News from the world of maths

In the 1930s, astronomers discovered that many galaxy clusters observable from Earth have a much stronger gravitational field than they should have given their predicted mass. Further astronomical observations only added to this puzzle. After much consideration, it was concluded that something mysterious called dark matter must be involved. Dark matter is in all respects invisible and can only be detected by its gravitational effect on normal matter. If this new theory was right then dark matter would make up most of the mass of the universe.

However, in February three scientists claimed that dark matter was not necessary and in fact by slightly altering Einstein's equations for general relativity they could account for the acceleration. Not everyone was convinced by the new explanation though, and now new evidence has been put forward in support of dark matter through studying the "bullet" galaxy cluster with the Chandra X-ray telescope.

The cluster was created when two separate clusters smashed together. Tremendous amounts of energy were released in this collision; enough in fact to tear the normal matter and the dark matter apart. Even though dark matter is invisible, scientists were able to see the effect by measuring how the mass of the cluster was distributed.

The data gathered supported a model involving dark matter but not an altered form of general relativity as was previously proposed. No doubt the argument over the existence of dark matter will continue but supporters of the dark matter model believe this provides the most conclusive evidence yet.

You can read the full story on Science Daily

The buzz is building in the mathematical community. It looks more and more likely that Grigori Perelman's proof of the Poincaré conjecture is correct — and that he has solved a problem that has eluded the best mathematical minds for more than a century. When Perelman first posted his proof on the web in 2002 many thought this would be just another failed attempt, but since then it has survived intense mathematical scrutiny and appears to close to being accepted as correct.

Now the rumour mill has gone into overdrive. Word on the mathematical street is that he will receive the Fields Medal (thought of as the maths equivalent to the Nobel prize) next week at the International Congress of Mathematics in Madrid. And not only mathematical glory awaits him. The Poincaré Conjecture is one of the seven Millennium Problems named by the Clay Institute, and if Perelman has proved it he is eligible for the $1 million prize. So if the rumours are right, Perelman's fame and fortune are just around the corner.

That is all very exciting, but there is something that may get Perelman even more column inches in the press (he has already made the front pages): Perelman has a history of not accepting prizes. It seems that not only may he refuse the $1 million Clay prize, he may refuse the Fields medal too. This would make him the first to refuse the Field's Medal, and the first not only to win the Clay prize, but also the first to turn it down.

Regardless of whether Perelman accepts the accolades that may come his way, the biggest news for most mathematicians is whether his work is finally accepted as correct — and whether we can start calling the Poincaré Conjecture, the Poincaré Theorem, after all this time.

The world of mathematics waits with baited breath....

• You can read more about the Poincaré Conjecture, the Fields Medal and the Clay Millennium Problems on Plus
• There is an excellent article about the history of Perelman's proof in the latest issue of the Notices of the American Mathematical Society
• You can read more about the maths rumour mill in an article in the New York Times.

You'd think that boarding a plane is an easy thing to do: get on, find your seat and sit down. But reality is never like that: there's always that woman whose oversized make-up bag doesn't fit into the overhead locker, the business man who has to fold up his jacket with utmost precision and the family of five that try out every possible seating arrangement before settling down. But now some new research, reported in the New Scientist last week, shows that it's not all down to a few annoying individuals. "Enplaning", as airlines call it, really is a complicated business and it takes some complicated maths to model it: Einstein's theory of relativity.

Einstein's theory postulates that time passes differently depending on how you move through space: a person travelling in a space rocket at high speed, for example, will have aged less on his or her return than the people who stayed back on Earth (see Plus article What's so special about special relativity?). The key for airline boarding lies in the behaviour of an object in free-fall: Einstein's theory predicts that it will follow the path that takes longest to travel, where the time is measured from the point of view of the moving object.

Eitan Bachmat and his team from the Ben-Gurion University of Negev in Israel realised that, even though plane passengers usually aren't in free-fall, airline boarding involves maximisation of time in a way that can be modelled by Einstein's theory for free-falling objects. They applied their model not to the usual four-dimensional space (three space dimension and one time dimension) but to a new two-dimensional space based on the passengers' seat allocations and their position in the queue.

Having devised their model, the scientist checked to see if boarding passengers in a certain order, for example those in the last rows first, can make boarding any quicker. What they found is that the "last rows first" technique adopted by many airlines is no better than seating passengers randomly. In fact, getting passengers to queue in a random order is surprisingly efficient. The best option, in terms of boarding time, would be to assign to each passenger a specific place in the queue, but this is rather unrealistic as passengers are unlikely to respond well to such regimental techniques. In practise, the most efficient way of queuing takes into account the order within the rows: getting window seat passengers to board first and isle seat passengers to board last seems to work pretty well. The scientists also found that the time it takes to board a plane is proportional to the square root of the number of passengers.

This is the first application of Einstein's theory outside of physics and, according to Bachmat, one of the first scientific studies of airline boarding. So far, airlines have taken preciously little notice of the scientific studies, rather surprisingly given the great amount of money hinging on turn-around time of planes. But what many people would really like to know is whether the all-out-chaos approach of budget airlines that don't allocate seats is any more efficient than traditional ways of boarding. Unfortunately, Bachmat's model doesn't cover this. It's a whole different dimension.