**Terence Tao** is known to his friends and colleagues as **Terry Tao**. His father, Billy Tao, is a Chinese-born paediatrician who has undertaken research on educating gifted children and on autism. Terry's mother, Grace, was born in Hong Kong and has a university degree in physics and mathematics. Billy and Grace met while they were studying at the University of Hong Kong and they emigrated to Australia in 1972. Grace Tao taught physics, chemistry, science and mathematics in various secondary schools in Hong Kong before she emigrated to Australia and, once in Australia, also taught in secondary schools there. Terry, the subject of this biography, is their eldest child, having two younger brothers Trevor and Nigel.

When Terry was two years old his parents realised that he was different from other children. They saw him teaching five year old children to spell and to add numbers and, when they asked him how he had learnt these skills, he replied that he had been watching Sesame Street on television. When he was three and a half years old his parents sent him to a private school but, six weeks later, they realised that he was not ready for schooling and also that the teachers did not know how to teach someone like him. So they removed him from the school and he did not start schooling again until he was, like other children, five years old. The article [4] is an evaluation of Terry's mathematical abilities just before his eighth birthday by which time he was attending Blackwood High School, Adelaide. Ken Clements writes that when he went into his home, Terry was:-

... sitting in the far corner of a room reading a hardback book with the title 'Calculus'. Terence was small, even for a seven-year-old. After meeting his two brothers, I was accompanied by Terence to his father's study, where, after a brief chat, I began my usual assessment procedure for exceptionally bright primary school-age children.

Clements discovered that Terry knew the definition of a group and could solve graph sketching problems using the differential calculus. He wondered how much his mother was teaching him but found that her role [4]

... is more one of guiding and stimulating Terence's development than one of teaching him. She said that Terence likes to read mathematics by himself, and he often spent three or four hours after school reading mathematics textbooks.

By the time Terry reached the age of eleven, he was dividing his time between his studies at Blackwood High School and taking classes at Flinders University in Adelaide where he was taught by Garth Gaudry. Even earlier, at the age of ten, he began participating in International Mathematical Olympiads. He won a bronze medal in 1986, a silver medal in 1987 and a gold medal in 1988, becoming the youngest ever gold medalist in the Mathematical Olympiad. At the age of fourteen he began full-time university studies at Flinders University and was awarded a B.Sc. with Honours in December 1991. He continued to study at Flinders University for a Master's Degree advised by Garth Gaudry and was awarded the degree in August 1992 having written the thesis *Convolution operators generated by right-monogenic and harmonic kernels*. He was awarded the University Medal by Flinders University and a Fulbright Postgraduate Scholarship to enable him to undertake research in the United States.

Tao undertook research at Princeton University advised by Elias Stein. He was an assistant researcher at Princeton during 1993-94 and he was awarded a Sloan Postgraduate Fellowship in 1995. He was awarded his doctorate in June 1996 for his thesis *Three regularity results in harmonic analysis*. In 1996 his research papers began to appear in print, four papers being published in that year. These are: *Weak-type endpoint bounds for Riesz means*; (with Andrew C Millard) *On the structure of projective group representations in quaternionic Hilbert space*; *On the almost everywhere convergence of wavelet summation methods*; and *Convolution operators on Lipschitz graphs with harmonic kernels*.

Following the award of his doctorate, Tao was appointed Hedrick Assistant Professor at the University of California at Los Angeles, a position he held from 1996 to 1998. He continued as an assistant professor at the University of California at Los Angeles where, at the age of twenty-four, he was promoted to full professor in 2000. In 2007 he was named the James and Carol Collins Professor there.

It is very difficult to write a biography of someone who is at the height of their creative powers as Tao is. Anything that one writes about his research contributions will be quickly outdated as he is contributing major results in such a wide range of different areas. Yet he has produced such a fantastic collection of results, leading to the award of all the top prizes in mathematics, that one must try to at least give a vague picture of the work of this remarkable mathematician. Before looking at his contributions we note the prizes and awards he has received (although again this list is bound to become rapidly outdated as he continues to receive awards). These include: the Salem Prize (2000); the Bôcher Memorial Prize from the American Mathematical Society (2002); the Clay Research Award from the Clay Mathematical Institute (2003); the Levi L Conant Award from the American Mathematical Society (2005); the Australian Mathematical Society Medal (2005); the ISAAC Award from the International Society of Analysis, its Application and Computation (2005); the SASTRA Ramanujan Prize (2006); the Fields Medal (2006); the Ostrowski Prize from the Ostrowski Foundation (2007); the Alan T Waterman Award from the National Science Foundation (2008); the Onsager Medal(2008); the Information Theory Society Paper Award (2008); the Convocation Award from Flinders University Alumni Association (2008); the King Faisal International Prize (Mathematics) (2010); the Nemmers Prize in Mathematics from Northwestern University (2010); and the George Polya Prize from the Society for Industrial and Applied Mathematics (2010). In addition he has received a Sloan Foundation research Fellowship (1999-2001), a Foundation Fellowship from the David and Lucille Packard Foundation (1999-2006), and a MacArthur Fellowship from the MacArthur Foundation (2007-11). He has been elected to the Australian Academy of Sciences (2006), to a fellowship of the Royal Society (2007), to the National Academy of Sciences (2008), and to the American Academy of Arts and Sciences (2009). He was a finalist in Australian of the Year in 2007.

To gain some insight into his research contributions, let us first note that he received the 2006 Fields medal:-

... for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.

