News

Prime numbers have fascinated mathematicians since there were mathematicians to be fascinated, and the Prime Number Theorem is one of the crowning achievements of the nineteenth century. The theorem answers, in a precise form, a seemingly basic question: how many prime numbers are there?

Up to small thresholds, we may search exhaustively. Up to a hundred, there are 25 primes; up to a thousand, there are 168; up to a million, there are 78,498. The proportion of numbers that are prime seems to be decreasing – from 0.25, to 0.17, to 0.08 – but how quickly? In this podcast, Simon Myerson, Sofia Lindqvist, Jamie Beacom and host Aled Walker reveal the answer, and discuss the collection of mathematical ideas which combine to give the theorem’s first remarkable proof. Listeners who enjoyed Marcus du Sautoy’s ‘The Music of the Primes’ will find similar themes examined in greater detail, but those without any background will find all the necessary terminology developed from first principles.

The story begins with Euclid’s proof of the existence of infinitely many primes. Although this is an argument of infamous elegance, the quantitative aspects are embarrassingly poor. Indeed, the argument only shows that there are at least log log x prime numbers up to a threshold x, and in particular only 5 primes less than a million! In the middle of the nineteenth century, Chebyshev invented methods for detecting many more primes, but he still fell short of the conjectured level of precision. It would take a revolutionary insight of Riemann (pictured), connecting the discrete theory of primes to the continuous theory of mathematical analysis, to uncover the exact distribution of the primes, and to prove the Prime Number Theorem.

This podcast is part of the Secrets of Mathematics series where the pleasure (and occassional) pain of the subject is communicated to a wide audience.

The podcast also forms part of the In Our Spare Time series, in which Oxford Mathematician Aled Walker chairs discussions between various panels of DPhil students, drawn from all the different academic spheres of the university. Current topics range from Oscar Wilde to Dark Matter to Cicero to Medieval Song.

Modern software allows us to draw symbols (such as Chinese characters, or mathematical symbols) that the computer will then recognise and turn into type. How can these systems be improved, so that they run faster and more accurately?

A key tool is machine learning, whereby the software is 'taught' on a large set of examples, and then draws on its learning to make predictions for subsequent examples. This sort of approach is very widespread, and understanding the mathematical underpinnings is crucial to being able to improve the software in future. Oxford Mathematician Dr Hao Ni is part of a research group working at the frontiers of this subject.

Dr Ni recently spoke at the Oxford Mathematics North meets South colloquium, which was started during this academic year, in which two early career researchers give short talks introducing their research area to the whole department, with the aim of fostering understanding and collaboration between mathematicians working in the north (pure mathematics) and south (applied mathematics) wings of the Andrew Wiles Building, the home of Oxford Mathematics. Dr Ni described how the theory of rough paths can be applied to the study of non-parametric statistics on streamed data and particularly to the problem of regression where the input variable is a stream of information and the dependent response is also (potentially) a path or a stream.  

To find out more and to hear Dr Ni speak about her work to the public, come to SoapboxScience in Oxford on Saturday 18th June. SoapboxScience is a novel public outreach platform for promoting women scientists and the science they do.

The Mathematical Institute at the University of Oxford has been awarded a new Regius Professorship as part of the Queen’s 90th birthday celebrations.

Twelve new Regius Professorships – rare, sovereign-granted titles recognising the most outstanding levels of research in their fields – were awarded to leading British universities to mark the milestone. This is the first time since 1842 that Oxford has been awarded a Regius Professorship.

The Vice-Chancellor of the University of Oxford, Professor Louise Richardson, said: "2016 is proving to be quite a year for the Mathematical Institute at Oxford, with the Abel Prize presented to Sir Andrew Wiles and Nigel Hitchin recently announced as Shaw Prize laureate. Being awarded a Regius Professorship in Mathematics is wonderful news for the University and another mark of distinction for Oxford Mathematics."

Professor Martin Bridson, Head of the Mathematical Institute at Oxford, said: "This is a special moment in the history of Oxford Mathematics. The award of this Regius Professorship is a wonderful recognition of all that we have achieved and of the exciting future that lies before us.

A noteworthy feature of the award is that it recognises both our pre-eminence in fundamental research and the enormous benefits that flow to society from mathematics. Progress at the frontiers of science and technology has always made great demands of mathematics, and today it reaches more deeply than ever into the core of the discipline. Oxford is proud of the way in which it embraces the power of this interaction."

