Sunday, 5 April 2020

How do mathematicians model infectious disease outbreaks? ONLINE Oxford Mathematics Public Lecture 5pm, 8 April

Models. They are dominating our Lockdown lives. But what is a mathematical model? We hear a lot about the end result, but how is it put together? What are the assumptions? And how accurate can they be?

In our first online only lecture Robin Thompson, Research Fellow in Mathematical Epidemiology in Oxford, will explain. Robin is working on the ongoing modelling of Covid-19 and has made many and varied media appearances in the past few weeks. We are happy to take questions after the lecture.

Wednesday 8 April 2020

Watch live:

Oxford Mathematics Public Lectures are generously supported by XTX Markets

Sunday, 5 April 2020

The Oslo International Congress of Mathematicians in 1936 and the first Fields Medals

The International Congress of Mathematicians (ICM) that was held in Oslo in July 1936 was a unique event that took place in turbulent times, research by Oxford Mathematician Christopher Hollings has revealed. The Nazis had been in power in Germany since 1933, and their dismissal of Jewish scholars from university posts had already had a profound effect on academia the world over.  In March 1936, Germany had remilitarised the Rhineland, in violation of the Treaty of Versailles, and sought further to enhance its international standing by hosting the Summer Olympics in August that year.  In October 1935, troops from Mussolini’s Italy had invaded Ethiopia (then Abyssinia), to widespread international condemnation, and in the USSR, Stalin was strengthening his grip on power and was about to unleash his Great Terror.  All of these events, either directly or indirectly, had their impact on the Oslo ICM.

During the opening speeches of the congress, the spirit of international cooperation was strongly invoked, and yet the participants could not have helped but be aware of the ways in which world politics was affecting their meeting.  A number of German mathematicians who might have been expected to attend did not appear since the Nazi authorities had denied them the right to travel.  Other mathematicians who had been dismissed from their posts in Germany were a visible presence in Oslo, hoping that they might find jobs elsewhere.  Italian mathematicians, on the other hand, were conspicuous by their absence – they too had been denied the right to travel to the congress, in response to Norway’s involvement in the sanctions that had been imposed on Italy by the League of Nations.  Soviet mathematicians were also kept at home – many of them were then involved in the so-called ‘Luzin affair’, an ideological attack launched by the Academy of Sciences against the Moscow function theorist N. N. Luzin.

These absences had a noticeable effect in particular on the mathematical profile of the congress: algebraic geometry – then a subject dominated by Italian mathematicians – was entirely absent, whilst the coverage of both probability and topology was rather narrower in scope than originally planned by the organisers, owing to the prominence of Soviet mathematicians in these fields.  On the other hand, number theory, hailed by some less-than-neutral commentators as a great German subject at this time, was very strongly represented, accounting for around one third of the plenary lectures.

Two topics that were certainly visible at the Oslo congress were those relating to the award of the Fields Medals.  The idea of an international prize in mathematics had first been suggested in the early years of the twentieth century, resulting in the creation of the so-called ‘Guccia Medal’ – but this was awarded only once: to the Italian mathematician Francesco Severi in 1908.  The Fields Medals, funded by and named for the Canadian mathematician John Charles Fields, and now presented every four years to researchers under 40, were awarded for the first time at the Oslo congress in 1936: to the American Jesse Douglas and the Finn Lars Ahlfors.  The latter received his Medal for work in the theory of functions of a complex variable, whilst Douglas’ award was for his solution of Plateau’s Problem, concerning the existence of a minimal surface for a given boundary – a problem that drew its inspiration from experiments with soap films carried out by the Belgian physicist Joseph Plateau.

