Results in mathematics come in several flavours — theorems are the big important results, conjectures will be important results one day when they are proved, and lemmas are small results that are just stepping stones on the way to the big stuff. Right? Then why has the Fields medal just been awarded to Ngô Bào Châu for his proof of a lemma?
It turn's out that Ngô's lemma, formulated in 1970 as part of the famous Langlands programme, wasn't so small after all. And after an enormous amount of mathematical theory came to rely on this unproven lemma, it got a promotion and became known as the Fundamental Lemma.
In the 1970's the mathematician Robert Langland had a grand vision that could bring together the seemingly unrelated fields of group theory, number theory, representation theory and algebraic geometry. Langlands work laid out a mathematical map connecting these diverse areas of mathematics which has lead to a large area of research known as the Langlands program. One of the most important tools in this work is the trace formula, an equation which allows arithmetic information to be calculated from geometric information, itself linking together the disparate concepts of the continuous (a property of things that can be divided into infinitisemal parts, which include geometric objects such as lines, surfaces and the the three-dimensional space we live in) and the discrete (describing things that come in whole indivisible parts, such as the whole numbers which are studied in number theory).
However in order for the trace formula to be applied in any useful way in the Langlands program, an apparently relatively simple condition, that two complicated sums were equal, needed to hold. Langlands and others assumed this condition was true and stated it as a lemma, and many results in the Langlands programme relied on this lemma being true. Langlands set a graduate student the exercise of proving this lemma, but when the student, and then many others failed to prove the result was true, it became known as the Fundamental Lemma. And it's unproven state became a thorn in the side of the Langlands programme.
Finally the thorn has been removed as Ngô has proved the Fundamental Lemma using some surprising methods. Unexpectedly he was able to use geometric objects, called Hitchin fibrations, solve this problem in pure mathematics. And not only did he prove this result, he provided a deeper understanding of this area of mathematics. Ngô not only revealed the the mathematical iceberg of which the tip was the Fundamental Lemma, he also provided a way to understand the whole ice field, said James Arthur, who explained Ngô's acheivements to the International Congress of Mathematicians after Ngô received his Fields Medal.
Ngô's work highlights the unexpected nature of mathematics, how even the smallest steps in a proof can turn out to be giant leaps in knowledge and understanding. And Ngô's career as an explorer in the mathematical wilderness has only just begun, and we all look forward to the new frontiers he will go on to discover.
To kick off our ICM adventure, Plus attended the International Conference of Women Mathematicians, which started yesterday in Hyderabad. Women from around the world gathered to present their mathematical work to each other, but mostly to network and exchange experiences. It was great talking to women whose experience as professional mathematicians is quite different from what we're used to in Europe. One Indian delegate told us that the immediate problems facing women are not things like glass ceilings or sexual harrassment, but far more elementary challenges like university departments that don't have toilet facilities for women. Another difference between India and Europe, which we're really jealous of, is the fact that mathematics in India doesn't suffer an image problem. People see it as a solid career foundation which allows people to prove they've got brains, rather than a subject for boring geeks.
You can hear our conversations with conference delegates in our podcast, which also contains conversations with Ulrike Tillmann who's on the ICWM organising committee and Gwenoline Michaud from sponsor Schlumberger.
The day of maths talks was rounded off by a panel discussion on the state of female mathematics around the world. Sylvie Paycha from European Women in Mathematics asked which European country would be best for female mathematcians to work in. The answer is difficult. While the proportion of female mathematicians is higher in the South and East of Europe, women in these countries receive lower wages, have a higher teaching load and less time for research, and enjoy less prestige. In the North and West, opportunities, money and prestige are better, but child care is often scarce and expensive and women who leave their children with carers all day might feel stigmatised. So there's no easy answer.
The situation in India is similar, as Geetha Venkataraman described. Although there are now more female maths undergraduates (a proportion of 40% to 50%) and a high proportion of women teaching maths undergraduates (in some departments over 50%), there seems to be a glass ceiling. You don't actually need a PhD to teach undergrads. The proportion of female maths PhD and those actively persuing research is nowhere near the positive figures for undergraduates.
