# How brilliant is Terence Tao

## Awarding of the Fields Medals in Madrid

The Fields medals were awarded in Madrid on Tuesday. One of the winners is the Russian Grigory Perelman, who caused a sensation with the proof of the Poincaré conjecture.

The winners of the Fields medals were announced at the International Congress of Mathematicians in Madrid on Tuesday. The winners are the Russians Grigory Perelman and Andrei Okounow, who works at Princeton University, Terence Tao from the University of California in Los Angeles and the German-born Frenchman Wendelin Werner, who teaches at the Université Paris-Sud and the Ecole Normale Supérieure. The Fields Medals, awarded every four years, are the highest honors in mathematics and are considered to be equivalent to the Nobel Prize. In order to encourage the creativity of young researchers, the medals are awarded to mathematicians who must not be older than 40 years.

### Award for a loner

Much has already been written about the audience-shy Perelman, whose proof of the hundred-year-old Poincaré Conjecture made headlines in 2003. Known as a loner, the mathematician resigned from the Steklow Institute in St. Petersburg last December and has been in an unknown location ever since. A few weeks ago, Richard Hamilton gave a lecture at the Pauli Lectures at ETH Zurich on the evidence given by the award committee as the reason for the award. The work, which Perelman published in three parts only on the Internet, goes beyond the Poincaré problem. It proves a much more extensive conjecture about the geometry of three-dimensional bodies.

The 38-year-old award winner Werner works at the interface between mathematics and physics. He finds the inspiration for his work in statistical physics. In it, the theory of probability is used to study complex macroscopic systems, which consist of many (in the limit even an infinite number) microscopic particles. Although the behavior of the particles is determined by the random principle, according to the law of large numbers, the overall system can act deterministically. On the other hand, as with Brownian motion, it can remain unpredictable.

With critical values of certain parameters, such systems often show phase transitions. So water starts to boil at 100 degrees Celsius. As it turns out, many physical systems behave very similarly near their critical parameters. Physicists formulated explanations for this, but they could not strictly prove and their general validity was therefore not given. It was Werner - in collaboration with colleagues Greg Lawler and Oded Schramm - who developed a new approach to the study of critical phenomena in two dimensions. This allowed mathematically rigorous proofs for the conjectures of the physicists.

The Russian Okounkow was honored for his bridging between probability theory, algebraic geometry and representation theory. The latter deals with the study of algebraic objects with the help of matrices. By relating these branches of mathematics to one another, new knowledge for the solution of physical problems could be obtained. The laudation particularly mentions Okounkov's contribution to research into matrices filled with random numbers. These are widely used in both mathematics and physics. Okounkow used ideas from quantum field theory to obtain new knowledge about the so-called eigenvalues of these matrices.

At the age of 31, Terence Tao is the youngest of the current award winners. However, he has always been an early starter. At the age of 8 he was called a child prodigy after a brilliant result on a math test for freshmen, at the age of 13 he won a gold medal at the International Mathematical Olympiad, at the age of 21 he received his doctorate from Princeton University, and was just four years later he was appointed professor at the University of California in Los Angeles. Tao's versatility is proverbial. His research includes partial differential equations, combinatorics, harmonic analysis, and number theory. The laudation describes Tao as a mathematician who combines extraordinary technical skills with an unusual acumen for new ideas.

### Solution to a 90 year old problem

In 2004 his proof (with the Englishman Ben Green) that there are arbitrarily long arithmetic sequences of prime numbers received international attention. Another example of his work is the solution - in 1999 with Allen Knutson - of a problem that was almost ninety years old at the time and was known as the "Horn's conjecture". The question was what the eigenvalues of a sum of two "Hermitian" matrices are if the eigenvalues of the individual matrices are known.

George Szpiro

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