A few surviving documents hint at possible intrigue connected with the award of the prize to Douglas.  Not all members of the Fields Medal Committee were happy with the way in which the decision-making process had been handled, and there are suggestions that the congress organisers may have tried to suppress a contributed talk that took the study of Plateau’s Problem somewhat beyond Douglas’ work.  Whatever the truth of this, the award of these first prizes to two US-based mathematicians signalled the fact that American mathematics had now quite definitely stepped out from beneath the shadow of its European counterpart.  The way was paved for the next ICM to be held in the USA – although this didn’t take place until 1950, and then under very different circumstances.

Christopher Hollings is Departmental Lecturer in Mathematics and its History, and Clifford Norton Senior Research Fellow in the History of Mathematics at The Queen’s College, Oxford.  Further background to the ICMs can be found in a prior research case study. A detailed account of the Oslo ICM, both mathematical and political, can be found in the book 'Meeting under the Integral Sign? The Oslo Congress of Mathematicians on the Eve of the Second World War' by Christopher Hollings and Reinhard Siegmund-Schultze (American Mathematical Society, 2020).

Monday, 30 March 2020

Mobile phone data and Covid-19 - missing an opportunity?

The Coronavirus disease pandemic (COVID-19) poses unprecedented challenges for governments and societies around the world. In addition to medical measures, non-pharmaceutical measures have proven to be critical for delaying and containing the spread of the virus. However, effective and rapid decision-making during all stages of the pandemic requires reliable and timely data not only about infections, but also about human behaviour, especially on mobility and physical co-presence of people. 

Seminal works on human mobility have shown that mobile phone data can assist the modelling of the geographical spread of epidemics, and several initiatives involving researchers and governments have started to collaborate with private companies, most notably mobile network operators, to estimate the effectiveness of control measures or the real-time mobility flows in a country. However, in most instances, too little has been done too late, and there is currently hardly any coordination or information exchange between national or even regional initiatives.

In collaborations with researchers from NGOs, mobile phone companies and Universities around the globe, Oxford Mathematician Renaud Lambiotte explores in what ways and how mobile phone data can help to better target and design measures to contain and slow the spread of the COVID-19 pandemic. It has long been recognised that mobile ​phone traces can be exploited to infer human mobility and social interactions. Researchers have used various types of mobile network data (e.g. Call Details Records, x Data Records and passive records), as well as GPS and mobile application data. Critically, some of this data, such as CDRs, are collected by mobile network operators for billing reasons, and each record contains information about the time and (approximate) geographical location of customers' interactions with the phone, therefore allowing us to estimate individual or aggregated trajectories. These data would have been, and still are, useful in helping predict and control the course of the epidemic, for instance by providing dynamical origin-destination matrices, hotspot identification, or contact matrices. 

At present, there is still a scarce strategic use of mobile phone data for tackling the COVID-19 pandemic, except for notable successes in China and South Korea. This failure originates from a combination of factors, including the difficulty to access data, a lack of communication between the research community and governments, and privacy concerns. Yet, it is not too late to exploit the potential of mobile phone data to flatten the curve and help prevent second waves from emerging, and the authors propose a range of concrete steps in a call for data-driven action against COVID-19 both now and in case it should re-emerge in a second wave, including:

* Early liaison with governments, data protection authorities and civil liberties advocates

* A strategy for data preparation for all stages of the pandemic

* International collaboration on good practices and code sharing

Read the full article here

Wednesday, 25 March 2020

Exercise of employee stock options with privileged information

Executive stock options (ESOs) are contracts awarded to employees of companies, which confer the right to reap the profit from buying the company stock (exercising the ESO) at or before a fixed maturity time $T$, for a fixed price specified in the contract (the strike price of the ESO). ESOs are used to augment the remuneration package of employees, the idea being to give them an incentive to boost the company's fortunes, and thus the stock price, making their ESO more valuable.

It has been argued that holders of ESOs could exploit their inside information on the company to advantageously exercise in advance of upcoming bad news that would cause the stock price to fall, news which is yet to be publicly revealed. Empirical studies of ESO exercise have indicated that such options tend to be exercised prior to poor stock performance. The question is whether this is due to some informational advantage.

