contact with the substrate or not. In both stable states, the adhesion force is found to be always larger than an equivalent rigid case, suggesting that deformation can be good—the surface tension deforms the sheet, increasing the force provided by surface tension: a positive feedback. Further, when the sheet is in contact with the substrate, the adhering force can be orders of magnitude larger than the non-contacting state.

Experiments performed by Finn Box and Thomas Robert show a good qualitative agreement with the theory, as well as demonstrating that adhesion is possible in a lab scale model. However, it was found that adhesion was only maintained if the initial attachment phase took a sufficient length of time. Further study of the dynamical theory suggests this bottleneck is likely to be due to the formation of a fluid dimple as the deformable sheet tries to contact the base, but is resisted by fluid viscosity.

Despite this caveat, the significant increase in adhesion compared to the rigid case means that deformable capillary adhesives are a fruitful area for further investigation. Of particular interest with this system, is that being able to set the tension and separation may allow for high adhesion, whilst also giving easy control of detachment when required. It is hoped that further investigation of similar devices that exploit deformability may lead to advancements in the world of adhesion science.


Monday, 17 June 2019

1000 people enjoying Maths? It must be the Oxford Maths Festival 2019

Getting tied up in knots, experimenting with bubbles, playing board games, doing origami, experiencing dimensions in virtual reality, exploring historical mathematical instruments and sorting out teddy bears. These were just some of the mathematical activities enjoyed by over 1000 visitors to the Oxford Maths Festival during the weekend of 11-12 May 2019.  

On the first day, Saturday, a group of staff and student volunteers from Oxford Mathematics and Oxford's Dept of Statistics were in Templars Square Shopping Centre in East Oxford. Shoppers were rather surprised to find themselves exploring playful and unexpected aspects of maths at the various stalls. Barney Maunder-Taylor, of House of Maths, had eager audiences of children for his highly engaging maths shows and the historical mathematical artefacts brought by the team from the Oxford History of Science Museum certainly caught the imagination. Shoppers were also able to don virtual reality headsets to explore dimensions or try out the many puzzles and mathematical curiosities on show at the different stalls. 


As a visitor said "It was inspiring & impressive to see people of all ages so engaged. All those running exhibits were ace at making maths accessible to all. The 4yo loved it, didn't want to leave & giggled his way through." And the student volunteers were equally pleased (and a little surprised perhaps) “The local people seemed to be really interested - the response is way more positive than I had expected.”

Saturday evening saw the action move to the Andrew Wiles Building, home of the Oxford Mathematical Institute, as teams raced to complete the first ever Oxford Mathematics Escape Room. They just made it in time for day two of the Festival, which saw around 600 visitors enjoy a range of activities in the Mathematical Institute. The team from the Oxford Maths Observatory (our secret maths lab) welcomed visitors on the famous Penrose paving with experiments giving a glimpse of the power of maths to understand bubbles. Once inside the building, children and adults alike could choose from a wide range of activities. The mathematical craft room was very popular, with a constant stream of participants wanting to try their hands at mathematical origami, curved stitching, mathematical braiding and mathematical colouring designs from books by Alex Bellos and Edmund Harriss. The board games were no less popular, with visitors able to play popular games such as SET, Hanabi, NMBR9, Tantrix and many more. Games designer Educational Games were on hand again with City of Zombies.

The OR Society team's Lego Factory was also very popular, and the student volunteers had their hands full all day with the Hands-on Family Maths activities, many from the NRICH website.  Matt Parker, of Numberphile fame, gave a typically witty and brilliant talk, and Kyle D Evans' Maths Madness family show received rave reviews. 

Local MP Anneliese Dodds clearly enjoyed herself: "After two days of intensive outreach activity (one of which I enjoyed today), I hope all involved with the ⁦@OxMathsFest⁩ are now putting their feet up! Great to see maths being brought to life for all ages."


Over the course of the weekend, over 65 staff and student volunteers were involved, in addition to all those who helped with the planning and preparation. One of the volunteers said “Most people think mathematics is really boring and that when you're studying the subject you just sit in the library reading some books without any real purpose. This event shows both us, mathematicians, and the participants that it can give you a lot of entertainment. Even if we encouraged just one person to do mathematics, that can really mean a lot for someone and change their life.”

Professor Alain Goriely, Director of External Relations for Oxford Mathematics said: "Mathematical knowledge is critical to the future of our society. But too often we are told that people's mathematical curiosity is lost at an early age. The Oxford Maths Festival is doing its bit to put that right." 

