Elon Lindenstrauss got the Fields Medal for developing tools in the area of dynamical systems and using them to crack hard problems in number theory.
Elon Lindenstrauss, Princeton University Fields medallist 2010.
As the name suggests, number theory studies the basic properties of numbers. The whole numbers 1, 2, 3, etc are probably the first thing that spring to mind when you think about numbers. Close to follow are the rational numbers: these are the fractions, numbers of the form , where and are both whole numbers. But there also irrational numbers, which can't be written as fractions. An example is the number : some people write it as 22/7, but that's just an approximation: it's close to , but not exactly equal to it. In fact, there isn't any fraction that's exactly equal to .
In turns out that you can approximate an irrational number, call it , by a fraction to any degree of accuracy. If you give me a really small number , then no matter how small is, I can find you a fraction that's within of . But some approximations are better than others. The fraction 2147865/68341 is a tiny bit closer to than 22/7, but it's also much more horrible to write down because it has such a large denominator (and as a result a very large numerator). So what's the ideal relationship between the accuracy of approximation and the denominator of a fraction?
In the 19th century the German mathematician Johan Dirichilet came up with a notion of this ideal relationship. He decided that an approximation of an irrational number should be no further from than . In other words, if the denominator is large, (so that is even larger and therefore very small), then the fraction should make up for this by being close enough (within ) of . Dirichilet proved, and the proof wasn't very hard, that given any irrational number , you can always find infinitely many fractions which satisfy this criterion. So there's a "nice" approximation, in Dirichilet's sense, for any level of accuracy.
It turns out that something similar is true for pairs of irrational numbers and . There are infinitely many fractions and which are nice simultaneous approximations of and : the difference between and times the difference between and is less than Put in the form of an equation, this is
Since pairs of numbers can be interpreted as the coordinates of a point on a 2D plane, this result gives a measure of how well points with irrational coordinates can be approximated using points with rational coordinates that have the same denominator.
In the twentieth century the mathematician John Littlewood decided that we should be able to do even better than this. Given any two irrational numbers and and an that's as small as you like, there should be fractions and so that
The statement seemed like an easy generalisation, but no-one has so far been able to prove it. It's become known as the Littlewood conjecture.
Number theory is littered with statements that look like they should be easy to prove but turn out to be incredibly hard. In these cases you have to look for clever tools to help you find a solution. In his work Elon Lindenstrauss did just that, using tools from dynamical systems theory. As an example of a dynamical system, think of the 2D plane in which every point is defined by its co-ordinates, a pair of numbers . Now take any such point and shift it by a certain distance to the right and up by another distance . This rule gives you a dynamical system. You can apply it again and again and see what happens to the trajectories of various points.
In the case of the plane, nothing very interesting happens, as trajectories just move further and further away from the centre of the plane, given by the coordinates . If, however, if you look at the surface of a doughnut, things get more interesting. You can make such a surface by taking a square from the plane, turning it into a cylinder by gluing together the left and right edges, and then bending it around and gluing together the circles on either end of the cylinder. In this way, the doughnut's surface inherits the coordinates defined on the original square. Things now become more interesting as you shift points around as before, using the numbers and . Trajectories can travel round and round and visit the same patch of doughnut lots of times.
It turns out that if your two numbers and are irrational, then the dynamical system is what's called ergodic: loosely speaking, trajectories will visit every patch of the doughnut surface and patches of equal area will see comparable rates of traffic. And here is the connection with the Littlewood conjecture: suppose that the pair of numbers and , the distances by which you're shifting points, are the pair of irrational numbers you're trying to simultaneous approximate by fractions. It turns out that proving the Littlewood conjecture is equivalent to showing that you can get every point sufficiently close to the point , just by shifting along using the numbers and a suitable number of times. The number of times you need to shift along gives you the denominator you're after.
Using a more complicated dynamical system, Lindenstrauss and his colleagues made massive progress towards a proof of the elusive Littlewood conjecture. They showed that if there are any pairs of numbers that can't be approximated in the nice way stipulated by the conjecture, then they make up only a negligible portion of the plane in which they live. There are pairs for which the conjecture isn't yet proven, in fact there's infinitely many of them, but as Lindenstrauss showed, collectively they are nothing more than drops in the ocean of the 2D plane.
It's this progress on Littlewood's conjecture that forms part of the body of work for which Lindenstrauss is being honoured. You can find out more about his work in this excellent description on the ICM website.
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.