Terence Chi-Shen Tao

Quick Info

17 July 1975
Adelaide, South Australia, Australia

Terence Tao is an Australian-American mathematician who won a Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory. He has continued to prove remarkable results on an exceptionally wide range of topics.


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 (two years younger than Terry) and Nigel (four years younger than Terry).

Both Trevor and Nigel are extremely talented, and they went on to be awarded Ph.D.'s and have research careers. Trevor Tao (born 22 November 1977) was diagnosed from age two as autistic. He studied at the University of Adelaide for a double degree, B.Sc. (Mathematics and Computer Science) with B.Mus. (double major in piano performance and composition), which he completed in 2000. He is an international chess master, highest chess rating 2440, who also won prizes in mathematics and music (classical music composition) when he was a teenager. Nigel was awarded a BSc (Hons) from the Australian National University in Computer Science in 2000. He worked for WhizBang! Labs in Pittsburgh, but they closed in May 2002. In 2006 he began working for Google in Sydney. Let us return to out biography of Terry Tao.

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 [21] gives 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 [21]:-
... 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.
In 1984, when Tao was nine years old, his father took him on a trip to meet some of the world's top mathematicians. They went to the Institute for Advanced Study at Princeton where Tao was interviewed by Charles Fefferman and Enrico Bombieri. Fefferman asked Tao the following question [8]:-
Imagine that you are trapped in a room with a hungry lion. Both you and the lion are represented as points in space. Suppose the lion can run faster than you. Suppose you can run faster than the lion. Suppose you and the lion can run at exactly the same speed. How do you avoid being eaten?
Although nobody can now remember Tao's answer, Fefferman said [8]:-
I was impressed that a 9-year-old kid could come up with ideas to a maths problem that wasn't a conventional thing he had learned in any class.
By the time Terry reached the age of eleven, he was dividing his time between his studies at Blackwood High School, which he attended from 1983 to 1988, and taking classes at Flinders University in Adelaide where he was taught by Garth Gaudry (1941-2012). 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. But we should not get the idea that every school subject was easy for Tao. He was brilliant at subjects with precise rules, like mathematics and Latin, but when his English teachers asked the class to write an essay about "home", he was totally stumped.

In 1989, at the age of fourteen he began full-time university studies at Flinders University and, in 1990 when a 15 year old schoolboy, wrote his first book Solving mathematical problems. A personal perspective. In it Tao writes that a solution to a problem:-
... should be relatively short, understandable, and hopefully have a touch of elegance. It should be fun to discover.
The first edition was published in 1992 with a second edition in 2006; you can read the 1990 Preface, the Preface written in 2005, and extracts from reviews of the 2006 edition at THIS LINK.

