Tuesday, 14 August 2018

Flux-dependent graphs - enabling a better understanding of cellular metabolism

How does cellular metabolism change in different environments? Metabolism is the result of a highly enmeshed set of biochemical reactions, naturally amenable to graph-based analyses. Yet there are multiple ways to construct a graph representation from a metabolic model. 

This work, a collaboration between Oxford Mathematician Mariano Beguerisse and colleagues in mathematics and bioengineering from Imperial College and the Polytecnic University of Valencia, proposes a principled framework to construct genome-scale models of metabolism using modern network science. These models resolve various challenges, such as the incorporation of pool metabolites, directionality of metabolic flows, and scenario-specific flux information. 

This framework abandons metabolic descriptions that are generic blueprints in favour of tailored metabolic descriptions under any specific context of interest. For example, this model predicts the way in which the metabolism of Escherichia coli re-routes metabolic flows as environmental conditions change from being oxygen-rich (aerobic) to oxygen-poor (anaerobic). In many situations the reactions that constitute metabolic pathways form tighly-knit clusters, but in other cases the reactions may be dispersed, or not even connected to the network. Creating scenario-specific metabolic networks allows us to study how pathways behave in different conditions, and understand the role that individual reactions play. An analysis of the metabolism of human liver cells with a rare metabolic disease identifies several key reactions that flux-only analyses miss because they do not incorporate the rewiring of the metabolic network.

The method can be integrated into pipelines based on flux balance analysis and provides a systematic framework to explore changes in network connectivity as a result of environmental shifts or genetic perturbations. The paper giving more details on the research appears in the journal NPJ Systems Biology and Applications.


The image above shows two metabolic reaction networks of E. coli. The top network shows the metabolic connectivity under normal conditions. When the oxygen is removed from the environment, the cell must drastically re-wire its metabolism in order to survive (bottom). Click to enlarge.

Friday, 10 August 2018

How to control chaos in your body

In many natural systems, such as the climate, the flow of fluids, but also in the motion of certain celestial objects, we observe complicated, irregular, seemingly random behaviours. These are often created by simple deterministic rules, and not by some vast complexity of the system or its inherent randomness. A typical feature of such chaotic systems is the high sensitivity of trajectories to the initial condition, which is also known in popular culture as the butterfly effect. A better understanding of chaos has been the subject of intensive mathematical research for many decades.

The time lags in non-monotone feedback may also cause chaotic behavior, a notable example being the so-called Mackey-Glass equation. This nonlinear delay differential equation was proposed by Canadian scientists Leon Glass and Michael Mackey as a prototype equation for physiological regulatory processes. Variants of this equation have been successfully used to better understand and treat various types of disorders of the hematopoietic system.

The equation has the following form:

\[ \frac{\mathrm{d}}{\mathrm{d}t}x(t)=\beta\frac{x(t-\tau)}{1+x(t-\tau)^n}-\gamma x(t) \]

where all parameters are positive. The apparent simplicity of this model equation is rather deceiving: the Mackey-Glass equation encapsulates incredibly rich mathematical structures embedded into an infinite dimensional phase space, and has provided a lot of work for researchers over the past forty years: to date, more than 4000 papers have cited the original study, yet the emergence of its complex dynamics is not fully understood.

The reason for the presence of the time delay in the model is that following a loss of blood cells, it can take many days before new blood cells can be produced through the activation, differentiation, and proliferation of the appropriate blood stem cells. The unimodal shape of the production term is due to the fact that when the blood cell count is very high, the body does not need more blood cells hence production is low; while when the cell count is low, then your body is in bad shape and hence unable to produce enough. There is an intermediate level when the cell production runs at maximal rate. The interplay of the non-monotone feedback and the time delay makes the behaviour of solutions really tricky.

A specific area of ​​chaos theory research is concerned with the so-called chaos control. Since chaotic behaviour is often undesirable, it is important to understand the ways in which chaos can be avoided. The traditional way of controlling a chaotic system is to pick one of the infinitely many unstable periodic solutions from the chaotic attractor, and add a small perturbation which has the effect of stabilizing this periodic solution. In a recent study, Oxford Mathematician Gergely Röst with Gábor Kiss from Szeged, Hungary, were able to control Mackey-Glass chaos with a completely different approach, which does not require a previous determination of the unstable periodic orbits of the system before the controlling algorithm is designed. The idea is to push all solutions into a domain of the phase space where the feedback is monotone, and keep them there. This way, the long term behaviour is governed by monotone delayed feedback, for which a Poincaré-Bendixson type theorem holds, ensuring that all solutions will converge either to an equilibrium or to a periodic orbit. Hence, chaotic behaviour is excluded. The researchers have shown that several popular control mechanisms can be used to control Mackey-Glass chaos, if the parameters are properly chosen. This way some previously empirically observed controls have been mathematically explained, and also new ways of control have been found such as state dependent delay control.