Perhaps the Oxford Maths Festival 2019 is best summed up by this visitor, who said: “Thank you for organising and holding such a wonderful event that promotes the importance and fun of exploring maths. My family had a fantastic time, and we can't wait for next year.”

Oxford Mathematics is committed to communicating the breadth and depth of mathematics to a wide audience, and the Oxford Maths Festival is fast becoming a central part of the offering for the local community. We are grateful to the supporters and sponsors whose help and funding enabled the event to go ahead: Mathematical Institute, Department of Statistics, Van Houten Fund, Olamalu and the Public Engagement with Research Seed Fund. We are also grateful to Barney Maunder-Taylor, Matt Parker, Kyle Evans and the OR Society, and to NRICH for the quality hands-on activities.

If you or your company are interested in sponsoring a future Oxford Maths Festival, then please contact Mareli Grady.

Monday, 17 June 2019

BEM on Screens: not just science fiction from the 50s

From nanophotonics to aeroplanes, there are many applications that involve scattering in unbounded domains. Typically, one is interested in situations and geometries where there are no known analytical solutions and one has to resort to numerical algorithms to solve the problem using a computer. Such numerical algorithms should give physically meaningful solutions and hopefully obtain them with the minimal computational cost and time.

Boundary element methods (BEM) are broadly used to model such scattering problems and lead densely populated linear systems, which are usually solved by means of iterative solvers. This means that their computational efficiency relies on clever matrix approximation and compression techniques to reduce memory costs (like fast multipole methods or hierarchical matrices), and usually also on preconditioners, which reduce the number of solver iterations needed to reach an approximated solution. A good preconditioner should be easy to compute and is often labelled as optimal when its application makes the condition number remain low and almost constant regardless of how big the linear systems are.

Of course, scatterers have many different shapes and topological properties, depending on the application, and they pose different mathematical challenges (and fun). For instance, when the scatterer has corners, the solutions can have singularities and whatever method one is using to compute a solution should take this into account and be as accurate as possible.

Oxford Mathematician Carolina Urzua-Torres and her collaborators were interested in scatterers that are (bounded) infinitely thin structures, so-called screens or open surfaces in the literature, and which model, for instance, the microstrip patch antennas used in wireless communications. They have edges and they might have corners or even holes. In other words, they have a lot of particular features that differ from the standard case and that are difficult to deal with numerically. Although a lot of progress had been made in their mathematical and numerical understanding, it was still not known how to build optimal preconditioners for that case.

“A very popular and robust preconditioning approach in BEM is the so-called Calderón preconditioning that exploits the Calderón identities between the boundary integral operators (BIOs) to arrive to a well-conditioned system. Unfortunately, these identities do not hold on screens. Thus, we needed to find operators satisfying similar identities and that would give us the same nice properties. We chose to start by attempting this in the unit disk.

We had to dig into the mathematical properties of the Sobolev spaces and operators on these geometries in order to find closed-form formulas for the inverse operators of the BIOs for the Laplacian on the unit disk. We also took care that these formulas were amenable to Galerkin discretization, so we could actually use them in our code. Then we combined this new knowledge and the mathematical structure of the problems to come up with a new approach to precondition systems arising from Laplace, Helmholtz, and time-harmonic Maxwell equations on different shapes of screens.

Besides the desired optimality, this new approach has several good numerical properties. First, we can use it on many types of locally refined meshes, which is very important in resolving the singularities that the solutions have. Second, it fixes the low-frequency breakdown in the case of Maxwell equations, which is relevant for many applications. Moreover, it is ‘bandwith robust’, meaning that the preconditioner can be used for different frequencies and the numerical method will not breakdown, although it should be noted that the preconditioner’s performance will deteriorate the higher the frequency. Finally, it is easy to incorporate in existing BEM codes, which is very convenient.

Indeed, last week I met an engineer who wants to use the new preconditioner for Maxwell in industry projects and for him this implementation simplicity and bandwith robustness were of great importance. The truth is that at the moment there are no preconditioners that work for all frequencies, which is the reason why there is still a lot of research activity designing solutions for different ranges of frequency. However, given the many complexities of the problems engineers are interested in, he explained to me that they want to avoid at all costs having different things for different cases. Therefore, the fact that our approach will work in all cases, even if not great, promises a reasonable trade-off and should be good enough for the problems they are interested in. I am really looking forward to hearing about his practical experience with our preconditioner. At the end of the day, we want people to test and use our work in real-life applications.”

