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'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.
Photographing artwork is a subjective business. The photographer has to make aesthetic decisions on lighting and the angle from which the picture is taken, based on his or her aesthetic instincts. But accurate representations of paintings and sculptures are important for scholars, collectors and other art lovers who view them in books, museum archives and on websites.
To get around the problem of subjective photography, the Andrew W. Mellon Foundation has awarded a grant of $855,000 to the Rochester Institute of Technology and the colour scientist Roy Berns. The scientists will use maths to build a system that photographers can use in situ to eliminate subjective decisions.
The main part of the project will be to build an instrument that can capture information about an object's geometry and colour and measure how these depend on lighting. This involves reducing the painting or sculpture to its most basic optical blueprint, and stripping it of any attributes that are down to subjective perception. Additional instruments that can gather information on the
gallery's shape will also be developed later on in the project. Using mathematical models from computer graphics software, the piece of art can then be rendered as it is when viewed in real life, bypassing any input of the photographer.
Fractals are abundant in Nature and even molecules can have a fractal structure. Now, scientists from the University of Akron in Ohio and Clemson University have created the largest man-made fractal molecule at the nanoscale. And they even captured an image of it. The molecule is made up of six rings, each composed of six smaller rings, each of which is in turn each made up of six smaller
rings, and so on, giving rise to the self-similarity and incredibly intricate structure that makes a fractal.
These new man-made molecules are extremely precise and could be used to engineer new kinds of photoelectric cells, molecular batteries and energy storage. The molecules are about twelve nanometres wide, a nanometre being a billionth of a metre. This is pretty small to you and me but, apparently, it's big on the nanoscale. One of the scientists involved even called it huge. To see an image of
the molecule — enlarged a million times — and to find out more, read the University of Ohio's news release. The scientists' work was published in the journal Science.
Computer networks, connecting people, organisations and places, are becoming increasingly complex. So complex, in fact, that many are starting to buckle under their load. The huge amount of information sent through networks and the increasing complexity of this information can overwhelm particular nodes in the network, and before you know it your files have disappeared or your printer
connection has failed.
Nature, on the other hand, has no such problems: ant colonies and bee hives are networks of thousands of individuals engaged in complicated communal tasks, and evolution has made sure that these networks function perfectly. The BISON project, funded under the European Commission's FET (Future and
Emerging Technologies) initiative of the IST programme, has been set up to learn from these little animals.
The project takes its inspiration from the techniques used by ants to find their way to and from food sources, and by fireflies' ability to synchronise their flashing. Using computer models of these animals' behaviour, the project scientists developed algorithms that can optimally route information through a constantly changing network, and synchronise important network functions.
Although most of the project's work is not yet ready for commercial use, initial tests seem very promising. Maybe a bug in the system isn't that bad after all. To find out more read the IST's news release.
If you'd like to have a go at solving one of the most tricky and important open questions in maths and get rich in the process, then the QEDen web site is worth a look. The site has been set up with the ambitious goal to solve the Millennium Prize Problems, seven of the most
difficult unsolved problems in maths. The Clay Mathematics Institute has promised a million Dollars to anyone who solves one of the problems, but so far solutions remain elusive.
QEDen is designed to be "an online playground for the mathematically and scientifically minded people on the internet to converge and wrap their minds around the toughest problems yet to face the planet". As yet, its "progress thermometer" is firmly stuck at zero. But, who knows, maybe you could deliver a vital clue. And, don't worry, if you solve the problem, all the money is yours