For more on this topic please click here.

A 2-dimensional projection of the Mackey-Glass attractor



 Regulation of a chaotic motion into a periodic one


Control is initiated at t=50, but when it is turned off at t=100, chaotic behaviour returns


Applying a control on short intervals making a complicated solution to behave in a periodic manner.




Thursday, 9 August 2018

Teaching the machines - topology and signatures for parameter inference in dynamical systems

Oxford Mathematician Vidit Nanda talks about his and colleagues Harald Oberhauser and Ilya Chevyrev's recent work combining algebraic topology and stochastic analysis for statistical inference from complex nonlinear datasets.

"It is not difficult to generate very complicated dynamics via very simple equations. Consider, for each parameter r > 0 and natural number n, the update rules

x_{n+1} = x_n + r y_n (1-y_n) mod 1, and
y_{n+1} = y_n + r x_n(1-x_n) mod 1,

where the "mod 1" indicates that we restrict to the fractional part of the number, so for instance 3.7656 mod 1 is just 0.7656. These equations constitute a dynamical system on the unit square, and it turns out that the value of r makes an enormous difference to the behavior of this dynamical system. Below are typical pictures of the orbits (x_n,y_n) obtained (at r = 2.5, 3.5, 4, and 4.1 respectively) by applying the update rules to random initial choices of (x_0, y_0). The task for our machine, then, is to determine which r value has produced a given picture. If you were to see a fifth picture generated at one of these four r-values, you would have no trouble whatsoever determining which r was used. But it turns out to be very hard to efficiently teach a machine how to accurately make the distinction.

The difficulty lies, of course, in the nonlinearity of the dynamical system at hand. While machine learning methods are essentially linear, the geometry of the patterns is decidedly more complicated. One way to capture coarse nonlinear geometry is via the methods of topological data analysis. These reduce complicated point clouds (such as the aforementioned orbit images) to persistence barcodes, which are simply collections of intervals [b,d) labelled by geometric dimension. In our case of 2D images, the only interesting dimensions are 0 and 1. An interval [b,d) in dimension 1 indicates that when the points were thickened to balls of radius b, a hole appeared in the image, and that this hole was filled in upon thickening further to a radius d > b. Fortunately, barcodes generated at different r-values are very different while barcodes generated at the same r-value are quite similar. Unfortunately, the space of barcodes itself is nonlinear, and hence not directly amenable to machine learning.

In order to allow machine learning methods to accept barcodes as input, we linearize the space of barcodes by turning them into paths. There are several nice ways of doing this, and the picture below indicates one of them: sort the intervals in a given barcode in descending order by their length, and construct the envelope curves obtained by joining all the successive b-values and d-values so obtained. Thus, each barcode produces two paths, and one can now compute the signature of those paths to obtain a (linear!) feature map that contains all the nonlinear geometric information necessary for our parameter-inference problem. On a standard benchmark dataset, this "barcode to path to signature" managed to correctly determine the r-value with an accuracy of 98.1%." 

For more on this work please click here .

Tuesday, 7 August 2018

What’s UNIQUE about UNIQ? - Opening up Oxford

Oxford University is committed to encouraging as wide a range of applicants as possible. Oxford Mathematics is part of that commitment. But what does that mean in practice? Well over the Summer months it means UNIQ, Oxford’s way of breaking down barriers and building bridges. A kind of construction work for the mind.

Over the last two weeks, ninety students from schools around the country have visited us in the Mathematical Institute on the UNIQ Summer Schools. These summer schools offer an impression of what it’s actually like to study Maths at Oxford. Places are given to students who are doing well at school, who are from areas of the country with low progression to university, or from low socio-economic status backgrounds. So far, so good, but what do they actually do?

Well, the week consists of taster lectures and tutorials, and, crucially, plenty of opportunities to talk about maths, both with each other and with our team of student ambassadors. Lots of the students say that meeting other people who are interested in maths is the best part of the summer school; for some of them, no-one else at their school or sixth form is as keen on maths as they are, (a refrain that persists well beyond school of course).

