Thursday, 6 May 2021

Endre Suli elected Fellow of the Royal Society

Oxford Mathematician Endre Süli's work is concerned with the analysis of numerical algorithms for the approximate solution of partial differential equations and the mathematical analysis of nonlinear partial differential equations in continuum mechanics. 

Born in Yugoslavia, he was educated at the University of Belgrade and did his graduate work as a British Council Scholar at the University of Reading and at St Catherine’s College, Oxford. He received his doctorate from the University of Belgrade in 1985. Endre is a Professor of Numerical Analysis at the University of Oxford and a Fellow of Worcester College, Oxford. 

Oxford Mathematics now has 29 Fellows of the Royal Society among its current and retired members: John Ball, Bryan Birch, Martin Bridson, Philip Candelas, Marcus du Sautoy, Artur Ekert, Alison Etheridge, Ian Grant, Ben Green, Roger Heath-Brown, Nigel Hitchin, Ehud Hrushovski, Ioan James, Dominic Joyce, Jon Keating, Frances Kirwan, Terry Lyons, Philip Maini, Vladimir Markovic, Jim Murray, John Ockendon, Roger Penrose, Jonathan Pila, Graeme Segal, Martin Taylor, Ulrike Tillmann, Nick Trefethen, Andrew Wiles, and Endre himself of course.

Monday, 3 May 2021

Zachary Chase shares Danny Lewin Best Student Paper Award 2021 from SIGACT

There are plenty of awards and prizes for senior mathematicians and scientists. But just as important, and maybe more so, are the awards for those just starting out.

SIGACT (Special Interest Group on Algorithms and Computation Theory) is an international organisation that fosters and promotes the discovery and dissemination of high quality research in theoretical computer science. The Danny Lewin Best Student Paper award is presented by SIGACT each year at the ACM Symposium on Theory of Computing.

Oxford DPhil Mathematician Zachary's winning paper is entitled “Separating Words and Trace Reconstruction.” A deterministic finite automaton is one of the most basic computational models in theoretical computer science. Telling two strings apart is one of the most basic computational tasks. In this paper, progress is made on an old problem of how efficiently one can tell two strings apart with a deterministic finite automaton. The proof methods surprisingly involve complex analysis and connections to other fundamental problems.

The 2021 SIGACT Symposium will take place online from 21-25 June.

Thursday, 22 April 2021

How compliance might help us get back to the workplace after lockdown

Oxford Mathematician Arkady Wey discusses a stochastic agent-based model of the workplace, developed to explore the importance of compliance with test and trace programs following a pandemic lockdown.

This study is inspired by the situation in the UK in the latter half of 2020. However, it is perhaps equally relevant now, as the nation attempts, once again, to restart the economy in the aftermath of a COVID-19 infection peak. One task that policy makers are currently faced with is to simultaneously permit the re-opening of workplaces whilst preventing a new surge in cases. There has been much recent evidence to suggest that rapid testing, contact tracing, and isolation of those in the contact group, is one of the most effective virus containment measures. Policy makers are interested in assessing how the degree of compliance in the population affects the success of this strategy.

In a workplace setting, one crucial measure of compliance is the transparency of the employee workforce. We consider a transparent workforce to be one made up of a large proportion of individuals that isolate at home upon receipt of a positive test, and notify other individuals, with whom they have been in contact, to do the same. A compliant employer might therefore be considered as one who encourages such transparent behaviour among employees.

In our study we assume a world in which rapid testing and results are available to anyone who is symptomatic. Routine contact tracing is used, where contacts of those testing positive are told to self-isolate. In this setting, we explore the effect of the presence of a proportion of non-compliant, or opaque, workers, who are not transparent about their infection-risk status, and do not go home when made aware that one of their colleagues has tested positive. Such opacity may be interpreted as a form of presenteeism, where workers continue to go to work despite possible infection or symptoms.