Until now, only 14 Regius Professorships had been granted since the reign of Queen Victoria, including 12 to mark the Queen’s Diamond Jubilee. As in 2012, recipients of new Regius Professorships have been selected by open competition, judged by an independent panel of business and academic experts.

Each institution will assign the title to an existing professor in the chosen department or will appoint a new professor to take the chair and hold the title.

John Penrose, Minister for Constitution, said: "It is a privilege and an honour to announce these new Regius Professorships in recognition of the truly outstanding work of our universities and as a fitting tribute to mark Her Majesty’s 90th birthday. The 12 institutions can consider themselves truly deserving of this great honour."

In the past, Regius Professorships were created when a university chair was founded or endowed by a royal patron. Previously, they were limited to a handful of the ancient universities of the United Kingdom and Ireland, namely Oxford, Cambridge, St Andrews, Glasgow, Aberdeen, Edinburgh, and Trinity College, Dublin.

Announced in the government’s Productivity Plan in July, the new Regius Professorships will celebrate the increasingly important role of academic research in driving growth and improving productivity over the past 90 years.

The creation of Regius Professorships falls under the Royal Prerogative, and each appointment is approved by the monarch on ministerial advice.

Chancellor of the Exchequer George Osborne said: "I am passionate about promoting science and economic growth right across the country. That’s why I promised to push for prestigious new Regius Professorships, not just in London and Oxbridge but in other great centres of learning, including the Northern Powerhouse, Wales, Scotland and Northern Ireland. I’m delighted that promise is being honoured today."

Jo Johnson, Minister for Universities and Science, said: "The success of our economy is underpinned by the exceptional science and research taking place in our world-leading universities up and down the country, and I’m delighted these 12 institutions have been recognised for their achievements. We’ll continue to make sure pioneering science is recognised and supported to help improve the lives of millions across the country and beyond."

Our latest Oxford Mathematicians are the three Savilian Professors of Geometry who dominated Oxford’s mathematical scene during the Victorian era: Baden Powell (1796–1860), Henry John Stephen Smith (1826–83) and James Joseph Sylvester (1814–97). None was primarily a geometer, but each brought a different contribution to the role. Find out more.

The Savilians are the fourth in our series exploring Oxford's mathematical heritage.

Oxford Mathematician Heather Harrington has been awarded a Royal Society University Research Fellowship. The fellowships recognise outstanding scientists in the UK who are in the early stages of their research career and have the potential to become leaders in their field. Heather's work covers a range of topics in applied mathematics, including algebraic systems biology, inverse problems, computational biology, and information processing in biological and chemical systems.

The Fourier transform is that rarest of things: a mathematical method from over 200 years ago which not only remains an active area of research in its own right, but is also an invaluable tool in nearly every branch of mathematics. Though originally developed by Fourier in 1807 to help solve certain partial differential equations, the transform is a living example of a remarkable feature of mathematics, that a tool created in one sub-discipline can break through these artificial classifications and become vital in another. Find out more about a method that has attracted the attention of mathematicians from Hardy and Littlewood to John Nash.

The Fourier Trasform is the latest in our Oxford Mathematics Alphabet, a sequence of 26 letters explaining key concepts and our latest research. 

Professor Nigel Hitchin FRS, Savilian Professor of Geometry in the Mathematical Institute, University of Oxford has won the prestigious Shaw Prize in Mathematical Sciences for, in the words of the Prize Foundation "his far-reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance."

Professor Frances Kirwan FRS, a colleague in Oxford, paid tribute: "Nigel Hitchin has made fundamental contributions to the fields of differential and algebraic geometry and richly deserves the award of the Shaw Prize. His work has influenced a wide range of areas in geometry and mathematical physics, including symplectic and hyperkähler geometry, the theory of instanton and monopole equations, twistor theory, integrable systems, Higgs bundles, Einstein metrics and mirror symmetry."

Professor Martin Bridson FRS, Head of the Mathematical Institute in Oxford, said: "'it is a real joy to see Nigel Hitchin's profound and influential work recognised by the award of the 2016 Shaw Prize. His inspiring intellectual leadership in geometry has been matched throughout his career by many services to the mathematical community in the UK and across the world, for which we are all deeply grateful. Oxford has been extremely fortunate to have Nigel with us for so much of his career, and we are very proud of him."