During the week the students have had a fascinating series of talks on topics including Benford’s Law, the Twin Paradox and the game theory of the TV show The Chase. But they have also been working together on group presentations on their favourite topics in mathematics and they’ve been working together modelling projects - open-ended problems which they’re free to approach with a variety of methods which give them an insight in to how maths actually works and enables them to spend time trying out different ideas, a luxury they may not get at school.

For example, groups have been comparing strategies to tackle malaria, investigating refraction, and optimising a bridge network. We use these projects to give the students an impression of what tutorials are like; each group has a half-hour tutorial on their project with a member of our faculty. By giving the students a first-hand experience of studying at Oxford, we can break down some of the myths, and make the whole system more transparent.

As well as giving the students a taste of the mathematics that they might study, the UNIQ summer schools also give the students a chance to experience life in Oxford. They’ve been staying in St. Anne’s College and New College, where they’ve had a quiz night, a scavenger hunt and a ghost tour, before a party on the last evening. Life in Oxford is not so different to anywhere else.

Throughout the week, the students have been helped and guided by a fantastic team of ambassadors, who are all current students or recent graduates of Oxford. One of the signs of success of the UNIQ summer schools is the high application rate to study at Oxford from UNIQ students on the summer school, and some of the ambassadors were themselves previously on UNIQ summer schools as students.

Thank you to everyone. There is much to be done, but in some not so small part of the mathematical world, progress is being made.


Photography by Ian Wallman

Wednesday, 1 August 2018

PROMYS Europe 2018 - nurturing our mathematical future

Each summer, a group of very enthusiastic teenage mathematicians come to spend six weeks in Oxford, working intensively on mathematics. They are participants in the PROMYS Europe programme, now in its fourth year and modelled on PROMYS in Boston, which was founded in 1989. One of the distinctive features of the PROMYS philosophy is that the students spend most of the programme discovering mathematical ideas and making connections for themselves, thereby getting a taste for life as a practising mathematician.

Mornings start with a number theory lecture followed by a problems sheet, which sounds very traditional. But at PROMYS Europe, the lectures are always at least three days later than the material comes up on the problems sheets! This allows the students to have their own mathematical adventures, exploring numerical data and seeking patterns, then proving their own conjectures before the ideas are discussed in a lecture. Another crucial part of PROMYS Europe is the community feel. This year there are 21 students participating for the first time, and six who have returned for a second experience. In addition, there are eight undergraduate counsellors, who mentor the students. Each counsellor gives daily individual feedback to their three or four students, allowing each student to progress at their own rate and to focus on their own particular interests. The counsellors are also working on their own mathematics - this year they are teaching themselves about p-adic analysis. The returning students are working in small groups on research projects, and this year are also exploring group theory. The PROMYS Europe faculty are also available to the students for much of the time, reinforcing the supportive and collaborative nature of the programme.

The occasional guest lectures give the participants glimpses of current research mathematics and of topics beyond the programme. So far, in the first two weeks of the 2018 programme students have learned about Catalan numbers and quivers from Konstanze Rietsch (King's College London), and Andrew Wiles (University of Oxford) spoke about using analysis to solve equations.

As Andrew said: "PROMYS has done very impressive work over many years in creating an environment in Boston in which young mathematicians from all over the United States can immerse themselves in serious mathematical problems over several weeks, without distraction. It is an exciting development that PROMYS and the Clay Institute have now opened up the same opportunity in Europe."

The programme is very intensive, and students spend a great deal of time grappling with challenging mathematical ideas through the daily problem sets. At the weekends, students have extra-long weekend problem sets, but also have time to explore Oxford and the surrounding area. So far this has included a tour of Oxford colleges, the chance to go punting, and a visit to Bletchley Park and the National Museum of Computing.

As in previous years, this year's group is very international, coming from 15 countries across Europe. Students have to demonstrate a sufficient command of English when they are applying, and the international language of mathematics soon transcends linguistic and cultural differences once participants arrive!

Students apply to attend PROMYS Europe, and are selected based on their mathematical potential, as displayed in their work on a number of very challenging problems. This year there were more than 200 applications for around 21 places: the students who are invited to participate have produced exceptional work on the application problems, and displayed significant commitment and mathematical maturity. The programme is dedicated to the principle that no student should be unable to attend PROMYS Europe due to financial need, and is able to provide partial and full financial aid to students who would otherwise be unable to participate.