We pose the questions: is there an incentive for employees to engage with transparent behaviour? And, is there an incentive for employers to encourage it? That is, is it valuable to engage with test, trace, and isolate procedures, in order to halt virus spread while also maximizing productivity?

To answer these questions, we present a discrete-time stochastic agent-based model for the workplace. The basic virus model we choose is a variation of the Kermack-McKendrick SEIR model.

In our model, nodes in a network are agents that represent individuals in the workplace. Each agent is in one of six viral states: Susceptible, Exposed, Infectious-Asymptomatic, Infectious-Symptomatic (Unwell), Quarantined, and Recovered. Agents are either Transparent, in which case they self-isolate at Home when they become unwell, and warn others to do the same, or they are Opaque, in which case they do neither, staying in Work. A diagram of the full set of possible state transitions is presented in Figure 1.



Figure 1: Sketch outlining the operation of the discrete-event model. Here is susceptible, E is exposed (not yet infectious),
A is asymptomatic infectious, U is unwell infectious, Q is quarantined and R is recovered (and immune).
Lines/arrows give all possible transitions between states. More details here.

Edges in the network represent contacts between agents when they are in Work. The kinds of workplaces considered are those with relatively static interaction networks. An example of such a working environment is an office that is split into teams that are physically co-located, with a number of middle managers and service personnel who naturally migrate between several teams. We therefore make the simplifying assumption that the workplace can be represented by a fixed interaction network, some examples of which are shown in Figure 2.

Figure 2: An example of a randomly generated workplace contact network of 15 workers. 1/3 of the workforce is opaque
(solid square) and 2/3 is transparent (semi-transparent circle). One individual is infected with the virus and is therefore
in the Exposed state (orange node labelled 0), whereas all others are Susceptible (green nodes labelled 1-14). More details here.

Our workplace networks are dynamic, though, in the sense that an agent's contacts change as other agents isolate at Home and return to Work. Moreover, we do not consider a completely closed facility, since each agent is assumed to go about their daily business outside of the workplace, where there exists some base level of infection rate.

The results of our simulations point to a double benefit of workforce transparency. That is, we find that a policy of high transparency is optimal for most workplace networks explored, not only for disease spread-prevention, which is perhaps unsurprising, but also for maximal productivity. This is despite the fact that a policy of transparency involves sending a worker home as soon as they are in contact with an infectious colleague. This result holds true in cases where working from home does not significantly diminish the productivity of each individual worker.

Additionally, our results suggest an inverse proportionality between workforce opacity and the average contact degree of the workplace network. That is, the greater the average number of connections to each worker in the workplace, the more transparent the workforce must be to avoid the disease taking hold. This suggests that contact-limiting measures should always be taken alongside transparency policies, since disease spread will not be sufficiently limited in highly connected workplace networks, even by a maximally transparent workforce.

Our study highlights the need for policies that encourage compliance in the workplace population. We outline guiding principles for these policies, such as the no detriment principle, whereby workplace populations engender a culture in which there is no perceived detriment to a worker who is transparent. One associated policy intervention might be the decision to remunerate workers if they are required to self-isolate and to penalize workers if they are found to be at work while knowingly infectious. Another might be the introduction of governmental statutory sickpay for all workers, irrespective of their contractual status, from day one of self-isolation.

Overall, we find that compliant behaviour in the workforce population is mutually beneficial for employees and the employer, and might be necessary for the safe re-opening of workplaces in the post-peak phase of pandemics such as COVID-19.                                 

Sunday, 18 April 2021

Oxford Mathematics Open Days 24th April and 1 May

Study Maths & describe the world.

Mathematics underpins so much of our understanding of our world, whether it be its close relationship with the other Life Sciences or its wider influence on such things as how cities grow or how social media networks operate.

At Oxford Mathematics Open Days you'll get a real sense of where a maths degree can take you. Indeed, as well as Maths you may also want to consider our joint schools in Maths and Statistics, Maths and Computer Science and Maths and Philosophy.

These events will be online-only, with live broadcasts, pre-recorded content, and question-and-answer sessions with our tutors and current students. No registration is necessary.

