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. 


Saturday, 14 July 2018

The Mathematics of Smoothies - the Dynamics of Particle Chopping in Blenders and Food Processors

Have you ever forgotten to replace the lid of the blender before beginning to puree your mango and passion-fruit smoothie? If you have, you'll have witnessed the catastrophic explosion of fruit and yoghurt flung haphazardly around the kitchen in an unpredictable manner. This is a consequence of the complicated and turbulent fluid dynamics present within the machine, the exact behaviour of which is unknown. Sharp, angular blades rotate at extremely high speeds to mix and chop the fruit into a puree consisting of particles that are ideally as small and uniform in size as possible. But what characteristics of the blender are responsible for the outcome? While experimental evidence gives intuition into blade and vessel design, along with operational parameters such as speed and blend time, there is a knowledge gap surrounding the precise impact on the particle and fluid dynamics.

Oxford Mathematicians Caoimhe Rooney, Ian Griffiths and Colin Please worked with Chuck Brunner, James Potter and Max Wood-Lee from SharkNinja, the company responsible for Nutri Ninja blenders, to understand the chopping dynamics that take place in a blender, with the aim of shedding light on this complex process.

The team derived an integro-differential-equation system, inspired by Becker-Döring and Smoluchowski theory, which provides a predictive model for the resulting size distribution of particles comprising a smoothie after blending an initial mixture of fruits (such as the contents of the blender shown in the figure) for a given amount of time.

The results of the model were found to agree well with experimental trials performed in house (see figure). An unexpected result was the emergence of a second peak in the size distribution of chopped pieces. This is attributed to the fact that each time the blade slices through a piece of fruit, some small debris is also formed. The team modified their model to account for this additional feature, which enabled the second peak to be predicted.    

The taste and texture of a smoothie is heavily dependent on the size and distribution of the pieces from which it is composed. Given an initial selection of fruit pieces, along with the blend time and blend speed, the model is able to predict how the distribution of particle sizes and the most common piece size changes with time during blending. This provides guidance on the optimal blend time to maximize the taste experience.

The work performed by the team forms a foundation for the exploration and optimization of food blenders. In particular, this work paves the way for understanding the complex interplay between fluid dynamics and chopping within a blender. Ultimately, these models will allow us to determine the precise operating regime that will create the most homogeneous smoothies in the most efficient manner. 


For more information click here.

The images above feature the NutriNinja blender and a comparison between theoretical prediction (red) and experimental data (grey) for the distribution of different particle sizes in a blender after 50 seconds of blending.

Friday, 29 June 2018

Oxford Mathematician Heather Harrington awarded Whitehead Prize

Oxford Mathematician Heather Harrington has been awarded a Whitehead Prize by the London Mathematical Society (LMS) for her outstanding contributions to mathematical biology which have generated new biological insights using novel applications of topological and algebraic techniques. 

In the words of the citation Heather "has made significant advances through the application of ideas originating in pure mathematics to biological problems for which the techniques of traditional applied mathematics are inadequate. This has involved in particular the development of methods in algebraic statistics which allow one to characterize the qualitative behaviour of dynamical systems and networks, adapting approaches from algebraic geometry to test whether a given mathematical model is commensurate with a given set of experimental observations."