Alumni of PROMYS in Boston have gone on to achieve at high levels in mathematics. More than 50% of PROMYS alumni go on to earn a doctorate, and 150 are currently professors, many at top universities in the US. PROMYS Europe alumni are also proving to be dedicated to pursuing mathematical studies, with several now studying at the University of Oxford.  Of this year's eight counsellors, seven previously participated in PROMYS or PROMYS Europe as students, and four are Oxford undergraduates.

PROMYS Europe is a partnership of PROMYS, Wadham College and the Mathematical Institute at the University of Oxford, and the Clay Mathematics Institute.  The programme is generously supported by its partners and by further financial support from alumni of the University of Oxford and Wadham College, as well as the Heilbronn Institute for Mathematical Research.

Thursday, 26 July 2018

Oxford Mathematics London Public Lecture: 'To a physicist I am a mathematician; to a mathematician, a physicist' - Roger Penrose in conversation with Hannah Fry

'To a physicist I am a mathematician; to a mathematician, a physicist'

7.00pm, 30 October 2018, Science Museum, London, SW7 2DD

Roger Penrose is the ultimate scientific all-rounder.  He started out in algebraic geometry but within a few years had laid the foundations of the modern theory of black holes with his celebrated paper on gravitational collapse. His exploration of foundational questions in relativistic quantum field theory and quantum gravity, based on his twistor theory, had a huge impact on differential geometry. His work has influenced both scientists and artists, notably Dutch graphic artist M. C. Escher.

Roger Penrose is also one of the great ambassadors for science. In this lecture and in conversation with mathematician and broadcaster Hannah Fry he will talk about work and career.

This lecture is in partnership with the Science Museum in London where it will take place. Please email to register.

You can also watch online:

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

Thursday, 26 July 2018

The journey of the applied mathematician - retiring Sedleian Professor Sir John Ball reflects

John Ball is retiring as Sedleian Professor of Natural Philosophy, Oxford oldest scientific chair. In this interview with Alain Goriely he charts the journey of the applied the subject has developed over the last 50 years.

Describing his struggles with exams and his time at Cambridge, Sussex and Heriot-Watt before coming to Oxford in 1996, John reflects on how his interests have developed, what he prizes in his students, as well as describing walking round St Petersburg with Grigori Perelman, his work as an ambassador for his subject and the vital importance of family (and football).





Monday, 23 July 2018

How are trading strategies in electronic markets affected by latency?

Oxford Mathematicians Álvaro Cartea and Leandro Sánchez-Betancourt talk about their work on employing stochastic optimal control techniques to mitigate the effects of the time delay when receiving information in the marketplace and the time delay when sending instructions to buy or sell financial instruments on electronic exchanges.

"In order driven exchanges, liquidity takers face a moving target problem as a consequence of their latency – the time taken to send an order to the exchange. If an order is sent aiming at a price and quantity observed in the limit order book (LOB) then by the time their order is processed by the exchange prices could have worsened, so the order may not be filled; or prices could have improved, so the order is filled at a better price.

Traders can mitigate the adverse effects of missing a trade by including a price limit in their orders to increase the probability of filling the order when it is processed by the Exchange. This price limit consists of the best price seen by the trader in the LOB plus a degree of discretion that specifies the number of ticks the order can walk the LOB and still be acceptable. In other words, for a buy order, the number of ticks included in the discretion specifies the maximum  price the trader is willing to pay to fill the order. Similarly, for a sell order, the number of ticks included in the discretion specifies the minimum price the trader is willing to receive to fill the order. This discretion does not preclude the order from being filled at better prices if the LOB is updated with more favourable prices or quantities.

In our paper we show how to choose the discretion of orders in an optimal way to improve fill ratios over a period (days, weeks, months), while keeping orders exposed to receiving price improvement. Increasing fill ratios is costly. Everything else being equal, the chances of filling an order increase if the order can walk the LOB. Thus, there is a tradeoff between ensuring high fill ratios and the execution costs borne by the trading strategy. In our approach, the dynamic optimisation problem solved by the trader balances this tradeoff by minimising the discretion specified in the marketable orders, while targeting a fill ratio over a trading horizon. The trader's optimal strategy specifies the discretion for each transaction depending on the proportion of orders that have been filled, how far the strategy is from the target fill ratio, the cost of walking the LOB, and the volatility of the exchange rate.