Find out more 

Wednesday, 14 April 2021

Bernadette Stolz awarded the 2021 Anile-ECMI Prize

The Anile-ECMI Prize is given to a young researcher for an excellent PhD thesis in industrial mathematics successfully submitted at a European university. It was established in honour of Professor Angelo Marcello Anile (1948-2007) of Catania, Italy and consists of a monetary prize of 2500 Euros and an invitation to give a talk at the ECMI 2021 conference on Wedneday 14 April.

In her DPhil (PhD), Bernadette investigated the use of topological data analysis for biological data. She developed methods to quantify the unique features of tumour blood vessel networks. Using persistent homology on experimental data from different imaging modalities, she validated known treatment effects on the networks and showed how the effects of radiation treatments alter the vascular structure.

In her thesis, Bernadette further applied persistent homology to functional networks from neuroscience experiments. To overcome computational challenges that are a major limitation in applications of persistent homology to real-world data, she researched the use of local computations of persistent homology and her results indicate that these can be used for outlier-robust subsampling from large and noisy data sets. In addition, she demonstrated that such computations can detect points located near geometric anomalies in data sampled from intersecting manifolds. This work has recently been published in Proceedings of the National Academy of Sciences of the United States of America (PNAS).

Bernadette developed her research in close collaboration with Oxford Mathematicians Heather Harrington, Jared Tanner, Vidit Nanda, and Helen Byrne as well as Mason Porter (UCLA), biological collaborators from Oxford Radiation Oncology and industrial researchers from Roche.

In her current postdoc at Oxford Mathematics' Centre for Topological Data Analysis, Bernadette is looking at applying persistent homology to quantify the output from mathematical models of angiogenesis.

Saturday, 10 April 2021

Colourings without monochromatic arithmetic progressions

Oxford Mathematician Ben Green on a tale of conjectures, mistaken assumptions and eventual solutions: a tale of mathematics.

"The famous discrete mathematician Ron Graham sadly passed away last year. I did not know him well, but I had the pleasure of meeting him a few times. On the first such occasion, in Vancouver in 2004, he mentioned one of his favourite open questions over lunch. This concerns the size of certain "van der Waerden numbers", a kind of arithmetic variant of graph Ramsey numbers.

Fix a positive integer $k \geq 3$, and let $N(k)$ be the smallest value of $N$ such that the following is true: however you colour the integers $\{1,\dots,N\}$ red and blue, there is always either (i) a blue arithmetic progression of length 3 or (ii) a red arithmetic progression of length $k$ (or both). That there exists such a value of $N(k)$ is not a trivial fact, but it is a consequence of a celebrated theorem of van der Waerden from the 1920s. In fact, it is now known that $N(k)$ grows at most exponentially as a function of $k$.

What about the true value? There is cast-iron numerical data up to about $k = 20$, and conjectured values up to about $k = 40$, obtained with computer searches. The numerics very strongly suggest that $N(k)$ is roughly $k^2$. For instance, it is believed that $N(20) = 389$ and $N(30) = 903$. The question Ron Graham asked me in 2004 was this: is it true that $N(k) < Ck^2$ for some absolute constant $C$, and for all $k$? When Ron asked me this question I immediately told him that I thought the answer was no, and I thought I would send him a proof within a few days. My reasoning was as follows. There are well-known examples of (for instance) subsets of $\{1,\dots, k^{10}\}$ of size $k^{9.9}$ with no three-term arithmetic progressions, provided $k$ is sufficiently large. Take a set like this, and colour it blue. Colour the remaining points red. There are quite a lot of red points but, unless something unexpected happens, one would still not anticipate red arithmetic progressions of length much longer than about $k^{0.1}$, and certainly nowhere near as long as $k$. (One can actually make this rigorous: if the blue points were a random subset of size $k^{9.9}$, then almost surely (as $k \rightarrow \infty$) what I just wrote is true.)