He was awarded a B.Sc. with Honours in December 1991. He said [109]:-
I still remember the realisation in college at Flinders University in Australia that mathematics was not just an abstract game of symbols but could be used as a tool to analyse and understand the modern world.
He wrote about this moment of realisation in [89]:-
Why are some statistics trustworthy and some not? Why are some investment strategies sound, and others risky? How come a computer can search the entire internet for you in a matter of seconds, but cannot read a printed word if it is even just slightly distorted or blurred? How come our modern array of satellites can tell millions of drivers their location with amazing accuracy, but cannot correctly predict the weather a fortnight into the future? Knowledge of mathematics can answer these questions, and make the world comprehensible and orderly rather than mysterious and capricious. Once I saw the power of this knowledge, and the satisfying feeling when everything "clicks" and one sees a confusing problem resolve itself into a clear solution, I was hooked for life. I wanted to use mathematics to explore and understand as much of the world as I could.
Not everything he did as an undergraduate went well, however [116]:-
At university, I had a Fortran class that I rebelled against, on the grounds that I already knew BASIC; I remember defiantly writing my final assignment code in BASIC instead of Fortran (and subsequently failing the class).
He continued to study at Flinders University for a Master's Degree advised by Garth Gaudry. Tao wrote in [88]:-
I knew Garth ever since I was an undergraduate at Flinders University. He was head of school then (a position roughly equivalent to department chair in the US), but still was able to spare an hour a week to meet with me to discuss real analysis, as I worked my way through Walter Rudin's "Real and complex analysis" and then Elias Stein's "Singular integrals", and then eventually completed a masters thesis under his supervision on Clifford-valued singular integrals.
Tao 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. He explained in [109]:-
I didn't really complete two degrees in four years; I attended Flinders part-time before my formal enrolment, and they decided to count those courses toward my degree.
Tao went to the United States in 1992 when he was sixteen years old to begin graduate school [8]:-
Graduate school was Tao's first time away from home, and the transition proved difficult. His father stayed with him for the first week of classes to teach his son how to perform such basic tasks as doing laundry and opening a bank account. In his free time, Tao joined the Film Club and played table football, online bridge, and computer games.
He began to undertake research at Princeton University advised by Elias Stein. At first things were difficult since Tao had found things so easy up until then that he had not developed any study skills. Scraping through an examination, he resolved to work harder. He writes in [90]:-
My weekly meetings with Eli would tend to go something like this: I would report on all the many different things I had tried over the past week, without much success, to solve my current research problem; Eli would listen patiently to everything I said, concentrate for a moment, and then go over to his filing cabinet and fish out a preprint to hand to me, saying, "I think the authors in this paper encountered similar problems and resolved it using Method X." I would then go back to my office and read the preprint, and indeed they had faced something similar and I could often adapt the techniques there to resolve my immediate obstacles (only to encounter further ones for the next week, but that's the way research tends to go, especially as a graduate student). Amongst other things, these meetings impressed upon me the value of mathematical experience, by being able to make more key progress on a problem in a handful of minutes than I was able to accomplish in a whole week.
At this stage in his career he spent time both in Australia and in the United States. He was an assistant researcher at Flinders Medical Centre (1992-94), working as a computer programmer in his father's research group, and also as an assistant researcher at Princeton University during 1993-94. In 1995 he was awarded a Sloan Postgraduate Fellowship. 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. This made him the youngest person ever to be appointed as a full professor at UCLA.

When he began lecturing at UCLA, he taught a class containing a student named Laura who [114]:-
... doesn't remember who instigated the coffee date they went on some time after she started slipping homework under her professor's office door. She liked how relaxed and stable Tao seemed.
She went on to graduate as an electrical engineer and worked at NASA's Jet Propulsion Laboratory at the California Institute of Technology. Tao and Laura were married in 2002 and they have two children, William Tao and Madeleine Tao. When the article [114] was written (March 2015), William Tao was eleven years old and Madeleine Tao was three years old. William, at age eleven, was [114]:-
... learning piano, guitar and clarinet. One day he was scouted by a talent agency in a mall - "We like his face," they said - and he has appeared in commercials for Ford and Disney. "He's different from me in some ways," says Tao, who likes to watch Doctor Who with his son. "He likes maths, he's good at it, but he likes free-form; he likes things that I was never into, like writing. I was never very good at school with ... humanities ... anything which was more a matter of opinion."
In 2007 Tao was named the James and Carol Collins Professor at UCLA.

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 continues to contribute major results in a remarkably wide range of different areas. Yet at a young age, he produced a fantastic collection of results, leading to the award of all the top prizes in mathematics, and this outstanding mathematician continues to make major advances in diverse areas of mathematics.

Before looking at his contributions we note the prizes and awards he has received (although again this list, compiled in November 2023 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); the George Polya Prize from the Society for Industrial and Applied Mathematics (2010); the Crafoord Prize (2012); the Breakthrough Prize in Mathematics (2014); the Royal Medal for physical sciences from the Royal Society of London; the PROSE award in the category of "Mathematics" (2015); the Riemann Prize (2019); the Princess of Asturias Award for Technical & Scientific Research (2020); the Janos Bolyai International Mathematical Prize by the Hungarian Academy of Sciences (2020); the United Sigma Intelligence Association Award for Mathematics (2021); the Education and Research Award by Advance Global Australian Awards (2022); Global Australian of the Year Award (2022), and the Grande Médaille of the French Academy of Sciences (2022).

For more information about these awards to Tao, see THIS LINK.

