Monday, 22 August 2016

Constructing reaction systems - an inverse problem in mathematics

There is a wide class of problems in mathematics known as inverse problems. Rather than starting with a mathematical model and analysing its properties, mathematicians start with a set of properties and try to obtain mathematical models which display them. For example, in mathematical chemistry researchers try to construct chemical reaction systems that have certain predefined behaviours. From a mathematical point of view, this can be used to create simplified chemical systems that can be used as test problems for different mathematical fields. From an experimental perspective, it is useful to create chemical systems that can be used as blueprints for constructing physical networks in synthetic biology, for example in the area of DNA computing.

Oxford Mathematicians Tomislav Plesa and Radek Erban, together with their colleague Tomáš Vejchodský of the Institute of Mathematics in the Czech Academy of Sciences, have recently published a paper in the Journal of Mathematical Chemistry developing the theory behind this. They have built on previous work to create an inverse problem framework suitable for constructing a reaction system, and have used this to construct certain two- and three-dimensional systems of particular interest. Their work concentrates primarily on networks modelled by systems of kinetic equations, a class of ordinary differential equations of a certain type. Such equations may display 'exotic phenomena', corresponding to specific biological functions in the underlying biochemical reaction networks, and so the inverse problem framework needs to be able to handle such behaviours.

Sunday, 21 August 2016

The mathematics of species extinction

Correctly predicting extinction is critical to ecology. Claim extinction too late, and you may be taking resources away from a species that actually could be saved. Claim extinction too early, and you may cause the true extinction due to stopping resources, such as removing protection of its habitat.

There is a balance to be sought, and it's clear that we're not quite there because every year several species that were thought to be extinct are rediscovered. This may seem good news, but it creates a lack of faith for the International Union for Conservation of Nature (IUCN) when it categorises a species as extinct.

Rediscovered species are dubbed Lazarus species, after the character in the bible that came back to life. We have data for some Lazarus mammals, and some mammals that are presumed extinct. What can we infer from the Lazarus mammals to help us establish which presumed-extinct mammals are actually still alive?

Oxford Mathematician Tamsin Lee and colleagues from Australia in a paper published in Global Change Biology use information about the size of the mammal, the search effort it has received, and whether the mammal lived in dense or sparse populations. Some traits, such as the body size, may affect the chances of extinction and rediscovery - large mammals are easier to hunt, but also easier to rediscover. Whereas, factors such as search effort, will only affect the chances of rediscovery. How can we separate and quantify these effects? 

To establish which mammals are likely to be rediscovered, and when, the researchers used a model that is commonly used in medicine. Suppose you're conducting a trial for a new medicine which may cure a terminal disease. You give this medicine to 100 subjects, and you make a note of the proportion which are cured. And among those which are not cured, you note how long it takes the patient to die from the disease. You also have notes about, for example, the age of the patients, their gender, cholesterol and whether they're a smoker. From these traits you can establish which patients are most likely to be cured - perhaps young female non-smokers with low cholesterol, followed by young male non-smokers with low cholesterol, and so on. Among those which are not cured, perhaps the medicine prolonged their life, So again, we need to establish which traits created a delay. 

Applying this model to the mammal data set, the researchers quantified the effect of traits such as body size, on extinction and rediscovery. They found that indeed, large mammals, such as the Tasmanian Tiger, are more likely to go extinct, as are those mammals that live in dense populations. The effect is compounded for mammals that are both large and live in dense population, such as the Saudi Gazelle which has a 95% chance of being extinct after missing for 79 years. This chance will keep increasing until it reaches 100% in 2039. 

Large mammals, which experience a medium search effort (3 to 6 searches) are likely to be rediscovered less than 50 years after they were last seen, whereas small rodent-sized mammals could be missing for over a hundred years, and still be rediscovered. These time limits can be decreased with higher search effort, but search effort has a stronger effect on large mammals. That is, when choosing to allocate resources, searching for a large mammal will enable us to determine the status of the species sooner than when searching for a small mammal. The Saudi Gazelle illustrates this well, since it is a large mammal, but has not reached 100% chance of extinction despite being missing for 79 years. This is because it has received a low search effort. 

The strong effect of search effort on large mammals bodes poorly for the Tasmanian Tiger, which was last seen in 1933. There has been a huge search effort, but they did not bear any certain sightings. (The question of certain and uncertain sightings is, as you can imagine, another huge topic in ecology). This implies that since 1983 the Tasmanian Tiger has been truly extinct. However, the Chinese River Dolphin, which has also received a high search effort, has only been missing for 9 years, so it has a 72% chance of being extinct, with this chance not reaching 100% until 2034. 

Ulitmately this model demonstrates how even ecology, a relatively new scientific field, is advancing by capitalising upon centuries worth of mathematics.