To find out more about the research please click the links below:
Inverses on disks
Preconditioning for Laplace (which also applies to Helmholtz) and also here
Compact-equivalent for Maxwell
Preconditioning for Maxwell - see Chapter 7 of Carolina's Thesis

By the way, for those of you who wondered, in the world of sci-fi BEM stands for bug-eyed monster, a particular feature of the genre in the 1950s. The image above comes from 'Astounding Stories, May 1931, Dark Moon' by Charles Willard Diffin.

Monday, 10 June 2019

Oxford Mathematics Open Day LIVE online on 3 July

Oxford Mathematics Open Day Live Stream - 3 July

On 3 July we shall we live streaming our Open Day for prospective applicants as part of our going Behind the Scenes' at Oxford Mathematics. This is our way of making the Open Day 'open' to everyone, wherever you are.

The running order:
10.00am - James Munro introduces you to Mathematics at Oxford

10.30am - Vicky Neale on Pure Mathematics at Oxford

11.00am - Dominic Vella on Applied Mathematics at Oxford


10.30am - we will be taking online questions

How to watch:


The films will remain available after the live stream and will join the student lectures and tutorial which are proving very popular on our Oxford Mathematics YouTube Channel.

Friday, 7 June 2019

The pros and cons of cell cannibalism - mathematics and medicine join forces to understand the causes of inflammatory and infectious diseases

Certain inflammatory and infectious diseases, including atherosclerosis and tuberculosis, are caused by the accumulation inside immune cells of harmful substances, such as lipids and bacteria. A multidisciplinary study published in Proceedings B of the Royal Society, by researchers from the Universities of Oxford and Sydney, has shown how cell cannibalism contributes to this process.

Hugh Ford, an applied mathematics PhD student with Prof. Mary Myerscough at the University of Sydney, conducted the research whilst a visiting student at Oxford Mathematics. With Prof Myerscough and Prof. Helen Byrne in the Oxford Mathematics’s Wolfson Centre for Mathematical Biology, Hugh developed a mathematical model that accurately describes the accumulation of harmful substances in macrophages, a type of white blood cell, which act as “waste-disposal” cells for the immune system. When there are too many of these substances for macrophages to handle, they are unable to remove the excess from the area before they die. This triggers a cycle in which macrophages that die while removing harmful substances from the arteries leave the substances in situ, causing more macrophages to be recruited to ingest them. The researchers’ model describes how this leads to an accumulation of macrophages. These ingest the dead cells, along with the cholesterol they have accumulated, causing the cycle to accelerate.

The researchers tested their model experimentally, in collaboration with Prof. David Greaves at the Dunn School of Pathology in Oxford. Hugh tracked the accumulation of plastic beads in thousands of macrophages in laboratory experiments, generating data to validate the cascading effect of substance accumulation predicted by the mathematical model. Dr Joshua Bull, an Oxford Mathematics postdoctoral researcher, assisted with the data analysis by adapting his existing image analysis software to enable automatic counting, from thousands of high-resolution images, of individual macrophages and the numbers of plastic microbeads contained within them.

Prof. Greaves from the Dunn School of Pathology said: “Hugh’s mathematical modelling allowed us to do a set of biology experiments that shed new light on the processes that drive diseases. Armed with these new insights we are keen to look for drugs that enhance tissue protection by changing cell behaviour.”

Hugh said the paper contributes to a growing body of evidence that casts cannibalistic cell removal as a double-edged sword. “While on the one hand this process is crucial for tissue stability and the resolution of inflammation, it also perpetuates subcellular accumulation of harmful substances that can then contribute to the development of diseases, such as heart disease and tuberculosis,” 

Wednesday, 5 June 2019

Oxford Mathematics Public Lectures: Marcus du Sautoy - The Creativity Code: How AI is learning to write, paint and think. Full lecture now online

Artificial Intelligence (AI) is a great asset. Artificial Intelligence is a threat to our freedom. Much of the debate around AI seems to focus on these two positions along with a third argument, namely AI could never replicate our creativity or capture what makes us human. We will never go to galleries to look at AI paintings or read AI poetry.

Or perhaps we might? In this fascinating and provocative lecture, Marcus du Sautoy both tests our ability to distinguish between human and machine creativity, and suggests that our creativity may even benefit from that of the machines.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.







Wednesday, 22 May 2019

Graham Farmelo - The Universe Speaks in Numbers. Latest Oxford Mathematics Public Lecture now available

An old-fashioned tale of romance and estrangement, Graham Farmelo's Oxford Mathematics Public Lecture charts the 350-year relationship between Mathematics and Physics and its prospects for the future. Might things be less dramatic in future? Might they just have to be 'going steady' for a while?