Delegates from other continents described similarly depressing pictures, though at least in some places trends are pointing upwards. Perhaps the most shocking statistic of the day came from Paraguay: here for every 100 men who can read and write, there are 88 illiterate women! The delegate from Japan, Basabi Charkraborty, reported on improved efforts to achieve gender equality in maths, but for a curious reason: due to Japan's ageing population, the work force is shrinking, so it's been decided that women might actually play a useful role in it.
The overwhelming message of the day was that young female mathematicians need more role models from their own countries and better support structures. These might involve improved child care, but also national female mathematicians' networks, conferences and seminars. Perhaps the fact that the International Confernce for Women Mathematicians is an integral part of this year's ICM is a step in the right direction.
The Fields Medal is the most prestigious prize in mathematics, akin to the Nobel Prize. It is awarded to up to four mathematicians at each ICM, which meets every four years. The prize is awarded to mathematicians under the age of 40 in recognition of their existing work and for the promise of their future achievements. You can read more about the Fields Medal on Plus.
And the Fields medal isn't the only prestigious prize being awarded at the ICM. The Rolf Nevanlinna Prize recognises achievements in mathematical aspects of computer and information science. The Carl Friedrich Gauss Prize, which was first awarded at the last congress in 2006, is for outstanding mathematical contributions that have found significant applications outside of mathematics. The first recipient of this prize was the Japanese mathematician Kiyoshi Itô, then aged 90, for his development of stochastic analysis. His work has allowed mathematicians to describe Brownian motion — a random motion similar to the one you see when you let a particle float in a liquid or gas. Itô's theory applies also to the size of a population of living organisms, to the frequency of a certain allele within the gene pool of a population, or even more complex biological quantities. It is also now integral to financial trading as it forms the basis of the Black-Scholes formula underlying almost all financial transactions that involve options or futures. (You can read more about the Black-Scholes formula in A risky business: how to price derivatives on Plus.)
This years ICM also sees the inauguration of a new prize, the Chern Medal, for an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics, regardless of their field or occupation. The medal is in memory of the outstanding Chinese mathematician Shiing-Shen Chern. Plus is looking forward to finding out the winners of all of the prizes at this year's ICM, and more importantly, to learning about their mathematical achievements and how they have contributed to mathematics and society at large. Stay tuned to our news section, our blog or follow us on Twitter to find out all the news first.
Plus is proud to host the 68th edition of the carnival of mathematics, celebrating mathematical blogging!
The carnival invites mathematical bloggers to submit the recent blog posts they're most proud of and the current host then publishes a list of the best ones on the first Friday of the month. (You can find out more at Walking randomly.) So here we go....
Renowned cosmologist and mathematician John D. Barrow has turned his attention to rowing, with intriguing results. As others did before him, Barrow noticed that the force generated by a rower in a boat has two components: one drives the boat forward and one to the side. Since the sideways motion represents wasted effort, rowers should be positioned in the boat so that it is minimised. So what exactly is the ideal positioning of rowers, the ideal rig?
It's a mathematical problem and Barrow has come up with solutions to an idealised version, including a rig that never seems to have been used before in competitive rowing. Last week the New Scientist put Barrow's ideas to the test in a little paddle down the Thames ... you can read about the results on the New Scientist website.
If you'd like to read more of John D. Barrow's work, have a look at his Plus column Outer space.
After several months of hard work, particularly by computer legend Owen Smith, the new Plus site is now live!
All the Plus content you know and love is still there, and is still accessible at the same URLs. And with a few changes to the layout and navigation, we hope the new site will be easier to use. All the different types of content we produce is listed in the top bar. Every article now has a comments feature at the bottom of the page, which you can use by registering with Plus — it's free and easy to do. Registering with Plus will also enable you to create pdf documents from our content. You can register here.
We've done our best to make sure everything on the new site works, but if you do come across any problems or something you don't like, then please get in touch, either by emailing us at email@example.com, or by posting a comment on the relevant page. We'd be really grateful for your feedback and we hope you like the new site!