There have been few if any mathematical models which have been able to capture such potential information effects. Oxford Mathematicians Michael Monoyios and Christoph Reisinger and co-authors have constructed models which incorporate privileged information on the evolution of a stock price, to analyse and illustrate exercise patterns in the presence of inside information.

One such model (1) features a stock whose expected return (or drift) parameter suffers a change point at an exponentially distributed random time $\theta$, independent of the Brownian motion driving the stock; its drift falls from its initial constant value $\mu_{0}$ to a lower constant value $\mu_{1}<\mu_{0}$.

In this scenario we studied two optimal stopping problems, to find the best exercise time for agents with different information on the change point. The two problems are distinguished by full information, in which the change point as well as the stock price is observed, or by partial information, in which the change point is not observable, and so is filtered (or estimated) from observations of the stock price. The fully informed agent is to be thought of as a senior executive, privy to upcoming bad news (think of the VW emissions scandal), while the partially informed agent is a less senior employee.

The information differential leads to the two agents having different perceptions of the stock price drift, so they perceive different dynamics for the stock price. Mathematically, the different information flows are modelled by filtrations: sets of events (increasing in size as time flows) which can be determined
to have happened (or not) by each agent. The stock drift for the fully informed agent is $\mu(Y)$, given by
\mu(Y_{t}) := \mu_{0}(1-Y_{t}) + \mu_{1}Y_{t} = \left\{\begin{array}{ccc}
\mu_{0}, & \mbox{on} & \{t<\theta\}=\{Y_{t}=0\}, \\
\mu_{1}, & \mbox{on} & \{t\geq\theta\}=\{Y_{t}=1\},
\end{array}\right. \quad t\in[0,T],                   
Y_{t} := \left\{\begin{array}{ccc}
0, & \mbox{on} & \{t<\theta\}, \\
1, & \mbox{on} & \{t\geq\theta\},
\end{array}\right. , \quad t\in[0,T],  
is the indicator process that the change point has occurred. On the other hand, the drift seen by the partially informed agent is a diffusion process given by
\mu(\widehat{Y}_{t}) := \mu_{0}(1-\widehat{Y}_{t}) +
\mu_{1}\widehat{Y}_{t},  \quad t\in[0,T],
with $\widehat{Y}$ a diffusion in $[0,1]$ representing the partially informed agent's best estimate of the process $Y$.

We analysed the optimal exercise problems for each agent in these models, both featuring random drift processes (and this is where the mathematical and numerical challenge lies), to examine how the different information could lead to advantageous exercise for the fully informed agent.

The fully informed agent's strategy is characterised by two exercise thresholds, a pair of ordered, non-increasing, time-dependent exercise boundaries $x^{*}_{0}(t)\geq x^{*}_{1}(t),\,t\in[0,T]$, such that optimal early exercise occurs in the state where the drift is $\mu_{i},i\in\{0,1\}$, as soon as the stock breaches $x^{*}_{i}(\cdot)$ from below, or if such a breach is triggered by the change point.

On the other hand, the partial information exercise boundary is a surface $x^{*}(t,y),\,t\in[0,T],\,y\in[0,1]$, with an additional spatial dependence on a variable $y\in[0,1]$, arising from the dependence of the drift on the filtered change point process $\widehat{Y}$, and such that the partial information exercise surface lies between the full information exercise thresholds.

We showed that this can lead to an interesting range of possible exercise patterns, such as immediate exercise by the fully informed agent in response to the change point, a strategy unavailable to the agent who does not see the jump in drift. In this vein, the figure below shows some simulations which illustrate the possible exercise patterns, indicating how complete information on the change point leads to a greater exercise profit.


Image above: simulations of stock price path, full information exercise boundaries (blue) and a cross-section of the partial information exercise boundary (red). Exercises of the fully informed agent are marked with circles, with those of the partially informed agent marked by squares. In each panel, the shaded background indicates the switch in drift regime to $\mu_1 < \mu_0$.

