One of the world’s most celebrated mathematicians, Sir Andrew Wiles, is widely known as the man who cracked Fermat’s Last Theorem.
In 1994 Andrew’s proof catapulted him to international fame. He went on to earn some of mathematics’ highest honours, including the Abel Prize and a Copley Medal from the Royal Society. Both his peers and the wider world were gripped by his solving of what was widely believed to be an 'impossible' problem. In 1637 Fermat had stated that there are no whole number solutions to the equation xn + yn = zn when n is greater than 2, unless xyz=0. Fermat went on to claim that he had found a proof for the theorem, but said that the margin of the text he was making notes on was not wide enough to contain it.
At a rare public appearance earlier this week, at the Inaugural Oxford Mathematics London Public Lecture, held in the Science Museum, the audience were treated to revealing insight in to the man behind the maths.
Hosted by mathematician and broadcaster Dr Hannah Fry of University College London, the event featured a lecture followed by an interview including Andrew taking questions from the floor. In front of an audience that included the TV presenter Dara Ó Brian, Andrew addressed a range of questions and shared his thoughts and feelings on everything from his own personal career challenges to the role of maths in society at large.
Answering the question ‘is mathematical skill more nature than nurture?’ he drew a comparison with the film Good Will Hunting, which it is implied that if you are born with a natural aptitude for mathematics, it is easy. ‘There are some things that you are born with that might make it easier, but it is never easy", he said.
‘Mathematicians struggle with mathematics even more than the general public does. We really struggle. It’s hard. But we learn how to adapt to that struggle.’ Of his own struggles, Andrew revealed his belief in the ‘three Bs: Bus, bath and bed.’ Time when you can give into your subconscious and the mind is free to wander away from the immediate problem.
Asked by the comedian Dara Ó Brian, to describe his progress tackling Fermat’s problem, he compared it to three flagpoles in a row, ‘each one higher than the other’ but the ropes that join them ‘sort of sink’ with a ‘particularly big sag in the last one.’
Andrew also talked about his motivations. On the influences that drove his success in proving Fermat’s Last Theorem, and that continue to drive him today, he said: ‘I am always quite encouraged when people say something like: ‘You can’t do it that way.’
When asked his opinion on the skills required to be a great mathematician, he said that he believes that success in the field is more down to character than technical skill. ‘You need a particular kind of personality that will struggle with things, will focus and won’t give up.’ Sometimes the best young mathematicians can’t adapt to this need for struggle, he said. They want to solve things in days. Sometimes it takes much, much longer.
As a specialist in number theory, was he ever tempted to explore other mathematical fields? The answer was an unequivocal ‘no.’ He fell in love with his chosen specialism at a young age and his commitment has never wavered. ‘I confess that I was addicted to number theory from the time I was 10 years old and I have never found anything else in mathematics that appealed quite as much.’
When pressed as to whether his loyalty may be due to feeling less able in other areas, he added ‘definitely true.’
On the subject of exactly what it was that intrigued him most about Fermat’s theorem, he said that he was most captivated by the fact that Fermat wrote down the problem in a copy of a book of Greek Mathematics. ‘It was only found after his death by his son.’
He was so intrigued by the problem that as a student he would ‘sneak off to the library’ to read Fermat. But getting to grips with Fermat did not come easily to him. ‘He had this really irritating habit of writing in Latin.’ To this day Andrew’s Latin remains ‘minimal’.
Pressed on his view on one of the most challenging educational issues, the quality of and attitude towards mathematics at school and in the wider population, he cautioned that the issue is not that children are less interested in maths. ‘Most young people do have a real appetite for mathematics but they are put off by uninspiring teaching.’ When the teachers are not interested in the subjects they teach, it shows and rubs off and ‘gets passed on’ to their students. This is a particular problem at primary level where few of the teachers are mathematicians. As for a potential solution to the ongoing teaching crisis Andrew suggested ‘pay them more.’
Encouraging great mathematicians of the future, he advised them to tackle the ‘impossible problems’ while they were young (teens or undergraduates) and get a taste for research. But, they should buckle-down and ‘be responsible’ when they start their careers.
Andrew also discussed the Millennium Prize Problems, seven mathematical problems that were set by the Clay Mathematics Institute 17 years ago, offering $1 million dollar rewards for each one solved.