Unfortunately, something unexpected does happen. For all the well-known examples of large sets free of three-term progressions, their complements contain extremely long arithmetic progressions. In particular, none of these examples provide a disproof of Ron Graham's conjecture. I apologized to Ron for not believing his conjecture, and to make amends I repeated it myself in print.

However, in recent work I have shown that the conjecture is, after all, false. Not only is $N(k)$ not bounded by a quadratic, but in fact it is not bounded by any polynomial. It grows at least as quickly as roughly $e^{(\log k)^{4/3}}$.

The red-blue colouring which shows this is rather elaborate. Very roughly, one sets up a discrete-time dynamical system on a high-dimensional torus given by an irrational rotation. On the torus one takes a large, randomly-chosen, collection of very thin ellipsoidal annuli, all with the same eccentricity, but with this eccentricity also chosen randomly. Then, we colour $n$ blue if $n$ is a return time of our dynamical system to this set of annuli. All other $n$ are coloured red.

By far the hardest part of the proof is to show that (with high probability) this colouring does not contain long red progressions. This occupies about 50 pages and uses tools from harmonic analysis, the geometry of numbers, combinatorics and random matrix theory.

Despite this new result, the gap between the known upper and lower bounds for $N(k)$ remains close to exponential and seems likely to remain so for the foreseeable future."

Saturday, 3 April 2021

The many faces and hands of Online Student Lectures

It is a cliche that crises create opportunities. But they certainly demand innovation (and a lot of hard work). In Oxford Mathematics, in line with many others departments and universities, we have had to switch from in-person teaching to online in most cases. This has been 100% the case in terms of undergraduate lectures which normally take place in large lecture theatres where a whiteboard, a marker pen (or two or three) and a mathematician take centre stage.

However, the online world is a much more varied place. Lectures tend to be shorter (though the courses are the same length), some lecturers write as they go using tablets while some use pre-prepared slides. Some are in shot, some are not. However, as you can see from the image above and the lecture below, some lecturers, in this case André Henriques (and also Artur Ekert in other lectures), are trying different approaches, taking advantage of latest technologies. The lightboard is not new, but this might be its teaching moment.

Then again, the important thing is that the teaching is up to scratch. You can judge for yourself via the lecture below. You can also watch a range of student lectures on our YouTube channel as we show what we do and the increasing variety of ways we do it.

Thursday, 25 March 2021

The Oxford SIAM Student Chapter 3 Minute Thesis Competition

Postgraduate students are mathematics' future. 

The Oxford University Society for Industrial and Applied Mathematics Student Chapter 3 Minute Thesis Competition saw 10 of our postgraduates present their latest research to a panel of our judges. Topics included Langland's Grand Unified Theory; Quantum irreversibility; and using magnets and maths to deliver stem cell therapy.

You can watch the competition via the video below. 
Wednesday, 24 March 2021

One Term in 5 Minutes

To gain an insight in to mathematical student life under lockdown, we asked Oxford Mathematics and St Peter's College 2nd Year Undergraduate Matt Antrobus to provide us with one-minute updates over the course of last term.

So he did in a very personable and honest way, describing the maths he is doing, how he is doing it and how much work is involved. Matt also reflects on the stark fact that over half his time in Oxford has been under the cloud of Covid.

You can watch all five films by scrolling through our Twitter, Facebook or Instagram pages.



Friday, 19 March 2021

How growing nerves in the brain behave like light rays

During the early growth of the brain, an extraordinary process takes place where axons, neurons, and nerves extend, grow, and connect to form an intricate network that will be used for all brain activities and cognitive processes. A fundamental scientific question is to understand the laws that these growing cells follow to find their correct target.

A well-known observation is that multiple axons bundle and migrate together towards other neurons to make connections, sometimes very far from the cells where they originate. During this trip, the tip of each axon can only rely upon its near environment to find its path. For example, chemical guidance (chemotaxis) is a major modality of axon navigation: when a "smell" is sensed, the axon tip pulls itself on the substrate to move and elongate the trailing axon towards or away from the signal (1).

