Sunday, 3 May 2020

Invariant theory for Maximum Likelihood Estimation

Oxford Mathematician Anna Seigal talks about her work on connecting invariant theory with maximum likelihood estimation.

"A widespread problem in statistics is to fit a model to data. Given a model, which is believed to describe some data, the point in the model that best fits the data is sought. This point is called the maximum likelihood estimate (or MLE) given the data. For example, the probability of surviving a disease can be estimated from a sample of people who have had the disease: the MLE is obtained by dividing the number of people who survived by the total number of people in the sample. Another example of an MLE is to find a line of best fit that relates two variables, assuming one variable depends linearly on the other.

Often statistical models are more complicated structures than straight lines, as in the following picture. A statistical model, represented by the black curve, lies in a space, represented by the red triangle. The observed data gives the blue point $\bar{u}$ in the space. The MLE is a point in the model that is closest to the data, in the sense that it was most likely to give rise to the observed data.


Many different approaches can be used to search over a model to find a good fit to data. There is growing interest in understanding the mathematical structure that underpins maximum likelihood estimation. This structure allows us to obtain theoretical guarantees, and to answer questions such as: how can we test that we have reached an optimal point in the model? How many optimal points do we expect to find? How much data is required before the MLE can even exist?

In a recent preprint we apply algebraic methods from invariant theory to the problem of maximum likelihood estimation in various statistical models: log-linear models in the discrete setting and multivariate Gaussian models in the continuous setting. We describe algebraic approaches to finding the MLE, and we characterise when the MLE exists, in terms of orbits of points under group actions.

Classically, invariant theory studies orbits under group actions, orbit closures, and the equations that vanish on them. The orbit of a point under a group is the set of points that can be obtained from it by acting by a group element. We can summarise information about an orbit using notions of stability. For example, a point that can be scaled arbitrarily close to zero under a group action is called unstable. These invariant theory structures were studied classically by mathematicians including David Hilbert and Emmy Noether. More recently, numerical and algorithmic approaches to study the stability of points under group actions have become possible.

This picture describes our invariant theory set-up. We have an orbit of a point under the action of a group, represented by the black curve in the picture. The orbit lies inside a space, represented by the red square. For each point in the orbit, we can compute its distance to the origin. We seek the point in the orbit that is closest to the origin. Another way to say that a point is unstable is that this distance to the origin gets arbitrarily small, i.e. the orbit contains the origin in its closure.


In our preprint, we build a dictionary that translates between these two pictures: between properties of the MLE (such as existence and uniqueness) and stability of a corresponding orbit under a group action. This connection enables us to find new conditions for MLE existence, as well as suggesting a more general class of statistical models, which we call Gaussian group models.

This research is joint work with Carlos Améndola at TU Munich, Kathlén Kohn at KTH Stockholm, and Philipp Reichenbach at TU Berlin. Although we most enjoyed working on the project together in person, we were still able to have a good time when finishing our preprint under strict social distancing (with a closest pairwise distance of more than 300 miles)." 


Friday, 1 May 2020

High dimensional footballs are almost flat

Oxford Mathematician Ben Green on how and why he has been pondering footballs in high dimensions.

"A 3-dimensional football is usually a truncated icosahedron. This solid has the virtue of being pleasingly round, hence its widespread use as a football. It is also symmetric in the sense that there is no way to tell two different vertices apart: more mathematically, there is a group of isometries of $\mathbf{R}^3$ acting transitively on the vertices.

In a recent paper I showed that, perhaps surprisingly, high-dimensional footballs are almost flat. More precisely, a finite transitive subset of the unit sphere in $\mathbf{R}^d$ (the vertices of the football) has width bounded above by a constant times $1/\sqrt{\log d}$: this means that you can rotate the football so that the first coordinates of all the points in it satisfy $|x_1| \leq C/\sqrt{\log d}$.

The bound is sharp, because there does exist a $d$-dimensional football whose width is at least a constant times $1/\sqrt{\log d}$. Its vertices consist of all permutations and all sign combinations of $\frac{1}{\sqrt{H_d}}(\pm 1, \pm \frac{1}{\sqrt{2}}, \dots, \pm \frac{1}{\sqrt{d}})$, where $H_d$ is the harmonic mean $\sum_{i = 1}^d \frac{1}{i}$.