One of the problems, the Poincaré conjecture had already been solved in 2003, and of the remaining six, Andrew suggested the Riemann hypothesis is receiving the most attention. First identified by the German mathematician David Hilbert in 1900, the problem ‘says something about the way prime numbers are distributed’ a field where number theorists across the world have been collaborating with huge success over the past two decades.
Andrew shared his view of the beauty of mathematics and what this actually means. He compared solving a mathematical conundrum to walking down a path to explore a garden, designed by the great landscape architect Capability Brown and seeing a new and breathtaking view for the first time. The real thrill is in ‘this surprise element of suddenly seeing everything clarified and beautiful.’
But, like any thing of beauty, you should not stare at a mathematical problem for too long, otherwise ‘the majesty will fade, as is the case with great paintings and music.’ Instead, he prefers to ‘keep walking’ through the garden of mathematics, rediscovering his life’s passion, again and again.
Research led by Oxford University highlights the accelerating pressure on measuring, monitoring and managing water locally and globally. A new four-part framework is proposed to value water for sustainable development to guide better policy and practice.
The value of water for people, the environment, industry, agriculture and cultures has been long-recognised, not least because achieving safely-managed drinking water is essential for human life. The scale of the investment for universal and safely-managed drinking water and sanitation is vast, with estimates around $114B USD per year, for capital costs alone.
But there is an increasing need to re-think the value of water for two key reasons:
- Water is not just about sustaining life, it plays a vital role in sustainable development. Water’s value is evident in all of the 17 UN Sustainable Development Goals, from poverty alleviation and ending hunger, where the connection is long recognised - to sustainable cities and peace and justice, where the complex impacts of water are only now being fully appreciated.
- Water security is a growing global concern. The negative impacts of water shortages, flooding and pollution have placed water related risks among the top 5 global threats by the World Economic Forum for several years running. In 2015, Oxford-led research on water security quantified expected losses from water shortages, inadequate water supply and sanitation and flooding at approximately $500B USD annually. Last month the World Bank demonstrated the consequences of water scarcity and shocks: the cost of a drought in cities is four times greater than a flood, and a single drought in rural Africa can ignite a chain of deprivation and poverty across generations.
Recognising these trends, there is an urgent and global opportunity to re-think the value of water, with the UN/World Bank High Level Panel on Water launching a new initiative on Valuing Water earlier this year. The growing consensus is that valuing water goes beyond monetary value or price. In order to better direct future policies and investment we need to see valuing water as a governance challenge.
Published in Science, the study was conducted by an international team (led by Oxford University) and charts a new framework to value water for the Sustainable Development Goals. Putting a monetary value on water and capturing the cultural benefits of water are only one step towards this objective. They suggest that valuing and managing water requires parallel and coordinated action across four priorities: measurement, valuation, trade-offs and capable institutions for allocating and financing water.
Lead author Dustin Garrick, University of Oxford, Smith School of Enterprise and the Environment, explains: ‘Our paper responds to a global call to action: the cascading negative impacts of scarcity, shocks and inadequate water services underscore the need to value water better. There may not be any silver bullets, but there are clear steps to take. We argue that valuing water is fundamentally about navigating trade-offs. The objective of our research is to show why we need to rethink the value of water, and how to go about it, by leveraging technology, science and incentives to punch through stubborn governance barriers. Valuing water requires that we value institutions.’
Co-author Richard Damania, Global Lead Economist, World Bank Water Practice said: ‘We show that water underpins development, and that we must manage it sustainably. Multiple policies will be needed for multiple goals. Current water management policies are outdated and unsuited to addressing the water related challenges of the 21st century. Without policies to allocate finite supplies of water more efficiently, control the burgeoning demand for water and reduce wastage, water stress will intensify where water is already scarce and spread to regions of the world - with impacts on economic growth and the development of water-stressed nations.’
In conclusion, co-author Erin O’ Donnell, University of Melbourne adds: ‘2017 is a watershed moment for the status of rivers. Four rivers have been granted the rights and powers of legal persons, in a series of groundbreaking legal rulings that resonated across the world. This unprecedented recognition of the cultural and environmental value of rivers in law compels us to re-examine the role of rivers in society and sustainable development, and rethink our paradigms for valuing water.’