Saturday, 20 August 2016

Alison Etheridge named Fellow of the Institute of Mathematical Statistics

Alison Etheridge FRS, Professor of Probability in the University of Oxford, has been named Fellow of the Institute of Mathematical Statistics (IMS).  Professor Etheridge received the award for outstanding research on measure-valued stochastic processes and applications to population biology; and for international leadership and impressive service to the profession.

Each Fellow nominee is assessed by a committee of their peers for the award.  In 2016, after reviewing 50 nominations, 16 were selected for Fellowship.  Created in 1935, the Institute of Mathematical Statistics is a member organisation which fosters the development and dissemination of the theory and applications of statistics and probability. An induction ceremony took place on July 11 at the World Congress in Probability and Statistics in Toronto, Ontario, Canada.

Alison is also President Elect of the IMS.

Wednesday, 10 August 2016

Mathematics enables faster computer simulations of biology

Numerous processes across both the physical and biological sciences are driven by diffusion, for example transport of proteins within living cells, and some drug delivery mechanisms. Diffusion is an unguided process which is of great importance at small spatial scales. Partial differential equations (PDEs) are a popular tool for modelling such phenomena deterministically, but it is often necessary to use stochastic (probabilistic) models instead to capture the behaviour of a system accurately, especially when the number of diffusing particles is low, such as in gene regulation.

Exploring the underlying mathematics behind these models is an important current area of research. Mathematicians need to understand these models better, so that they can be applied more meaningfully and so that they can be made more efficient while still preserving their accuracy (as computational power and time are often limiting factors). Oxford Mathematicians Paul Taylor and Ruth Baker, working with colleagues Christian Yates of the University of Bath and Matthew Simpson of the Queensland University of Technology, have been seeking to explore stochastic models of diffusion that are 'compartment-based'. In their paper, published in the Journal of Royal Society Interface, the domain under consideration is discretized into compartments, with particles jumping between compartments, possibly with constraints such as that a compartment cannot contain more than a certain number of particles. Previous work by these authors has concentrated on situations where the compartments all have the same size, but these can be unhelpfully restrictive for some applications, where it is important to focus at a high resolution in some parts but impractical to apply this same high resolution across the whole domain. This latest piece of work brings together a number of aspects, including allowing different compartments to have different sizes.

Crucially, this research demonstrates that these new approaches will be of value to researchers working on multi-scale systems, as they can speed up simulations while preserving precision where needed.

Friday, 29 July 2016

Summer (school) time in Oxford and the livin' is mathematical

This summer, about 200 teenagers will take part in mathematical summer schools hosted by Oxford Mathematics in the Mathematical Institute. Here is their story.
First to arrive were the 24 students from 15 countries across Europe who are taking part in PROMYS Europe, a six-week mathematical programme run by a partnership of PROMYS, the Mathematical Institute in Oxford, the Clay Mathematics Institute, and Wadham College, University of Oxford. As one student attending for the first time put it, "At PROMYS we do not learn Maths; we discover it. This gives us a much better understanding of the basics on which all other Maths is built".  

PROMYS has run in Boston, USA for more than 25 years, and this year sees the second occurrence of the new PROMYS Europe programme. Thanks to generous support from the organising partners and donors, selection for the programme is needs-blind, with partial and full financial support for those participants who would otherwise not be able to attend. The super-keen students are joined this summer by 7 undergraduate 'counsellors', also from across Europe (and including two current Oxford students), with teaching from Glenn Stevens (Boston University), Henry Cohn (Microsoft Research), Vicky Neale (Oxford) and David Conlon (Oxford), and guest lectures by mathematicians from Oxford and beyond. One counsellor, who also attended PROMYS as a student, observed "Three years ago, when I entered the PROMYS family, I learned one of the most important lessons - one should be taught how to think, not what to think - and this is exactly what this program does."
The Andrew Wiles Building is an ideal venue for hosting summer schools such as these, and indeed with careful planning can accommodate not one but three simultaneous events. The university's one-week UNIQ summer schools are for UK students about to enter their final year at school, to give them a taste of what it is like to study at Oxford, with priority being given to applicants from low socio-economic backgrounds and/or from areas with low progression to higher education. Demand for mathematics and statistics is high, and this year Rebecca Cotton-Barratt, the Schools Liaison Officer and Admissions Coordinator in the Mathematical Institute, and Mareli Grady, the Schools Liaison Officer in the Statistics Department, have between them coordinated three UNIQ summer schools, giving over 80 students the inspiring experience of studying Mathematics in Oxford. Reflecting at the end of the week, students commented "I thought I wasn’t good enough to apply but I will be applying now as I have gained more confidence", and "really enjoyable with lots of variety in various fields", and, interestingly, "I don’t want to go home now".
Later in the summer, we are looking forward to welcoming two summer schools organised by the UK Mathematics Trust. The National Mathematics Summer School and Summer School for Girls are each for around 40 students aged 15 and 16, invited to participate on the basis of their outstanding performance in national mathematics competitions. They give students a taste of mathematics beyond the school curriculum, as well as exploring more familiar material in depth, with an emphasis on problem solving and collaborative work. The teams of staff leading these summer schools include alumni, students and staff from Oxford Mathematics, and we are delighted to host these events.