An earlier model for ESO exercise with inside information was constructed by Monoyios and DPhil student Andrew Ng (2). Here, the inside information was modelled by assuming the agent had advance information on the value of the stock at the ESO maturity time. Mathematically, one uses techniques known as enlargement of filtration to compute the dynamics of the stock price as seen by the insider. Once again, the informational advantage was evident in the exercise strategy.

Full details of these models appear in the two papers below:

[1] V. Henderson, K. Kladıvko, M. Monoyios, and C. Reisinger, Executive stock option exercise with full and partial information on a drift change point. arXiv preprint 1709.10141, 2020.

[2] M. Monoyios and A. Ng, Optimal exercise of an executive stock option by an insider, Int. J. Theor. Appl. Finance, 14 (2011), pp. 83–106

Monday, 23 March 2020

Coronavirus Modelling - why social distancing works

Social distancing measures to reduce the spread of the novel coronavirus are in place worldwide. These guideline are for everyone. We are all expected to reduce our contact with others, and this will have some negative impacts in terms of mental health and loneliness, particularly for the elderly and other vulnerable groups. So why should we follow measures that seem so extreme? The answer is simple. Social distancing works. It reduces transmission of the virus effectively and lessens the impact on already stretched healthcare services. Oxford Mathematics's Robin Thompson explains.


"To illustrate this point, we can consider the effect that a social distancing strategy can have. Without these measures, an individual with COVID-19 can be expected to infect around three others over the course of their infection. One “generation” of infection takes around one week. So after one week, one infected individual will have become four infected individuals (the original infected, plus the three individuals they have infected). After a further week, each of the three new infected individuals can be expected to cause three new infections, leading to 13 infections in total. This compounding effect continues, so that after six weeks the initial infected individual will have started a chain of transmission that has led to over one thousand infections.

In contrast, however, if everyone in the population reduces their contacts by one third, then each infected individual will only infect two others over the course of their infection. This means that after one week, there will be three infected individuals in total. This is similar to the first scenario. But, after six weeks, if everyone reduces their contacts by one third, the chain of transmission will have caused 127 infections. While this is still a large number of infected individuals, it is substantially fewer than the scenario of over one thousand infections that occurs without social distancing.

Of course, widespread control measures have major effects on our lives. A colleague of mine in China – where case numbers have now fallen dramatically due to severe interventions – described to me how he had his temperature taken whenever he was driving his car on the highway as part of a “spot check”, and how he has to apply for a permit to even walk across his university campus. Measures like those may not be enforced in the same way in the west. But, if everyone adopts them, social distancing efforts here to reduce transmission can still be expected to have significant effects. Social distancing is particularly important because we do not know how much transmission is occurring from individuals displaying no or few symptoms. As a result, we should not try and self-determine whether or not we pose an infection risk. Instead, we should all take social distancing seriously, and support each other to follow these guidelines during this challenging time.

The handling of this pandemic worldwide has not been perfect. Only a couple of weeks ago, Donald Trump stated that COVID-19 is no worse than influenza, despite the fact that it has been clear to experts for weeks that the death rate is ten times higher than either seasonal flu or the H1N1 “swine flu” virus that swept around the world in 2009. But let’s face it – when this virus surfaced in early January, none of us could have imagined the scenario we are in now. The best thing we can do going forwards is put these things behind us. This novel coronavirus will continue to have an impact over the next weeks and months, and possibly beyond that. While looking out for one another, particularly those in vulnerable groups, we should do everything we can to stop this virus. Social distancing is of clear public health importance."

Robin Thompson's work focuses on using mathematical models to represent the epidemiological or evolutionary behaviour of infectious disease outbreaks. He has made many media appearances in recent weeks as the world grapples with the implications of the Coronavirus. In January he produced an early case-study on the potential of the virus. And on 23 March he gave an interview to ITV News where he expanded on the points in this article.