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Deciding when to return to the GP when symptoms do not resolve is something many people struggle with, especially when the symptoms may not appear to be serious or life-threatening. Research with cancer patents in Denmark, England, and Sweden, published today in BMJ Open, indicates that small changes to how doctors conclude consultations with their patient could help to improve both survival rates and efficiency.
Led by Oxford University, the international team included researchers from Aarhus University in Denmark with Lund University, Umeå University and the Karolinska Institute in Sweden. The project was funded by Cancer Research UK.
The researchers interviewed 155 people aged between 35 and 86 years old, within six months of being diagnosed with lung or bowel cancer. Patients were invited to talk about events since they had first noticed a problem, including what influenced their decision to consult a doctor.
The study compared Denmark, England, and Sweden because survival rates for lung and bowel cancer between 1995 and 2007 were persistently higher in Sweden than in Denmark or England, particularly in the first year following diagnosis.
Notable differences between the three countries were found in patients’ accounts of the clarity and communication of action plans at the end of GP consultations. Patients in England and Denmark told the researchers that they had not known whether and when to return to their GP with symptoms, while Swedish patients described clear action plans with instructions from their GP to return in specified timeframes.
The study concluded that if clear action plans are used routinely in primary care consultations then uncertainty, false reassurance, and the inefficiency and distress of multiple consultations during cancer diagnosis could be reduced.
The authors recommend that every GP consultation should conclude with explicit advice about what to look out for and provide specific advice for patients to return if symptoms do not resolve, or re-occur.
Dr John MacArtney, Senior Researcher at Oxford University’s Nuffield Department of Primary Care Health Sciences, said: ‘Spotting cancer early can make the difference between survival and death. We already know that many people are reluctant to seek medical attention in the first place, for fear of unnecessarily bothering their doctor. If they have already seen a GP for the same or similar symptoms then they may wait longer to return, even if they have been given the general advice to “come back if things don’t get better”. This makes it important to understand why people might be reluctant to re-visit their GP.’
The study, which is the first to study how patients experience the pathway to cancer diagnosis, has helped to illuminate previous findings from other International Cancer Benchmarking Partnership studies, which have shown that no single factor could explain why differences in cancer outcomes persist between high-income countries with universal health coverage.
The study lead Sue Ziebland, Professor of Medical Sociology in Oxford University’s Nuffield Department of Primary Care Health Sciences, said, ‘Making the general public aware of the seemingly innocuous symptoms of common cancers is only one part of improving early diagnosis. It is just, if not more, important to remove the barriers patients face in getting to see their GP if they are concerned about their health. One of those barriers arises when patients are unsure about what to do when symptoms return or persist, because they have already seen a GP.
‘Recognising these issues exist, and encouraging GPs to communicate concrete and specific action plans, could be vital in improving patient survival, especially for those patients who worry about wasting the doctors time.’
Sara Hiom, Cancer Research UK’s director of early diagnosis, said: ‘Diagnosing cancer is complex and most often starts with someone presenting to their GP with signs or symptoms they’re not sure about. This fascinating study builds our understanding of what’s happening at a very personal level, at what can be an anxious and difficult time for patients. Having this insight across different countries and cultures helps build a clear picture of what’s most important about the conversation between the doctor and patient, and could well improve cancer outcomes. GPs across the UK may need more support to help them make the most of these critical consultations in the little time available. Charities such as Cancer Research UK can provide information to patients at this point in the diagnostic process.’
The full paper, 'Patients’ initial steps to cancer diagnosis in Denmark, England and Sweden: what can a qualitative, cross-country comparison of narrative interviews tell us about potentially modifiable factors?' can be read in BMJ Open.
In a guest post for Science Blog, Oxford DPhil student Anabelle Cardoso, from the Environmental Change Institute in the School of Geography and the Environment, writes about a citizen science project helping us better understand the endangered African forest elephant.
Anabelle and her colleagues need volunteers to help classify photos taken by cameras set up in a forest-savannah landscape in Gabon. Find out more about the Elephant Expedition project, and volunteer as a citizen scientist, here. The project is also on Twitter and Instagram.