The article [1], describing the award of the Fields Medal, gives this overview:-

Terence Tao is a supreme problem-solver whose spectacular work has had an impact across several mathematical areas. He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, " Why didn't anyone see that before?" At31years of age, Tao has written over eighty research papers, with over thirty collaborators, and his interests range over a wide swath of mathematics, including harmonic analysis, nonlinear partial differential equations, and combinatorics. " I work in a number of areas, but I don't view them as being disconnected," he said in an interview published in the Clay Mathematics Institute Annual Report. " I tend to view mathematics as a unified subject and am particularly happy when I get the opportunity to work on a project that involves several fields at once."

Let us note here that he has now greatly exceeded the eighty research papers mentioned in the 2006 article with MathSciNet recording a list of 224 publications between 1996 and 2010. The Press Release which announced the award of the Fields Medal to Tao listed his accomplishments in a number of areas which had led to the award of this most prestigious mathematical award. First it describes his work with Ben Green on the distribution of prime numbers. They proved the remarkable result that the primes contain arithmetic progressions of any length. To dismiss this fantastic achievement in a single sentence seems silly, but there is so much more to say. An area to which Tao has made many contributions is that of the Kakeya problem. This problem, originally posed in 2 dimensions, asked for the minimum area of a shape in which one can rotate a needle through 180° . The answer is rather surprising, in fact you can make the area less than any chosen number. Tao has worked on the *n*-dimensional Kakeya problem where again the minimum volume can be made as small as one chooses, but the fractal dimension of the shape is unknown. This problem sounds rather specialised, but on the contrary there are surprising connections to Fourier analysis and nonlinear waves. Another area in which Tao has worked is solving special cases of the equations of general relativity describing gravity. Imposing cylindrical symmetry on the equations leads to the "wave maps" problem where, although it has yet to be solved, Tao's contributions have led to a great resurgence of interest since his ideas seem to have made a solution possible. Another area where Tao has introduced novel ideas, giving the subject a whole new look, is the theory of the nonlinear Schrödinger equations. These equations have considerable practical applications and again Tao's insights have shed considerable light on the behaviour of a particular Schrödinger equation.

One might imagine that with his remarkable output of research papers, Tao would not find time to write books. However, this would be entirely wrong since he has produced both research monographs and undergraduate texts. Let us now look at these. In 2006 Tao published a 2-volume textbook *Analysis*. The publisher describes the work as follows:-

This two-volume introduction to real analysis is intended for honours undergraduates, who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning - the construction of the number systems and set theory, and then goes on to the basics of analysis(limits, series, continuity, differentiation, Riemann integration), through to power series, several-variable calculus and Fourier analysis, and finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The course material is deeply intertwined with the exercises, as it is intended for the student to actively learn the material and to practice thinking and writing rigorously.

Also in 2006, Tao published *Nonlinear dispersive equations*. Sebastian Herr begins a review as follows:-

This monograph is a remarkable introduction to nonlinear dispersive evolution equations, in particular to their local and global well-posedness and scattering theory.

Yet a third 2006 publication was *Solving mathematical problems*. The publisher, Oxford University Press, describes the book as follows:-

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving mathematical problems includes numerous exercises and model solutions throughout. Assuming only basic high-school mathematics, the text is ideal for general readers and students of14years and above with an interest in pure mathematics.

But, amazingly, this still does not complete the list of Tao's 2006 books for in that year, in collaboration with Van Vu, he published *Additive combinatorics*. Serge Konyagin and Ilya Shkredov begin their detailed review of the book by describing the area:-

The subject of the book under review is additive combinatorics - a young and extensively developing area in mathematics with many applications, especially to number theory. Roughly speaking, one can define this area as combinatorics related to an additive group structure. Modern additive combinatorics studies various groups, from the classical group of integers to abstract groups of arbitrary nature. ... The monograph is designed for a wide mathematical audience and does not require any specific background from a reader. However, everybody who intends to read this book should be ready to study tools and ideas from different areas of mathematics, which are concentrated in the book and presented in an accessible, coherent, and intuitively clear manner and provided with immediate applications to problems in additive combinatorics.

It will come as no surprise to learn that Tao, who is such an innovator in everything he does, has created a new style of book. The textbook and research monographs described above are innovative in their approach but are traditional type of books. In 2008 Tao published the book *Structure and randomness. Pages from year one of a mathematical blog* and, in 2009, two similar books *Poincaré's legacies, pages from year two of a mathematical blog* Part I and Part II. Tim Gowers writes in a review:-

Textbooks and popular science are still the two obvious niches for mathematics in the book market, but the advent of the Internet has brought about a sudden change in the possibilities for mathematical exposition, because now anybody can put anything they like on the Web. As a result, there has been a rapid rise in a form of mathematical exposition that is too technical for the layperson, but much easier to read and enjoy for mathematicians than a textbook. A medium that is particularly well suited to this is the blog, and the undisputed king of all mathematics blogs, with thousands of regular readers, is that of Terence Tao. Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others. Some of these are areas to which he has made fundamental contributions. Others are areas that he appears to understand at the deep intuitive level of an expert despite officially not working in those areas. How he does all this, as well as writing papers and books at a prodigious rate, is a complete mystery. It has been said that Hilbert was the last person to know all of mathematics, but it is not easy to find gaps in Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later. Now, in an interesting experiment, several of Tao's blog posts have been tidied up(partly in response to comments from others on the posts)and published as books.

In 2010 the next in Tao's series was published *An epsilon of room, I: real analysis. Pages from year three of a mathematical blog*. One can anticipate a long and fascinating series of books that will appear over the next years.

**Article by:** *J J O'Connor* and *E F Robertson*