I didn't set out to study high-dimensional footballs for their own sake. The question came up in some work that David Conlon and Yufei Zhao (then both at Oxford) were doing on eigenvalues of quasirandom graphs, a topic in combinatorics.

The proof that high-dimensional footballs are almost flat doesn't use any geometry. Rather, it relies on a bit of representation theory, some inequalities having their origin in work of Selberg on the large sieve in number theory, and most importantly some group theory closely related to work done by Oxford's Michael Collins 15 or so years ago. In particular, it depends on the Classification of Finite Simple Groups."

Wednesday, 29 April 2020

Oxford Mathematician Ehud Hrushovski elected Fellow of the Royal Society

Congratulations to Oxford Mathematician Ehud Hrushovski who has been elected Fellow of the Royal Society (FRS). Ehud is Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He studied in the University of California, Berkeley, and worked in Princeton, Rutgers, MIT and Paris and for twenty five years at the Hebrew University in Jerusalem before coming to Oxford.

Ehud's work is concerned with mapping the interactions and interpretations among different mathematical worlds. Guided by the model theory of Robinson, Shelah and Zilber, he investigated mathematical areas including highly symmetric finite structures, differential equations, difference equations and their relations to arithmetic geometry and the Frobenius maps, aspects of additive combinatorics, motivic integration, valued fields and non-archimedean geometry. In some cases, notably approximate subgroups and geometric Mordell-Lang, the metatheory had impact within the field itself, and led to a lasting involvement of model theorists in the area. He also took part in the creation of geometric stability and simplicity theory in finite dimensions, and in establishing the role of definable groups within first order model theory. He has co-authored papers with 45 collaborators and has received a number of awards including the Karp, Erdős and Rothschild prizes and the 2019 Heinz Hopf prize.  

Oxford Mathematics now has 27 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, Ioan James, Dominic Joyce, Jon Keating, Frances Kirwan, Terry Lyons, Philip Maini, Jim Murray, John Ockendon, Roger Penrose, Jonathan Pila, Graeme Segal, Martin Taylor, Ulrike Tillmann, Nick Trefethen, Andrew Wiles, and Ehud himself of course.

Wednesday, 22 April 2020

Smartphones, Open Book Exams and Cats - how Oxford Mathematics is teaching and assessing in a lockdown world

How do you handwrite maths during a video teaching session?  How do students submit handwritten work electronically?  Are cats allowed to attend tutorials?

These are the kinds of questions that many mathematicians in universities around the world are suddenly grappling with, as we shift our teaching and learning, and our assessment, online.  We in Oxford Mathematics together with colleagues across the University have had to move very quickly to find new ways to teach and assess to the standards which we and our students expect; just as importantly, we need to support and keep in touch with our students who are now separated from us and each other in many countries across the world.

So what have we done?
In line with University guidance, we have made modifications to plans for exams for our third-year, fourth-year and MSc students, so that they can still complete their courses and (where relevant) graduate this summer.  They will sit their exams remotely, as open book exams.  And yes, importantly, they can write their solutions by hand as usual, and then submit them as a pdf.  Our second-year students will sit their exams in the next academic year instead, while our first years will progress to next year automatically, but still have the opportunity to demonstrate and receive feedback on their progress and achievement at the end of this academic year.

Many of our undergraduate students have returned home, although some remain in Oxford and are being supported by colleges.  Consequently we have been considering students' differing circumstances when planning our teaching and learning activities for the term: students will have a variety of devices and levels of internet access, and are in time zones right round the world.  Our lectures for first-year and second-year students will be delivered by prerecorded videos, available for students to watch at any time.  Gone are the days of 9 o'clock lectures (or 9.05am lectures if we are honest)!  Colleges are making provision for online tutorials and classes.  We will support third-year, fourth-year and MSc students through a mix of written content, prerecorded video, and live, interactive sessions. 

