Saturday, 22 April 2017

In Praise of Plato - Willow Winston's Sculptures in the Andrew Wiles Building

The Andrew Wiles Building, our home here in Oxford, is very much a public space with its large exhibition and conference facilities and public cafe. We have hosted theatrical productions, most recently Creation Theatre's stark production of Orwell's '1984' and in particular we have provided an outlet for artists and photographers to display their work.

Yet we are of course primarily a mathematics building - mathematics and mathematicians are evident everywhere you go from Roger Penrose's tiling at the entrance to the mathematical-shaped crystals at the heart of the building. 

Our latest exhibition, Sculptor Willow Winston's 'In Praise of Plato' represents all those elements. An artistic exploration of symmetry, it is a marriage of mathematics and art in a public setting. In Willow's own words "experimenting with geometrical form fabricated in metals, much based on work I have done with Plato's Perfect Solids, I use reflective materials allowing a union between real and virtual worlds, enhancing our ability to climb into the imagination. "

The exhibition, in the Mezzanine space of the Andrew Wiles Building, is open from 8am-6pm Monday to Friday and runs until 21 May. 

Friday, 21 April 2017

The mathematics of why our grandmothers love us

Michael Bonsall, Professor of Mathematical Biology at Oxford University's Department of Zoology, discusses his research in population biology, what it tells us about species evolution and, in particular, why grandmothering is important to humans. His research was done in conjunction with Oxford Mathematician Jared Field.

"What is mathematical biology?

It is easy to get lost in the details and idiosyncrasies of biology. Understanding molecular structures and how systems work on a cellular level is important, but this alone will not tell us the whole science story. To achieve this we have to develop our insight and understanding more broadly, and use this to make predictions. Mathematics allows us to do this.

Just as we would develop an experiment to test a specific idea, we can use mathematical equations and models to help us delve into biological complexity. Mathematics has the unique power to give us insight in to the highly complex world of biology.  By using mathematical formulas to ask questions, we can test our assumptions. The language and techniques of mathematics allow us to determine if any predictions will stand up to rigorous experimental or observational challenge. If they do not, then our prediction has not accurately captured the biology. Even in this instance there is still something to be learned. When an assumption is proven wrong it still improves our understanding, because we can rule that particular view out, and move on to testing another.

What science does this specialism enable – any studies that stand out, or that you are particularly proud of?

Developing a numerate approach to biology allows us to explore what, at face value, might be very different biologies. For instance, the dynamics of cells, the dynamics of diseases or the behaviour of animals. The specialism allows us to use a common framework to seek understanding. The studies that stand out in my mind are those where we can develop a mathematical approach to a problem, and then challenge it with rigorous, quality experiments and/or observation. This doesn’t have to happen in the same piece of work, but working to achieve a greater understanding is critical to moving the science forward.

You recently published a paper: ‘Evolutionary stability and the rarity of grandmothering.’ What was the reasoning behind it?

Grandmothering as a familial structure is very rare among animals. Whales, elephants and some primates are the few species, besides humans, to actually adopt it. In this particular study we took some very simple mathematical ideas and asked why this is the case.

Evolution predicts that for individuals to serve their purpose they should maximise their reproductive output, and have lots of offspring. For them to have a post-reproductive period, and to stop having babies, so that they can care for grandchildren for instance, there has to be a clearly identified benefit.

We developed a formula that asked why grandmothering is so rare in animals, testing its evolutionary benefits and disadvantages, compared to other familial systems like parental care and co-operative breeding for instance (when adults in a group team up to care for offspring). We compared the benefit of each strategy and assessed which gives the better outcomes.

What did the findings reveal about the rarity of grandmothering and why so few species live in this way?

Our maths revealed that a very narrow and specific range of conditions are needed to allow a grandmothering strategy to persist and be useful to animals. The evolutionary benefits of grandmothering depend on two things: the number of grandchildren that must be cared for, and the length of the post-reproductive period. If the post-reproductive period is less than the weaning period (the time it takes to rear infants) then grandmothers would die before infants are reared to independence.

We made the mathematical prediction that for grandmothering to be evolutionary feasible, with very short post-reproductive periods, it is necessary to rear lots of grandchildren. But if this post-reproductive period is short, not many (or any) would survive. Species with shorter life spans, like fish, insects and meerkats for instance, simply don’t have the time to do it and focus on parental-care. Evolution has not given them the capability to grandparent, and their time is better spent breeding and having as many offspring as they can. By contrast long-lived animals like whales and elephants have the time to breed their own offspring, grandparent that offspring and even to great-grandparent the next generation.

How do you plan to build on this work?

Although grandparenting isn’t a familial strategy that many species are able to adopt, it is in fact the strongest. Compared to parental-care and co-operative breeding, grandparenting has a stronger evolutionary benefit – as it ensures future reproductive success of offspring and grand-offspring – giving a stronger generational gene pool.

Moving forward we would like to test mathematical theories to work out if it is possible for species to evolve from one familial strategy to another and reap the benefits. Currently for the majority of species rearing grandchildren instead of having their own offspring is not a worthwhile trade-off.

What are the biggest challenges?

Ensuring that the mathematical sciences has relevance to biology. Biology is often thought to lack quantitative rigour. This would be wrong. The challenge is to show how the mathematical sciences can be relevant to, and help us to answer critical questions in biology. This will continue to be a challenge but will yield unique insights along the way in unravelling biology.
What do you like most about your field?