Tuesday, 17 March 2020

Konstantin Ardakov awarded the 2020 Adams Prize

Oxford Mathematician Konstantin Ardakov has been awarded the 2020 Adams Prize. The Adams Prize is awarded jointly each year by the Faculty of Mathematics, University of Cambridge and St John’s College, Cambridge to UK-based researchers, under the age of 40, doing first class international research in the Mathematical Sciences. This year’s topic was “Algebra”, and the prize has been awarded jointly to Konstantin and Michael Wemyss (University of Glasgow).

Professor Mihalis Dafermos, Chair of the Adams Prize Adjudicators, said: "Prof Ardakov has made substantial contributions to noncommutative Iwasawa theory, and to the p-adic representation theory of p-adic Lie groups. In a long-term collaboration with Simon Wadsley, he has developed a p-adic analogue of the classical theory of D-modules, of significance both in representation theory and to the local Langlands program.

The Adams Prize is named after the mathematician John Couch Adams and was endowed by members of St John’s College, Cambridge. It is currently worth approximately £15,000. It commemorates Adams’s role in the discovery of the planet Neptune, through calculation of the discrepancies in the orbit of Uranus.

Monday, 16 March 2020

How to reduce damage when freezing cells

Oxford Mathematician Mohit Dalwadi talks about his work on the modelling of cryopreservation.

"While Captain America was able to survive being frozen for over six decades in the Marvel Universe, we are a long way from this type of technology in real-life. At the moment, people can only preserve small numbers of cells when freezing them. This technology is known as cryopreservation, and it works because metabolic processes grind to a halt at low temperatures, essentially keeping cells in suspended animation. Cryopreservation has the potential to help protect endangered species, preserve tissue for organ transplants, and improve food security.

The major challenge in cryopreservation is to ensure that cells are not damaged during the freezing or thawing processes. One source of damage is ice formation; larger, spikier ice crystals are more likely to rip the cell apart. The size and shape of ice crystals depends on how quickly you cool the mixture you are trying to freeze, and so this is one key factor that needs to be understood to improve cryopreservation protocols.

Incredibly, some cold-blooded animals are able to naturally survive being frozen. There are certain species of frogs and fish that can survive being frozen overnight. They do this by releasing sugars into their bloodstream, which act as anti-freeze and reduce the ice formation within their bodies. Humans are trying to develop technology that mimics these frogs by adding cryoprotective agents (CPA) into solutions containing cells they want to freeze. While this does help to reduce ice formation, it comes with its own problems – CPA can be toxic to cells that have not evolved to deal with it.

This means that if we want to try and freeze cells, we have to be very careful about getting the right balance of CPA addition and cooling rate – there is a delicate balance between the two. As you might imagine, it takes a lot of time and effort to determine this optimal balance through experiments. And just because something works for one type of cell, there is no guarantee that it will work for another. By developing mathematical models of these cryopreservation procedures, we can quickly evaluate their efficacy, and determine the optimal protocols for given cell characteristics.

As the solution mixture of cells and CPA is cooled, ice will form and decrease the volume of liquid in the system. This will concentrate the CPA and other solutes, causing cells to dehydrate through osmosis. To account for this mathematically, we must track the heat and mass transfer in the system, which are coupled due to the depressing effect of CPA on the freezing temperature, and account for the moving boundaries at the freezing front and the cell membrane. While these moving boundaries can be computationally challenging to simulate, we are able to systematically reduce the complexity of the model for the freezing of a single cell (e.g. a human egg) by exploiting the disparate timescales over which transport mechanisms occur.