This all sounds fine, but of course all involved need to feel comfortable to the point where they can concentrate on the mathematics and not worry about the technology. We have created advice for staff and students on teaching online, and many have already attended online practice sessions to connect and to explore the different solutions available. For example, one good tool for many will be the smartphone-as-visualiser, with the key step being to get the pile of books just the right height before the phone is balanced on top of it. Who says smartphones have taken over our lives (see photo)?

We recognise the demands that these changes are placing on students and staff, and we are aware that we will only know how well they are working once they start - they can't replace face-to-face engagement but equally they might broaden our thinking about how we do things in future, especially as we turn our thoughts to the next academic year. However, there is a distinct and heartening community spirit as we come together to face these challenges.  All this alongside caring responsibilities for many, and ongoing research in all aspects of mathematics, including those relevant to COVID-19.  Teaching might not look quite the same this term, but the mathematics will be as good as ever.  And yes, cats are welcome at tutorials. As are dogs, rabbits...

Tuesday, 21 April 2020

Oxford Mathematics Online Open Days Saturdays 25 April & 2 May

The show goes on and that means the vital role of explaining what we do and what you need to do to join us as a student in Oxford Mathematics.

Our two Open Days will do just that. Admissions Guru James Munro will be live, talking about life in Oxford, explaining the Admissions process and, together with some of our students, answering any questions you want to ask. In addition there will be talks covering different aspects of the curriculum.

So please join us. All you have to do is go to this page a few minutes before 10.30am on each of the next two Saturdays (25 April & 2 May) and all will be explained, including how to ask questions in real time. The talks will all remain available after the livestream finishes.

Take care all

Thursday, 16 April 2020

The varied world of Gaussian Fields

Oxford Mathematicians Dmitry Belyaev and Michael McAuley explain the ubiquitous role of Gaussian Fields in modelling spatial phenomena across science, and especially in cosmology. This case-study is based on work with Stephen Muirhead at Queen Mary University of London (QMUL). 

Smooth Gaussian fields are a type of random mathematical object, which are used throughout the sciences for modelling spatial phenomena. As an example, imagine looking at a region of the ocean fixed at some moment in time. The surface of the water, which will be random due to currents and waves, can be thought of as a Gaussian field.

Gaussian fields are particularly important in the study of cosmology. The Cosmic Microwave Background is weak electromagnetic radiation which can be detected from any point in space. This is a remnant of the extremely intense radiation emitted during the Big Bang, and can therefore be used to study the early universe. The intensity of this radiation depends on the direction it comes from, and so can be thought of as random. Cosmological theories predict that observations of this radiation on Earth can be modelled as a Gaussian field. So the intensity of the radiation at each point on Earth is analogous to the height of the ocean at each point in the previous example (see Figure 1 above). Studying the properties of Gaussian fields mathematically, can therefore help in developing and testing cosmological theories.                                                          

(Figure 1 above: an observation of the Cosmic Microwave Background. Red regions have higher intensity of radiation. Source: Planck 2018).

There are many other applications of Gaussian fields, in subjects as diverse as medical imaging, quantum mechanics and machine learning. One of the beautiful aspects of mathematics, is that we can simultaneously study all of these very different situations through abstraction.

One interesting property of Gaussian fields, is the number of regions where it exceeds a certain value, which can be thought of as the number of `hotspots' (on Figure 1 this would be the number of red regions). This has some particular applications in testing cosmological theories.

Many geometric properties of Gaussian fields are said to be `local', which means that they can be studied by breaking the domain into tiny pieces and adding up the result. For example, the total area of all hotspots is local, because it can be found by adding up the area of hotspots contained in each piece of the domain. Local quantities can be understood very well using a tool known as the Kac-Rice formula. The number of hotspots, however, is non-local, because when looking at two tiny pieces of the domain, we do not know whether the red regions in those two pieces are connected. This makes the number of hotspots difficult to study mathematically, as one must account for interactions between different regions of the Gaussian field.

In the last five years, it has been shown that for stationary Gaussian fields (i.e. those which are statistically the same at every point) the average number of hotspots is proportional to the area of the domain. So when looking at a Gausssian field on a large square of area $R^2$, the average number of hotspots is proportional to $R^2$.

