Wednesday, 12 April 2017 
The British Applied Mathematics Colloquium (BAMC), held this year at the University of Surrey, has awarded its two talk prizes to Oxford Mathematicians Jessica Williams and Graham Benham. Their colleague in Oxford Mathematics Ian Roper won the poster prize. All three are part of the Industrially Focused Mathematical Modelling Centre for Doctoral Training (InFoMM) a partnership between EPSRC, the University of Oxford, and a large number of industry partners.

Wednesday, 12 April 2017 
Oxford Mathematicians Tamsin Lee and Peter Grindrod discuss their latest research on the brain, part of our series focusing on the complexities and applications of mathematical research and modelling.
"The brain consists of many neurons arranged in small, strongly connected directed networks, which in turn are connected up by a few directed edges. Let us call these small, strongly connected directed networks of neurons 'subgraphs.' Each subgraph receives messages from some upstream subgraph, and sends messages out to downstream subgraphs. Within each subgraph, when a single neuron fires a 'message' it goes into a refractory period. That is, it cannot send nor receive a message for a given period of time. Additionally, each connection from one neuron to another has a unique delay time, that is, a message from neuron A fired to B and C at the same time, will arrive at B and C at different times.
Within these dynamics we find that the system settles to a quasiperiodic state with almost periodic cyclic firings. Taking a closer look at the differences in firing times we find a quasi periodic pattern. This time series can be embedded in an mdimensional space using Taken's Theorem. A key example of Taken's Theorem uses the infamous Lorenz attractor, which plots three variables over time. The theorem shows that by plotting only one of these variables against itself, but shifted at three different time intervals, the result has the same topology as the original Lorenz attractor. To apply Taken's Theorem we create a matrix with our time series, but shifted at different intervals. Signaltonoise separation can be obtained by simply locating a significant break in the ordered list of eigenvalues of this matrix (pink or white noise would produce a natural decay or plateau of the spectrum, without such large breaks). This break gives an upper bound on the number of dimensions required to 'plot' our time series, which is essentially a proxy for the complexity of the behaviour of a single subgraph system.
To recap, neurons in a subgraph receive a message from some upstream subgraphs. This sets off a firing pattern across the subgraph that settles to a system such that the differences in firing times can be embedded in an mdimensional space, where m is a proxy for the complexity of the system.
Our work suggests that the complexity, m, of the subgraph dynamics only increases logarithmically with its size, n. This is a profound result as it states that a brain composed of many small, strongly connected, subgraphs is considerably more efficient that one composed of large, strongly connected, subgraphs. And brains are of course limited in terms of both volume and energy. This is akin to a computer using several small core processors instead of using one large core processor."

Wednesday, 12 April 2017 
Oxford Mathematician Doireann O'Kiely has been awarded the biennial LighthillThwaites 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 fulltime 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 realworld 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 
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 HeathBrown 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 $0,1,\ldots, p1$.
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 
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 – noone 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, coauthor,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 
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 
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 policymakers.
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 interdisciplinary 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 multidisciplinary 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 wellconnected region has many options, but one with only few peripheral industries has limited opportunities.
Such models aid policymakers, 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 www.datlascolombia.com.
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 knowhow within the Colombian economy, and can be used to model the path dependent process of industrial diversification for urban centres.

Thursday, 9 March 2017 
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 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 highranking 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 
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 axisymmetric 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 sufficientlyhigh loads, a radial wrinkle pattern forms as the sheet buckles outoftheplane. 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.