Recently, it has been demonstrated that the mechanical environment (stiff or soft) is also critical. In particular, the ability of an axon to progress depends on the stiffness of the substrate (2), potentially allowing for durotaxis, i.e., migration along stiffness gradients (3). In a recent paper published in Phys. Rev. Lett, Oxford Mathematicians Hadrien Oliveri and Alain Goriely in a joint work with neurophysicist Kristian Franze found a surprising connection between classic optic ray theory and axonal migration.

They started with the simple question: if each axon feels the stiffness of the substrate, what will be the overall group behaviour of the bundle? What path will a bundle follow if each axon produces a different force, depending on medium stiffness? Working in the theory of growing filaments (4), they modelled the path of a tip-growing bundle subject to differential traction forces. Then, they considered an idealised system where a bundle moves from a soft to a hard medium. In each separate domain the bundle follows a straight trajectory. However, when the axons approach the interface (say with incident angle $\theta_1$) part of the bundle will grow on the hard domain, while some axons are still on the soft domain. This results in a torque that forces the bundle to turn until all axons have passed the interface, at which point the bundle stops turning and follows a new straight trajectory ($\theta_2$, Fig 1).



                                                                                         FIgure 1: The Snell law of axon durotaxis.

The theory leads to a surprising relationship: $$ n_1 \sin\theta_1 \simeq n_2 \sin \theta_2 $$ where $n_i$ are related to the stiffness of each medium. The same mathematical relationship appears in a completely different setting. In optics, if $n_1$ and $n_2$ are the refractive indices and $\theta_1,\theta_2$ the angle that a light ray makes with the interface, this law is known as Snell's law (or Descartes's law in French). It governs the deflection of light rays at the interface between two refractive media (for example air and water) and is a consequence of Maxwell's equations for electromagnetic radiations. This law explains, for instance, why a wood stick appears broken when partially submerged in water.

Using this theory, the authors showed how durotaxis can be used to explain the guidance of xenopus retinal ganglion cell axons, originated in the retina (3, Fig 2).

This analogy between the path of a light ray and the path of axons in the developing brain is potentially powerful. Indeed, we know from the theory of optics that ingenious devices such as lenses, mirrors, optical guides, collimators, binoculars, periscopes, telescopes and microscopes can be built to control the path of light rays and collect information. Similarly, the authors show how one can design substrate with different stiffness that would induce lensing effects, or create the equivalent of an optic fibre with a soft corridor to guide axons during development or during regeneration. Their new work provides a foundation for a general theory of axon guidance and control.


Figure 2: the role of tissue stiffness in Xenopus lævis visual system development. Left: cartoon showing the path of the axons from the retina, through the optic nerve to the optic tectum. Right: numerical simulation of retinal ganglion cell axons undergoing a sharp caudal turn at their arrival in the mid-diencephalon ($N=5$ representative trajectories). The authors show how durotaxis has the potential to contribute to this turn.


1] K. Franze, “Integrating Chemistry and Mechanics: The Forces Driving Axon Growth,” Annual Review of Cell and Developmental Biology, vol. 36, Oct 2020.

[2] C. E. Chan and D. J. Odde, “Traction dynamics of filopodia on compliant substrates,” Science, vol. 322, pp. 1687–1691, Dec 2008.

[3] D. E. Koser, A. J. Thompson, S. K. Foster, A. Dwivedy, E. K. Pillai, G. K. Sheridan, H. Svoboda, M. Viana, L. da Fontoura Costa, J. Guck, C. E. Holt, and K. Franze, “Mechanosensing is critical for axon growth in the developing brain,” Nature neuroscience, vol. 19, no. 12, p. 1592, 2016.

[4] D. E. Moulton, T. Lessinnes, and A. Goriely, “Morphoelastic rods. Part I: A single growing elastic rod,” Journal of the Mechanics and Physics of Solids, vol. 61, pp. 398–427, Feb 2013.