The natural next step in understanding the number of hotspots is to study its variance, which is a non-negative number describing how much this quantity typically differs from its average due to randomness. Physicists have used non-rigorous methods to predict that, for a stationary Gaussian field, the variance of the number of hotspots in a square of area $R^2$ should be proportional to $R^2$.

Our work confirms half of this prediction: we show that for many stationary Gaussian fields, the variance is at least of order $R^2$. This bound is believed to be optimal for general fields. We also show that for one special field (see Figure 2), which has applications in quantum mechanics, the physics prediction is incorrect: the variance of the number of hotspots in a square of area $R^2$ is at least of order $R^3$. 


Figure 2: the hotspots of a particular Gaussian field above a given intensity are shown in black. This intensity increases throughout the animation, so that the hotspots shrink.

Thursday, 9 April 2020

The modelling of infectious diseases - Robin Thompson answers your questions

Yesterday, April 8th, Oxford Mathematician Robin Thompson gave a hugely well-received Oxford Mathematics Online Public Lecture on how mathematicians model infectious diseases such as COVID-19. We hope that it will continue to provide a useful introduction to mathematical models of infectious disease outbreaks (and how they can inform public health measures). It would be impossible to answer all of the questions that have been submitted, but we have selected eleven at random (we are mathematicians after all), and Robin has answered them here.


Thanks for the lecture. Just a quick question: in the models, why do social distancing measures affect the infection rate (beta)?
Chris, via email

Thanks for your question, Chris. The parameter beta represents the infection rate between pairs of infectious and susceptible hosts. Beta therefore depends on the contact rate between infectious and susceptible hosts, as well as the probability of infection per contact. If a social distancing strategy is introduced, then the contact rate between infectious and susceptible hosts decreases (everyone in the population has fewer contacts). As a result, beta decreases.


Is it clear that the R state exists for coronavirus?
Nic, via Vimeo live chat

The epidemiology of the novel coronavirus is still not fully understood. However, it is unlikely that individuals who have recovered from COVID-19 can be reinfected soon afterwards, due to the body’s antibody response. How long this antibody response lasts for is as yet unknown, but immunologists think that it is likely to be months or years.


Why is the contact matrix not symmetric?
Jerome, via Vimeo live chat

A few different people asked this question. Any two specific individuals will of course have the same number of contacts with each other. However, in general, an individual of age x may have a different number of contacts with individuals of age y than an individual of age y has with individuals of age x. This is because there are different numbers of individuals in different age groups.

For example, imagine a tiny population of five people, consisting of a grandparent and their four grandchildren. Suppose that the grandparent contacts each grandchild once per week. Then, in this small population, the grandparent would have four contacts per week with younger individuals, but each younger individual would only have one contact per week with grandparents.


How do you estimate uncertainty in your parameter estimation?
Alexey, via Vimeo live chat

Great question, Alexey. I am guessing that you are a mathematician, so I can give a relatively technical answer! There are a number of ways to include uncertainty in estimates of the parameters governing disease transmission. For the stochastic simulation models, one way to do this is to estimate parameter values using a statistical inference technique such as Markov chain Monte Carlo, which generates a (joint) posterior distribution for the parameter values. Then, in each forward simulation, we sample the parameter values at random from the posterior, giving a wide range of possible future dynamics. It is really important that this entire range of forecasts is communicated to policy-makers who are making decisions about which public health measures to introduce.


Thank you from Spain. I’m a mathematician, not an expert in this area, and I would like to ask for some bibliography regarding epidemiological models, and their mathematical properties. I’m mainly interested in deterministic models.
Jorge, via Facebook

There are some excellent resources about epidemiological modelling that are available. One book that I have found particularly useful is Keeling and Rohani’s 'Modeling Infectious Diseases in Humans and Animals'. Another useful book for mathematicians about the mathematical properties of epidemic models is 'Mathematical Epidemiology of Infectious Diseases' by Diekmann and Heesterbeek. But there are lots of other resources out there – some of which are online and available for free!