Our Oxford Mathematics Public Lectures are aimed at a general audience who are curious about maths and its many facets. They are all live streamed and available afterwards on our YouTube Channel. For a full list of forthcoming lectures please click here. You are all very welcome.

Oxford Mathematics Public Lectures are generously supported by XTX Markets.







Monday, 13 May 2019

The science of jumping popper toys

Snap-through buckling is a type of instability in which an elastic object rapidly jumps from one state to another. Such instabilities are familiar from everyday life: you have probably been soaked by an umbrella flipping upwards in high winds, while snap-through is harnessed to generate fast motions in applications ranging from soft robotics to artificial heart valves. In biology, snap-through has long been exploited to convert energy stored slowly into explosive movements: both the leaf of the Venus flytrap and the beak of the hummingbird snap-through to catch prey unawares.

Despite the ubiquity of snap-through in nature and engineering, how fast snap-through occurs (i.e. its dynamics) is generally not well understood, with many instances reported of delay phenomena in which snap-through occurs extremely slowly. A striking example is a children’s ‘jumping popper’ toy, which resembles a rubber spherical cap that can be turned inside-out. The inside-out shape remains stable while the cap is held at its edges, but leaving the popper on a surface causes it to snap back to its natural shape and leap upwards. As shown in the figure, the snap back is not immediate: a time delay is observed during which the popper moves very slowly before rapidly accelerating. The delay can be several tens of seconds in duration — much slower than the millisecond or so that would be expected for an elastic instability. Playing around further reveals other unusual features: holding the popper toy for longer before placing it down generally causes a slower snap-back, and the amount of delay is highly unpredictable, varying greatly with each attempt.

In a series of videos launching The Mathematical Observer, a new YouTube channel showcasing the research performed in the Oxford Mathematics Observatory, Oxford Mathematician Michael Gomez (in collaboration with Derek Moulton and Dominic Vella) investigates the science behind the jumping popper toy. Episode one discusses why the popper toy snaps, and the important role played by the geometry of a spherical cap. Episode two focuses on how fast the popper toy snaps, and how its unpredictable nature can arise purely from the mathematical structure of the snap-through transition.



Thursday, 9 May 2019

The third in our series of filmed student lectures - Ben Green on Integration

Back in October, for the first time, we filmed an actual student lecture, Vicky Neale's lecture on 'Complex Numbers.' We wanted to show what studying at Oxford is really like, how it is not so different to school while at the same time taking things to a more rigorous level. Since we made the film available, over 375,000 people have watched some of it. 

Emboldened, we went one stage further in February and live streamed a lecture (and made it available subsequently), James Sparks on 'Dynamics.' But in addition to the lecture, we also filmed the subsequent tutorial which all students receive, usually in pairs, after lectures, and which is the essential ingredient of the Oxford learning experience. Both have been huge successes.

So we come to the third in our series of filmed student lectures. This is the opening lecture in the 1st Year course on 'Analysis III - Integration.' Prof. Ben Green both links the course to the mathematics our students have already learnt at school and develops that knowledge, taking the students to the next stage. Like all good lectures it recaps and points forward (the course materials accompanying the Integration lectures can be found here).

The lectures and tutorial are all part of our going 'Behind the Scenes' at Oxford Mathematics. We shall we filming our Open Days in July and more lectures in the Autumn. Please send any comments to

Wednesday, 8 May 2019

Matthew Butler awarded the Lighthill-Thwaites Prize for 2019

Oxford Mathematician Matthew Butler has been awarded the biennial Lighthill-Thwaites Prize for 2019. 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.

Matthew's research focuses on fluid dynamics, particulary flows at low Reynolds number involving surface tension and interactions with elastic boundaries. His talk at the British Applied Mathematics Colloquium 2019 where the prize was awarded was entitled 'Sticking with droplets: Insect-inspired modelling of capillary adhesion" and focused on how having a deformable foot can be beneficial when trying to adhere to a substrate using the surface tension of a fluid droplet. In his PhD Matthew is studying insect adhesion, and in particular how insects can utilise physical laws to improve their ability to stick to surfaces.

Oxford Mathematician Doireann O'Kiely won the prize in 2017 and Laura Kimpton, also from Oxford, won it in 2013. Oxford Mathematician Jessica Williams was also a finalist this year.