A sphere is a sphere, right? Yes, if you mean a globe or a beach ball 鈥� what mathematicians call a two-dimensional sphere 鈥� but not if you are talking about a sphere in seven dimensions.
Now the mathematician who discovered that spheres start to behave differently in higher dimensional space 鈥� an insight that seeded a whole new field of mathematics 鈥� has been by the Norwegian Academy of Science and Letters.
of the Institute for Mathematical Sciences at Stony Brook University in New York, was recognised for his 鈥減ioneering discoveries in topology, geometry and algebra鈥�.
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鈥淚t feels very good,鈥� Milnor told New 杏吧原创, though he says the award was somewhat unexpected: 鈥淥ne is always surprised by a call at 6 o鈥檆lock in the morning.鈥�
Inflated cube
Topologists like Milnor study shapes whose mathematical properties aren鈥檛 changed by stretching or twisting, but they aren鈥檛 concerned with the exact geometrical properties of a particular shape, like lengths or angles. For example, you can turn a cube into a sphere by inflating it, so the two shapes are topologically identical. But you can鈥檛 turn a sphere into a doughnut without tearing a hole, so they are topologically different.
It is also possible to apply stricter rules to these transformations by making them much 鈥渟moother鈥� 鈥� what mathematicians call differentiable. For shapes in three dimensions or less, those that share a topological geometry 鈥� for example a sphere and a cube 鈥� also have the same differentiable structure.
But mathematicians also study shapes in higher dimensions 鈥� even if they鈥檙e difficult to imagine. 鈥淵ou can often think of analogous things that are small enough to visualise,鈥� explains Milnor. 鈥淭he human brain is amazingly able to tackle all sorts of things.鈥�
Tangled sphere
Milnor did just that in 1956 when he discovered a seven-dimensional mathematical object that is identical to a seven-dimensional sphere under the rules of topology, but has a different differentiable structure. He called this shape an 鈥渆xotic sphere鈥�.
This was the first time a shape had been found sharing the topological properties 鈥� but not the differentiable structure 鈥� of its lower-dimensional counterpart. It led to the field that is now known as 鈥渄ifferential topology鈥�.
What does an exotic sphere look like? It鈥檚 difficult to imagine but bear in mind that it鈥檚 possible to tangle up a higher-dimensional sphere in a way that isn鈥檛 possible in two.
Imagine splitting an ordinary sphere into two halves along the middle, so that each half has a copy of every point on the equator. Now rejoin the two halves so that the southern copy of a point doesn鈥檛 join its northern counterpoint. In two dimensions, there鈥檚 only one way to do this: by twisting the sphere. But in seven dimensions the points can be mixed up with respect to each other in multiple different ways.
Smooth Poincar茅
It turns out there are a total of 28 exotic spheres in seven dimensions, and they also exist in other dimensions. Dimension 15 has as many as 16,256, while others like dimensions five and six only have the ordinary sphere. Mathematicians don鈥檛 yet know whether exotic spheres exist in four dimensions 鈥� a problem known as the smooth Poincar茅 conjecture, and related to the generalised Poincar茅 conjecture, which was solved in 2003.
鈥淗e鈥檚 been a great inspiration to many, many mathematicians,鈥� says , a mathematician at the University of Cambridge who gave a talk on Milnor鈥檚 work following the prize announcement.
Milnor is also well known for teaching other mathematicians about his idea. 鈥淓very time he writes a book, it turns into a classic,鈥� adds Gowers.