We employ a data set of foreign exchange trades to analyse the performance of the optimal strategy developed here. The data are provided by LMAX Exchange ( We use anonymised transaction data for two foreign exchange traders to compare the fill ratios they achieved in practice to those attainable with the optimal strategy derived in the paper. The data spans a set of dates from December 2016 to March 2017. During this period both traders filled between approximately 80% and 90% of their liquidity taking orders in the currency pair USD/JPY.

We find that the effect of latency on trade fills is exacerbated during times of heightened volatility in the pair USD/JPY. When volatility is arranged in quartiles, we find that between 36% and 40% of unfilled trades occur in the top quartile of volatility.

We employ the optimal strategy developed in our paper to show the tradeoff between increasing fill ratios through the use of discretion and the costs incurred by the strategy. We show that traders could have increased the percentage of filled trades, during the period 5 December 2016 to 30 March 2017, to 99% for both traders. In this example, the average cost incurred by the traders to fill trades missed by the naïve strategy was between 1.24 and 1.76 ticks. On the other hand, the cost of returning to the market 20ms and 100 ms later to fill trades missed by the naïve strategy is between 2.01 and 2.75 ticks respectively.

The performance of the optimal strategy is more remarkable during times of heightened volatility of the exchange rate. In the top quartile of volatility, the average cost of filling missed trades using the optimal strategy is approximately 1.87 ticks, while the mark-to-market average cost of filling the missed trades employing market orders that walk the LOB until filled 100ms later is between 3 and 3.3 ticks.

Finally, we build a function that maps various levels of latency to the corresponding percentage of filled orders. We use this mapping to calculate the shadow price of latency that a particular trader would be willing to pay to reduce the latency of his connection to an exchange. We show that the trader would be better off employing the latency-optimal strategy developed in our paper, instead of   investing in hardware and co-location services to reduce latency. The latency-optimal strategy is superior because it not only achieves the same fill ratios as those obtained with better hardware and co-location, but it scoops price improvements that stem from orders arriving with latency at the Exchange."

The full paper may be downloaded here.

Monday, 23 July 2018

How do Nodal lines for eigenfunctions bring together so many facets of mathematics?

Oxford Mathematician Riccardo W. Maffucci is interested in `Nodal lines for eigenfunctions', a multidisciplinary topic in pure mathematics, with application to physics. Its study is at the interface of probability, number theory, analysis, and geometry. The applications to physics include the study of ocean waves, earthquakes, sound and other types of waves. Here he talks about his work. 

"Of particular interest to me are the lines that remain stationary during membrane vibrations, the so-called `nodal lines'.


Figure1: Nodal lines


The study of these lines dates back to pioneering experiments by Hooke. The alternative name, `Chladni Plates,' derives from Chladni's work (18th-19th century). One wants to understand the fine geometric properties of the nodal lines. In several cases we introduce a randomisation of the model, to examine events occurring with high probability. The number theory aspects of this problem are related to which numbers are representable as a sum of two squares. For instance one may write 10 = 1+9 but 7 is not the sum of two squares. Their understanding is tantamount to the study of integer coordinate points (`lattice points') on circles.


Figure 2: Lattice points on circles (see larger title image for detail)


The merge of ideas from these disciplines has been brought together by the new and exciting research area of `arithmetic random waves'. There are natural generalisations of these two-dimensional concepts to higher dimensions. For instance in dimension three, one is interested in the `nodal surfaces'.


Figure 3: Nodal Surfaces


Here the number theory is related to integers expressible as a sum of three squares, and to the lattice points on spheres. For instance, one question concerns the distribution of the lattice points on the surface of the sphere, and in specific regions of it, as in the picture.


Figure 4: Lattice points on spheres


This is a new and exciting field of research with several recent breakthroughs. The group of academics working in this area is growing rapidly. Watch this space."

For more on this subject click here and here.

Wednesday, 18 July 2018

Oxford Mathematician Ian Griffiths wins Vice Chancellor's Innovation Award for his work on mitigation of arsenic poisoning

Oxford Mathematician Ian Griffiths has won a Vice Chancellor's Innovation Award for his work on mitigation of arsenic poisoning. This work is in collaboration with his postdoctoral research associates Sourav Mondal and Raka Mondal, and collaborators Professor Sirshendu De and Krishnasri Venkata at the Indian Institute of Technology, Kharagpur.

As part of this award a short video was produced explaining the problem and its possible mathematical solution.