For data on values such as Beta and Lambda, do researchers rely on pre-existing processed data or do they gather data in real time and process it?
Omar, via Facebook

This is a great question, Omar. Usually, epidemiological modellers rely on both of these approaches – some parameter values are estimated using existing data (or observations from previous outbreaks, for diseases that cause recurring outbreaks) and others are estimated and updated in real-time as an outbreak is ongoing. This real-time estimation is usually carried out by fitting the transmission model to data on, for example, the numbers of cases or deaths per day.


The parameter beta in the SI model is the same for the S-equation and the I-equation – why is that?
Ana, via Facebook

Hi Ana, thanks for your question. The idea there is that individuals move from the susceptible class (S) to the infectious class (I) when they contract the virus – so the same number of individuals leave S as enter I. The parameter beta determines the rate at which individuals leave S and enter I, and so it is the same in both equations (one equation for leaving S, and the other for entering I).


How do we/can we understand the different outcomes between a relatively light and relatively strict lockdown?
Andrew, via Twitter

Models can be used to explore how case numbers are likely to change under different potential control measures. To consider the difference between a light and strict lockdown, the key change is likely to be the number of contacts that individuals in the population make. This can be adjusted in the models by changing the value of the infection rate parameter, beta.


Why don’t the models of lockdown account for the economic impact and the downstream suicide rate?
Richard, via Email

This is a very important question. The potential economic impacts of control interventions and mental health effects should definitely be factored into decisions being made by policy-makers. Outputs from the models presented here could in theory be taken and used for additional analyses assessing the economic impacts and downstream suicide rates. Crucially, the output from models like those presented here represents only one of a range of factors that policy-makers should consider when deciding which interventions to introduce. Responses to infectious disease outbreaks rely on expertise from individuals in a range of fields.


Superb talk! How is R0 affected by COVID-19’s ability to remain infectious on surfaces?
Sarah, via Twitter

This is a great question – thanks Sarah! COVID-19 infections can occur via a number of different routes, including inhalation of droplets, transfer via contaminated surfaces, and possibly faecal-oral transmission. In principle, R0 can be split up according to each of these different components. R0 can then be calculated as the sum of the reproduction number values for each mode of transmission. 

An excellent study by Christophe Fraser’s team here in Oxford looked recently at dividing the reproduction number up between asymptomatic transmission (i.e. transmissions from infectious individuals that never show symptoms), presymptomatic transmission (i.e. transmissions from individuals that show clear symptoms, before those symptoms develop), symptomatic transmission and environmental transmission.


What it the rate of transmission of COVID-19?
Amaan1001, via Instagram

The transmissibility of the novel coronavirus is governed by the reproduction number, which represents the average number of individuals that an infectious host is likely to infect over their course of infection. Initial reproduction number estimates for COVID-19 were roughly in a range of between 2 and 3, although it depends on the precise setting. However, the number of individuals that any infectious host is likely to infect can be reduced substantially by public health measures such as social distancing, which is why we must all follow social distancing guidelines. You might be interested in this tracker of reproduction number estimates through time in different countries (full disclosure: I am involved in it, but the hard work is being done by Dr Sam Abbott and the rest of Dr Seb Funk’s excellent team at LSHTM!).

Tuesday, 7 April 2020

Life under lockdown - Oxford Mathematics Alumni Stories

Oxford Mathematicians don't stop being Oxford Mathematicians when they leave us. Of course not everyone puts their experience to direct use and some may prefer to forget it, but what they all have in common is stories. And more than ever at this time, we need stories.
Alumni Stories:
Like many of the Alumni, Patricia Phillips is, in her own words, confined to barracks. 