Nigel said on news of the award: "I am delighted and honoured to be awarded this prize. Since most of my working life has been spent in Oxford, it is also a recognition of the support I have received here. I was pleased to note that my “twin” in New College, the Savilian Professor of Astronomy, won the Shaw prize a few years ago.”

The Shaw Prize is an annual award first presented by the Shaw Prize Foundation in 2004. Established in 2002 in Hong Kong it honours living individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The 2016 prize is worth US$1.2m to each winner.

Congratulations to Oxford Mathematics' Zubin Siganporia who has won the award for Outstanding Tutor for the Mathematical, Physical and Life Sciences Division in the 2016 Oxford University Student Union Student Led Teaching Awards.

The work of Oxford University Professor Sir Andrew Wiles was celebrated as having 'heralded a new era in number theory' as he received the top international prize for mathematics. 

Sir Andrew received the 2016 Abel Prize from Crown Prince Hakon of Norway at the prize ceremony in Oslo on 24 May. He was awarded the prize 'for his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory'.

The ceremony at the University Aula was attended by more than 400 guests, from members of the international mathematics community to local residents. 

Professor Ole Sejersted, President of the Norwegian Academy of Science and Letters, which presents the Abel Prize, said: 'Mathematicians have tried to prove Fermat's Last Theorem for 350 years, without success, indicating that mathematicians regard this as one of the great mathematical puzzles.

'The Abel Committee says that Sir Andrew's work has heralded a new era in number theory. To me, this indicates that the work on the theorem required the development of an entirely new mathematical foundation, the significance of which goes far beyond the actual proving of the theorem.'

Accepting the prize, Sir Andrew Wiles said: 'As a ten-year-old eager to explore mathematics I rummaged in the popular mathematics section of my local public library and found a copy of a book called The Last Problem by E.T. Bell. I did not even have to open the book. On the bright yellow front cover it told the story of the 1907 Wolfskehl prize offered for the solution of a famous mathematical problem. The problem itself was on the back cover. I was hooked.

'It was a wonderful find for me. Apparently inside mathematics there was hidden treasure! A little over 300 years previously a Frenchman by the name of Pierre de Fermat had solved a beautiful sounding problem, but he had buried the proof and now there was a prize for finding it!

'Fermat did not leave any clues because he did not have a solution, but nature itself leaves clues. I just had to find them. There was never going to be a one-line proof. Nor do proofs come just because one has been born with mathematical perfect pitch. There is no such thing. One has to spend years mastering the problem so that it becomes second nature. Then, and only then, after years of preparation is one's intuition so strong that the answer can come in a flash.

'These eureka moments are what a mathematician lives for; the bursts of creativity that are all the more precious for the years of hard work that go into them. The moment in the morning of September 1994 when I resolved my last problem is a moment I will never forget.'

Fermat's Last Theorem had been widely regarded by many mathematicians as seemingly intractable. First formulated by the French mathematician Pierre de Fermat in 1637, it states:

There are no whole number solutions to the equation xⁿ + yⁿ = zⁿ  when n is greater than 2, unless xyz=0.

Fermat himself claimed to have 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. After seven years of intense study in private at Princeton University, Sir Andrew announced he had found a proof in 1993, combining three complex mathematical fields – modular forms, elliptic curves and Galois representations.

Sir Andrew not only solved the long-standing puzzle of the theorem, but in doing so he created entirely new directions in mathematics, which have proved invaluable to other scientists in the years since his discovery. The Norwegian Academy of Science and Letters said in its citation: 'Few results have as rich a mathematical history and as dramatic a proof as Fermat's Last Theorem.'

The Abel Prize is named after the Norwegian mathematician Niels Henrik Abel (1802-29) and was established in 2001 to recognize pioneering scientific achievements in mathematics. Abel himself did some of the early work on the properties of elliptical functions. Previous winners of the Prize include Britain's Sir Michael Atiyah and the late US mathematician John Nash.

Accompanying the prize-giving ceremony is a series of 'Abel week' activities aimed particularly at young people, including the awarding of the Holcombe Memorial Prize for an outstanding teacher of mathematics and the UngeAbel contest for teams of secondary pupils. This year's winning teacher and young winners were in the audience for the Abel Prize ceremony. 

The rolling of dice in a casino, Heisenberg's uncertainty, the meaning of consciousness. All are explored as Marcus takes us on a personal journey into the realms of the scientific unknown. Are we forever incapable of understanding all of the world around us or is it perhaps just a question of language, not having the right words to describe what we see?