So many things. Firstly, the people. I work with a lot of very smart people, who I look forward to seeing each day. I also get to think about biology and look at it through a mathematical lens. Finally I think the specialism allows us to do fantastic science that has the potential to improve the world.

Is there any single mathematical biology problem that you would like to solve?

Developing a robust method to combine with biological processes that operate on different time scales - as this would have so many valuable, and to use one of my favourite words, neat, applications to our work.

Why did you decide to specialise in this area?

Because of the perspective that we can gain from it and because I love biology and maths. Unpicking the complexities of the natural world with maths and then challenging this maths with observations and experiments is super neat. And I can do (some of this) while eating ice-cream!

‘Evolutionary stability and the rarity of grandmothering’ by Jared M.Field and Michael B.Bonsall is available to download from the science journal Ecology and Evolution.

Monday, 17 April 2017

Mathematical physicist James Sparks talks about his research into exact results in the AdS/CFT correspondence

As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician James Sparks discusses his latest work.

"Two great successes of 20th century theoretical physics are Quantum Field Theory and General Relativity. 

Quantum Field Theory (QFT) is a framework for applying the principles of quantum mechanics to the classical theory of fields (such as the electromagnetic field). In QFT elementary particles, such as electrons, quarks, and photons, are quantum excitations of fields. Particle accelerators, such as the Large Hadron Collider at CERN, have tested the theoretical predictions of QFT to extraordinary precision.

General Relativity (GR) is Einstein's theory of gravity. In GR gravity is described by the bending/curving of space and time, and this geometry of spacetime is dynamical. Again, the predictions of GR have been tested in detail by many experiments, most recently in 2016 with the direct observation of gravitational waves (predicted by Einstein almost exactly a century earlier).

Despite these successes, our understanding of both theories is incomplete.

QFT is best understood in perturbation theory. Here one assumes there is a small parameter, called the coupling constant g. For g=0 the particles/fields do not interact at all. The results of particle interactions, such as collisions in a particle accelerator, may then be computed to any desired order of accuracy as a series expansion in g. However, what about phenomena in which there is no small parameter g in which to make such an expansion? In general there is very little theoretical understanding of QFT at so-called strong coupling: there are both conceptual and technical problems.

On the other hand, applying quantum mechanics to gravity has proved very difficult. Such a quantum theory of gravity is expected to be relevant for understanding black holes and the Big Bang singularity.

20 years ago Maldacena proposed a remarkable relation between these two problems. It goes under the general name of the AdS/CFT correspondence, or gauge/gravity duality. This is a conjecture asserting that certain QFTs have a completely equivalent description as a quantum theory of gravity! In particular, AdS/CFT typically relates strongly coupled QFTs (in the limit that some g tends to infinity) to classical GR. Even more bizarre is that the theories live in different numbers of dimensions: for a QFT in 4 spacetime dimensions (3 space and 1 time), the dual theory of gravity lives in 5 spacetime dimensions. In fact, in a precise sense, the QFT may be regarded as living on the boundary of the region in which gravity propagates.

After 20 years of intensive research we still don't understand AdS/CFT very well. However, what is clear from all this research is that it must be true!

As a mathematical physicist my work on this topic is at the more formal mathematical end. A key additional ingredient is supersymmetry. Whether this is actually a symmetry of Nature (above some energy scale) or not is somewhat irrelevant for my purposes: it is there to provide additional analytic control on the problem. Over the last few years new techniques have been developed in supersymmetric QFTs, that allow certain observables of the theory to be computed exactly i.e. exactly as a function of g, rather than in a series expansion. These go under the general name of localization techniques.

Broadly speaking my recent research on this has focused on two problems, which are related to each other by AdS/CFT.

The first problem involves studying supersymmetric QFT on certain curved spacetime backgrounds. Although the spacetime is curved, gravity is not dynamical in these models. Localization techniques allow for exact computations of QFT observables in many cases. Typically the infinite-dimensional path integral that defines the QFT observable reduces exactly to a finite-dimensional integral. The latter is often quite complicated e.g. involving an integral over the space of solutions to certain differential equations. Simple observables of interest are the partition function and certain supersymmetric Wilson loops. I have particularly focused on strong coupling limits of these computations, which are expected to be dual to semi-classical gravity via the AdS/CFT correspondence.

The second problem concerns the dual description of the first problem. Here one wishes to study certain classes of solutions to GR (or rather its supersymmetric cousins, called supergravity theories). Mathematically this is a filling problem, where one solves equations with fixed boundary data. My recent work on this has included the construction of new classes of solutions to supergravity, relevant for describing strongly coupled QFT observables. I am particularly interested in deriving general results. For example, provided the filling satisfies certain topological assumptions, one can often show that supersymmetric observables of interest may be computed without knowing the supergravity solution in detail. This has led to a number of general "exact results" in AdS/CFT, in which both sides of the correspondence can be computed independently, in a broad setting, and shown to always agree."

For more on James's research into the field see below.

Wednesday, 12 April 2017

Oxford Mathematics students win prizes at BAMC

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

Modelling the architecture of the brain

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 quasi-periodic 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 m-dimensional 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. Signal-to-noise 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 m-dimensional 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

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.