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Fields Medals at Madrid

Terence Tao

Tao was born in 1975 in Adelaide, Australia, and is currently at the University of California, Los Angeles.

His work cuts across several areas that are normally considered to be different, but Tao sees them all as part of one unified picture. They range from nonlinear partial differential equations to combinatorics.

A typical example is his dramatic solution, jointly with Ben Green, of a long-standing question about prime numbers. The sequence of primes has many intriguing properties, but no obvious pattern. Centuries ago, mathematicians noticed that sometimes a sequence of primes exists that forms an arithmetic sequence—a list of integers, increasing by a constant amount at each step. For instance, the numbers

 5, 17, 29, 41, 53

are all prime, and each is 12 larger than the previous one.

Most such sequences quickly hit a composite number and stop, but occasionally a sequence goes on for many steps. How long can an arithmetic sequence of primes be? The longest known arithmetic sequence of primes currently has length 23. It starts with 56211383760397 and repeatedly adds 44546738095860. There is even a ten-step arithmetic sequence of consecutive primes: it begins with the 93-digit prime

100996972469714247637786655587969

840329509324689190041803603417758

9043417 0334888215 9067229719

and repeatedly adds 210.

It has long been suspected that there is no upper limit to the length of an arithmetic sequence of primes. Green and Tao proved in 200? that this conjecture is correct—that is, arbitrarily long sequences of this kind always exist.

Another problem solved by Tao derives from the famous Kakeya Problem —an idealised mathematical form of the problem of turning a car in the opposite direction in a limited region. Here the ‘car’ is a straight line of unit length, and the main limitation on the region to be used is its area, which should be small. What shape is the region of smallest area inside which the line can be turned through two right angles?

It doesn’t just have to pivot about its centre: it can drive forwards and backwards, as well as turning. For a long time the answer was thought to be a deltoid, which is a curvilinear triangle with three sharp corners, or cusps—a sort of ‘three-point turn’.

But in 1919 Abram Besicovich proved that regions of smaller area also permit a complete reversal of the line. In fact, given any area greater than zero, there is a suitable region with that area. (In particular, there is no region of smallest area: given anything that works, something of smaller area also works.) As the area shrinks, the region becomes more and more complicated, extending ever-longer and ever-thinner tentacles in more and more directions, like an overexcited sea urchin. In fact, it effectively becomes a fractal—an infinitely complex shape.

Associated with every fractal is a number, known as its dimension. This is defined in such a general way that it need not be a whole number. For instance, the famous snowflake curve, a fractal formed form ever-smaller equilateral triangles, has dimension log 4 /log 3 = 1.2618.

What is the fractal dimension of any region in which a line can be reversed? The answer is known to be 2, meaning that the region is a fairly ‘thick’ subset of the plane. The same questions can be posed in higher dimensions—for instance, what can we say a bout a subset of three-dimensional space in which a line can be reversed? With more space to manoeuvre, it might be possible to reduce the fractal dimension below 2. Is that possible? No one knows, but Tao has been responsible for much of the recent progress on the n-dimensional Kakeya Problem and its deeper connections with less frivolous mathematics, such as harmonic analysis.