Our analysis shows that while the temperature of the liquid settles down to a spatially uniform value over a matter of seconds, the chemical concentrations take several minutes to do the same. Moreover, we find that the motion of the moving boundaries is strongly coupled to the chemical mass transport. While the freezing front motion and the CPA concentration are both forced by the decrease in temperature, over the timescale of a few minutes the motion of the freezing front is slowed down by CPA being pushed out and building up just ahead of the advancing front. The cell membrane motion occurs over a longer timescale of many minutes, and this motion is mainly governed by the concentration difference across the cell membrane. We are able to use these results to provide predictions on what sort of CPA levels and cooling rates cryobiologists should be trying to aim for, given the cell characteristics.

Find out more about this work which was carried out with colleagues Sarah Waters, Helen Byrne and Ian Hewitt."

Image above: Ice forming within a cell

Image above: model predictions for supercooling at the cell centre over time for different cooling rates.




Friday, 13 March 2020

Coronavirus (Covid-19): advice and updates

Andrew Wiles Building

The University has announced numerous steps to prioritise the health and welfare of staff, students and visitors in the light of the UK’s escalating coronavirus situation. This is an unprecedented and challenging time for our university and department community, and I would ask that you please support each other wherever you can, and follow University guidance, which is continuously updated. MI staff and students should also check their emails regularly for further guidance.

Mike Giles, Head of Department

Wednesday, 11 March 2020

Oxford Mathematicians win 2019 PNAS Cozzarelli Prize

Oxford Mathematicians Derek Moulton and Alain Goriely together with their colleague Régis Chirat (University of Lyon) have won the 2019 PNAS Cozzarelli Prize in the Engineering and Applied Sciences category for their paper 'Mechanics unlocks the morphogenetic puzzle of interlocking bivalved shells.'

The paper describes how two groups of animals—brachiopods and bivalve mollusks—sport interlocking shells that help guard against predators and environmental perturbations, and explains how those shells are formed.

The Cozzarelli Prize is awarded annually to six research teams whose PNAS (Proceedings of the National Academy of Sciences of the United States of America) articles have made outstanding contributions to their fields. Each team represents one of the six classes of the National Academy of Sciences.


Thursday, 27 February 2020

Corrections in infinite dimensions

Several well-known formulas involving reflection groups of finite-dimensional algebraic systems break down in infinite dimensions, but there is often a predictable way to correct them. Oxford Mathematician Thomas Oliver talks about his research getting to grips with what structures underlie the mysterious correction process.

"In algebraic language, a reflection is an order two symmetry. What this means in practical terms is that to invert a reflection, one needs to reflect again. Reflection groups are formed by composing reflections in different mirrors, the effect of which can be visualised using a kaleidoscope.

A Weyl group is a special type of reflection group. Weyl groups describe the reflective symmetries of root systems, which are configurations of vectors with prescribed geometric properties. Root systems and their Weyl groups played a pivotal role in a key achievement of modern mathematics, namely the classification of semisimple finite-dimensional Lie algebras.

There are more general theories of infinite-dimensional Lie algebras, which tentatively have applications to arithmetic and physics. The finite-dimensional theory is not exactly valid, but it can often be "corrected" in a predictable way. In fact several important equations need to be adjusted by the same correction factor.

When mathematicians see the same factors appearing in different contexts, they demand an explanation. In the case of the correction factor, their appearance is due to a new phenomenon for root systems of infinite-dimensional Lie algebras. This is the existence of imaginary roots, which have the unlikely sounding property of having negative length. This is nothing more than colourful language, in which "length" is a word for an inner product.

We calculated the correction factor as an infinite sum over the roots and analysed its "support", that is, the roots contributing non-zero terms to the sum. We found that the reflection group swapped roots around within the support, but never took one outside. That is, the support is Weyl group invariant. This corresponds to a key property of the imaginary roots: no matter what combination of reflections you try, you can never pass from an imaginary root to a real one. Because finite-dimensional Lie algebras do not have any imaginary roots, our analysis in fact gives a new proof of several classical formulas.

The research was carried out with Kyu-Hwan Lee and Dongwen Liu."

Image above: the action of reflection groups can be visualised like a kaleidoscope.