Alexandra Hewitt (Merton 1988) works for the Advanced Mathematics Support Programme (AMSP) where she is busy providing vital online support for teachers. She is also setting up virtual Church Services (in Zoom for those of us cursed/blessed by remote team working). 
Jonathan Frank (Jesus 1990) is Director of Big Give, the largest charity match funding platform in the UK. In partnership with the National Emergencies Trust they are running an emergency COVID-19 campaign to double all donations to their appeal made through the platform
Adrienne Propp (Corpus Christi 2017) now lives in Washington DC, and is working at nonprofit nonpartisan think tank RAND Corporation working on a project with the state of Virginia assessing the existing models of COVID-19 and integrating them.
After working for 40 years for the NHS as Clinician, Researcher and Educator, Michael Venning (Balliol 1966) is pondering a return to action as a Maths teacher. But in the meantime he has just had his offer to return to the NHS accepted.
Jeremy Penwarden (Merton 1979) is Director of D MacIntyre and Son and is building a new business from the ashes of a hairdressing supply business, including sourcing a test kit for COVID-19 from China, which he would like to bring to market.
Kei Davis (1985) is working from home on a magnetohydrodynamics code for inertial confinement fusion modelling. He works at Los Alamos National Laboratory in the USA.
Zar Amrolia (Christ Church 1983) is Co-CEO of XTX Markets who have generously donated £20m to three charities in the UK, New York and Paris in their fight against COVID-19.  
Kirill Makharinsky (St John's 2003) is CEO of Enki. They have a data course for teams that combines remote instructors with software to make it efficient and cost-effective.
Jonathan Farley (Lincoln 1991) is a Professor of Mathematics at Morgan State University. He has been turning his mathematical skills to figuring out smarter ways to allocate customer time in supermarkets in the era of social distancing.
Kevin Olding (Magdalen 2003), after a career teaching maths, now makes maths videos through his Mathsaurus website and YouTube channel.
Richard Cayzer (Balliol 1990) works to accelerate the data science capability of catastrophe threat modelling including pandemics through his range of contacts across financial services and law enforcement.
Kishor Kale (St John's 1983) has lobbied his local MP and local Liberal Democrat Prospective Parliamentary Candidate to ask for home-delivery volunteers to be given free FFP3 respirators and free training in their use.
Alok Gupta (Hertford 2006) is leading Data Science & Machine Learning for DoorDash (the largest food delivery company in the US). He heads a team of 20 statisticians, mathematicians, physicists & economists who are creating new algorithms and re-training models to keep up with the change in the market and updated delivery best practices.
Ilse Ryder (St Hugh's 1947) is turning the increase in time afforded by the lockdown into a return to Mathematics, pursuing an interest in number theory by an investigation of semi-primes.
Ronald Brown (New 1953), Emeritus Professor, Bangor University, has a website dedicated to his continued passion for mathematics, especially what has in the past seemed unfashionable.

Caroline Jackson (Queen’s 1989) works in the Ministry of Housing, Communities & Local Government and leads a team who are supporting domestic abuse refuges to stay open during the pandemic. Those who work in the sector are expecting a rise in demand as the lockdown continues. Caroline is also doing some maths home schooling with her two boys.