'Our project is set up in a mosaic landscape in Gabon, where tropical savannahs and forests interlock with one another, forming a habitat that supports a hugely diverse range of species, including the endangered African forest elephant. In this landscape, and in many other African sites, valuable savannah habitat is being lost to forest encroachment as a result of human-induced global change. Most people don't think of forest expansion as a problem, but when it expands into ancient savannah ecosystems you lose habitat diversity and it can be really detrimental to the ecological health of the landscape.
'My research focuses on better understanding the factors that affect how much forest encroachment a landscape experiences, and elephants can be a key determinant of this. In other parts of Africa we know that the bush elephant, which is a different species to the forest elephant, can help prevent this loss of savannah habitat, for example by knocking down trees. But nobody knows what the forest elephant does to trees in these forest-savannah mosaics. Do they behave like bush elephants? Or are they doing something completely different?
'To try and answer some of these questions, our team uses camera traps to monitor where and how many elephants there are in the landscape at different times of year, as well as why the elephants might be choosing these places, and what effects they are having on the trees in the places that they visit. Gabon is the perfect place to do this because it's home to most of the world's remaining forest elephants.
'Forest elephants are an endangered species as they are being heavily hunted for their ivory across central Africa. A better understanding of forest elephants can help to develop more effective conservation strategies and advocate more compellingly for their protection, both on a local and a global scale.
'Our 40 camera traps across the landscape set to take photos when they get triggered by motion or by heat. The camera traps allow us to monitor the elephants 24/7, and we can set up lots at the same time across a large area, which makes them an extremely effective scientific monitoring tool. Forest elephants have also had to deal with a huge amount of hunting pressure for their ivory, so they can get quite spooked and upset when strangers sneak up on them in the forest! The camera traps help with this, because they are unobtrusive and don't bother the elephants too much, which is ideal because we don't want to upset these beautiful animals.
'When we first set up the project, the plan was that I would go through all the photographs myself and count the elephants, which retrospectively seems almost laughably optimistic, because we definitely didn't anticipate just how many animals there were in the forest and how many thousands and thousands of photos we would end up needing to classify. Thankfully, through the University of Oxford we linked up with Zooniverse.org, which is a wonderful citizen science platform that helps connect projects like Elephant Expedition with a great group of dedicated citizen scientists.
'In Elephant Expedition we've created a platform for citizen scientists to go through each photograph taken by our camera traps and classify it. Photos are classified according to whether or not they have an animal in them, and what kind of animal this is. If the photo has an elephant in it, the citizen scientist also counts how many elephants they see. The platform is super easy to navigate, and it's really fun. It's kind of like going on a virtual safari because you never know what you're going to find next! What this does is create a database where all the images become linked with classification information, and then we can calculate how many elephants there were at a particular site at a particular time. This information is the core of our research, and without the citizen scientists this work would be impossible.
'We find that citizen scientists are really observant and engaged, so the quality of the information we get from their classifications is absolutely amazing. The platform that Zooniverse provides for this connection between the project and the citizen scientists is also really engaging, so there is a lot of interaction between citizen scientists and the research team, which is beneficial to both. I am definitely learning a lot from the project volunteers, and the feedback we get from them indicates they feel the same! It's been really encouraging for us to see how many volunteers post the project photos to their personal Twitter or Instagram accounts, showing that they care about the elephants as much as we do.
'At the moment, however, we have nearly 750,000 photos to classify, and our team just needs more hands on deck to get through all of them. Every month there can be in excess of 3,000 images on each camera, and there are 40 cameras, so it all adds up. The good news is that it's really easy to help with the project just by visiting our project page. You can classify as many images at a time as you want to, so absolutely anyone can help with the project, no matter whether you have a spare five minutes or five hours. Every single volunteer makes a difference.
'One of the best things about the project is that it isn't just elephants you spot in the photos. Our study site is filled with gorillas, chimpanzees, leopards, mandrills, pangolins, red river hogs, forest buffalo, monkeys, and lots of different antelope. So when you go on an elephant expedition it's really more of a virtual safari through the central African rainforest! Plus, the website has features to keep a collection of your favourite images, which you can share on social media or even print out for your fridge if you want to. The website is also applicable for all ages, so we encourage everyone from kids to grandparents to get involved.'
The research has been made possible thanks to the University of Oxford's Hertford College Mortimer-May fund and the support of Gabon's Agence Nationale des Parcs Nationaux (ANPN) and the University of Stirling.