All in all, the schools demonstrate that there is a passion for the subject of mathematics, a passion Oxford and its partners are keen to nurture for the long-term educational, scientific and economic benefits it will bring. 

Photo courtesy of Wadham College.

Tuesday, 19 July 2016

Dominic Joyce wins 2016 London Mathematical Society Fröhlich Prize

Oxford Mathematician Dominic Joyce FRS has won the 2016 LMS (London Mathematical Society) Fröhlich Prize "for his profound and wide-ranging contributions to differential and algebraic geometry." Dominic is Professor of Mathematics and Senior Research Fellow at Lincoln College. His research is, in his own words, "mostly in Differential Geometry, with occasional forays into some more esoteric areas of Theoretical Physics."

Tuesday, 19 July 2016

Oxford Mathematicians recognised at European Congress of Mathematics in Berlin

Oxford Mathematician James Maynard has been awarded a European Mathematical Society Prize at the 7th European Congress of Mathematics in Berlin. The prizes are awarded every four years in recognition of excellence in mathematics to ten individuals under the age of 35 living or working in Europe.

In the words of the judges James was awarded the prize for "his remarkable and deep results in analytic number theory, dealing especially with the distribution of primes. He is recognised in particular for his new proof, with improved estimates, of the 'small gaps between the primes theorem'."

In addition to James, Geordie Williamson, formerly a researcher in Oxford Mathematics was also awarded a prize for his "fundamental contributions to the representation theory of Lie algebra and algebraic groups, including his proof of Seorgel's conjecture on bimodules associated to Coxeter groups, and his startling counterexamples to the expected bounds in Lustig's conjecture on the characters of rational representations of algebraic groups." 




Thursday, 14 July 2016

Vicky Neale from Oxford Mathematics wins divisional teaching award

Vicky Neale from Oxford Mathematics has won an MPLS (Mathematical, Physical and Life Sciences) Teaching Award for her innovative and entertaining undergraduate teaching. Using blogs and tips to back up her lectures, Vicky's expansive approach has led to widespread praise from the toughest of critics, namely the students themselves.

Vicky is Whitehead Lecturer at Oxford, a post dedicated to the wider communication of mathematics. She regularly gives public lectures, including the prestigious London Mathematical Society Popular Lectures in 2013 and runs workshops for schools and teenagers including PROMYS Europe. She is also a regular guest on radio including BBC Radio 4's' Start the Week' and 'In Our Time'.

The MPLS awards are part of the University of Oxford's commitment to the highest standards of teaching across all its departments. 

Friday, 8 July 2016

Modelling genes: the backwards and forwards of mathematical population genetics - Public Lecture now online

How can we explain the patterns of genetic variation in the world around us? The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic, ecological and evolutionary mechanisms. How do they interact with one another, and what was their relative importance in shaping the patterns we see today? 

In our latest Oxford Mathematics Public Lecture Alison Etheridge FRS, Professor of Probability in the University of Oxford explores the remarkable power of simple mathematical caricatures in interrogating modern genetic data.


Wednesday, 6 July 2016

How weights and pulleys might explain the hunting techniques of toads

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. In their recently published paper Oxford Mathematicians Dominic Vella and Alain Goriely and colleagues looked at the dynamics of the chain when one of the weights is removed and thus one end is pulled with constant acceleration.

This simple change has dramatic consequences for the ensuing motion. At a finite time, the chain ‘lifts off’ from the pulley, and the free end subsequently accelerates faster than the end that is pulled. Eventually, the chain undergoes a dramatic reversal of curvature reminiscent of the crack or snap of a whip. A key to this dynamic is its geometry. The imposed rotation of the chain around the pulley enables the end of the chain to ‘beat’ the free-fall that drives its motion.

Such insights have enabled the researchers to speculate more widely, notably on the peculiar hunting techniques of a variety of amphibians. Instead of throwing their tongue in a straight motion (as observed in chameleons), certain species of toads and salamanders adopt an unfurling tongue strategy. Of course, the reasons for such a mechanism are many and varied, but the researchers believe that, since the increase of tip velocity observed in the case of a chain has its origin in geometry, a similar effect is likely to reappear in more complicated problems involving, for example, a finite bending stiffness. It is then natural to wonder whether the geometrical amplification of acceleration may be used by these amphibians to allow them to maximise their chances of capturing a prey.

Image: Deban Laboratory