Ronald Stamper (Univ 1955) is researching and developing technology for transforming traditional bureaucracies into CHIs or Collective Human Intelligences.
Tony Hill (Brasenose 1968) is working on his Diversity in STEM project, aiming to get more kids from disadvantaged backgrounds in to the subjects. He is looking at cost-effective ways to ramp up numbers and is in contact with Departments across Oxford and umbrella organisations across the education sector. He is keen to talk to potential partners.
Aaron Morris (St Anne's 2013) is joint founder of PostEraPostEra uses artificial intelligence and machine learning to discover new pharmaceutical drugs. Its COVID-19 Moonshot project bringing together collaborators in Open Source to work on COVID-19 anti-virals.
Robert Christie (Magdalen 1962) is Treasurer of the Fishbourne Village Volunteer Squad. They provide services to the people living in the Parish including shopping, picking up prescriptions, dog walking and telephone chats with the lonely. We suspect there are many more like him.
Nick Taylor (St Catherine's 2014) is taking a break from his PhD in Mathematical Epidemiology in Cambridge to work with colleagues on an interactive model designed to illustrate (rather than accurately predict) the effect of how various control strategies applied today might impact different countries' number of infections, hospitalisations, ICU bed requirements and deaths.
Dominic Elliott Smith (St Catherine's 2001) asks if there are any forums, slack chats or groups etc. which have been setup to discuss the ongoing pandemic amongst alumni groups?
Ethel Heyes (St Hilda’s 1960) is long retired from a career in computing and has been volunteering with Citizens Advice for 30 years. During lockdown she is taking calls from home on Citizens' Advice's telephone service, Adviceline.
Tom Collins (Keble 2005-08) is now a Lecturer in Music Technology at the University of York. During lockdown he has got involved in an AI Eurovision Song Contest (have a listen) "using the maths and stats that I learned at Oxford."
Keiann Yeung (Univ 2010) is working at City Mental Health Alliance Hong Kong as Business Development Lead, working on a weekly COVID-19 bulletin which aims to combat misinformation and offer support around COVID-19’s impact on individuals' mental health in the workplace.
Jin Ke (Ying) (Somerville 2011) is a secondary school maths teacher in Oxfordshire who is working from home and trying to keep students busy including developing a “Zoom Escape” activity where people can collaborate online with friends to “escape” their home virtually.
Dave Blake (Exeter 1978) works as PA to an Archdeacon in the Diocese of Lichfield, and is involved with drawing up lists of Cemeteries and Crematoria within and just outside the Diocese, plus lists of retired Clergy (under 70) who might be willing to help out with the expected glut of Funerals. He is also trying to interpret the varying advice over how many people can attend a funeral (the consensus is a maximum of 10 people).
John Harris (Magdalen 1978) now specialises in digital cartography and is using the spare lockdown time to do voluntary work on a Green Spaces web mapping application for a Monmouth Climate Action group.
Richard Chapman (Wadham 1968) is a long retired actuary, but in this lockdown has resurrected his interest in number theory and in Goldbach's Conjecture in particular. 
Brian Reade (Wolfson 1993) is a Fellow of the Institute of Chartered Accountants and an essential worker in Jersey's Finance Industry. He is currently living in a hotel room to shield Italian in-laws, while plotting COVID-19's progress with interest and nostalgia for his PhD in population dynamics of infectious diseases.
Rachel Harrison (New 1979) is Professor of Computer Science at Oxford Brookes University where she is currently re-writing lectures and assignments to make them suitable for online learning, and planning research to help defeat the Covid-19 virus via software and AI.

Richard Pinch (Christ Church 1977) is Vice-President of the Institute of Mathematics and its Applications (IMA), the UK professional and learned society for mathematics.  He is currently working with the other officers and staff to explore ways in which the IMA can deliver services and support to its members and the mathematical community online.
Chris Rimmer (Pembroke 1989) has been software developer, house husband, and is now a maths LSA (Learning Support Assistant) at a secondary school. With the enforced isolation, he has continued his passion for economics by adding a video to his YouTube channel about an economic model based on balance sheets and net worth for studying the effects of individual actions (such as QE) on the whole economy.
Abhav Kedia (Exeter 2013) is working with a non-profit and doctors to develop an application to enable antibody testing in India.
Juan Jimenez (Lincoln 1994) is a Software Development Manager for a multinational company working in the automotive industry. His evening non-paid job is working for kin-keepers, a start up trying to help the ageing population live longer on their own. He has recently made a YouTube video to help with the data gathering challenges posed by COVID-19 and is looking for support. 
Jacob Armstong (St Catherine's 2012) worked as a Data Scientist for Oxford University in infectious disease epidemiology, and is now pursuing a DPhil in Computer Science in Oxford. However, given recent events, he's working with his old team again studying and integrating the NHS’s COVID-19 data into the UK Biobank (a health research resource covering the entire medical histories of 500,000 participants, available to researchers globally).
Vlad Margarint (St John's 2015) is a Postdoctoral Fellow of Mathematics at NYU Shanghai. Together with a NeuroScientist he met in Oxford he has coordinated a project which translated the South Korean Guide on COVID-19 from English to Romanian. The work has been used by the Romanian Government and hospitals and relevant groups around the country.

John Hampson (Jesus 1992) works with data-collection and analysis, and in particular with large data-sets. He is working on a project for monitoring specifically the health situation of NHS employees which is currently lacking. John is looking for partners in the NHS and beyond to help him in the initiative.
Not bad at all. Well done all of you. Have you got a story?

P.S. The image above is of horses struggling with social distancing on Port Meadow where Tommy Gee (Brasenose 1942) remembers sailing Fairy Fireflies and Yorkshire Barrels during the Second World War.
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).