In addition he has received a Sloan Foundation research Fellowship (1999-2001), a Foundation Fellowship from the David and Lucille Packard Foundation (1999-2006), a MacArthur Fellowship from the MacArthur Foundation (2007-11), and appointed Simons Investigator (2012). 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?" At 31 years 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 353 publications between 1996 and 2023. 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. He gave the André-Aisenstadt Lecture on Long Arithmetic Progressions in the Primes at the University of Montreal on Monday 3 April 2006. Here is his abstract [4]:-
A famous and difficult theorem of Szemerédi asserts that every subset of the integers of positive density will contain arbitrarily long arithmetic progressions; this theorem has had four different proofs (graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic), each of which has been enormously influential, important, and deep. It had been conjectured for some time that the same result held for the primes (which of course have zero density). I shall discuss recent work with Ben Green obtaining this conjecture, by viewing the primes as a subset of the almost primes (numbers with few prime factors) of positive relative density. The point is that the almost primes are much easier to control than the primes themselves, thanks to sieve theory techniques such as the recent work of Goldston and Yildirim. To "transfer" Szemerédi's theorem to this relative setting requires that one borrow techniques from all four known proofs of Szemerédi's theorem, and especially from the ergodic theory proof.
This fantastic achievement, however, is only one of many, and 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 two 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 nn-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. Yet 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.

Let us mention briefly some other remarkable contributions by Tao. In 2015 he solved the Erdös discrepancy problem, proving the conjecture made by Erdös in 1932. In 2016 he produced a new approach to the Navier-Stokes equations which describe fluid flow. There are important questions regarding these equations which figure in the Clay Mathematics Institute's seven Millennium Problems. Tao has continued working on these equations making further major progress in 2019. Also in 2019, Tao made progress in solving the Collatz conjecture. Given any integer nin_{i}, then if nin_{i} is even, define ni+1=12nin_{i+1} = \large\frac{1}{2}\normalsize n_{i}, if nin_i is odd, define ni+1=3ni+1n_{i+1} = 3n_{i} + 1. The Collatz conjecture, made by Lothar Collatz in 1937, states that given any integer n0n_{0} the sequence given by the above procedure always reaches 1 in a finite number of steps [36]:-
Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades.
Despite Tao's partial result, the conjecture remains open.

In 2022, Tao, together with Rachel Greenfeld, found a counterexample to a tiling conjecture. In 2016 Siddhartha Bhattacharya proved that a single tile cannot tile the plane aperiodically but not periodically if only translations of the tile are allowed. It was conjectured that this would be true in higher dimensions, but Tao and Greenfeld found a single tile, full of twists and holes, which lives in a very high dimensional space (perhaps of dimension greater than 21001002^{100^{100}} ) which tiles that space aperiodically but not periodically.

A recent (2023) paper by Tao, in collaboration with Tamar Ziegler, proves the following theorem [91]:-
... there exist infinite sets A={a1,a2,...}A = \{ a_{1}, a_{2}, ... \} and B={b1,b2,...}B = \{b_{1}, b_{2}, ... \} of natural numbers such that ai+bja_{i} + b_{j} is prime whenever 1i<j1 ≤ i < j.
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 Solving mathematical problems which we mentioned above. In the same year he published a 2-volume textbook Analysis.

For more information about this book, see THIS LINK.

Also in 2006, Tao published Nonlinear dispersive equations, 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.

For more information about these two books, see THIS LINK.

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 types 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. You can read Tao's own thought on these blogs in the extracts from the Prefaces given at THIS LINK.

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. When we wrote the first version of Tao's biography in 2011 we anticipated a long and fascinating series of books that would appear over the next years. Indeed, we were correct! He published An introduction to measure theory (2011), Topics in random matrix theory (2012), Higher order Fourier analysis (2012), Compactness and contradiction (2013), Hilbert's fifth problem and related topics (2014), and Expansion in finite simple groups of Lie type (2015). We note that Hilbert's fifth problem and related topics was awarded the 2015 Prose Award for Best Mathematics Book. There is detailed information about these books, including Tao's Prefaces and extracts from reviews, at THIS LINK.

When asked about hobbies in the interview [116], he replied:-
With work and family there isn't as much time for fun these days - pretty much restricted now to activities that can be done on my phone in a few minutes. For instance in recent years I spend a few minutes each day learning a language (currently Hebrew) through an app. ... I play a little piano, although at this point I think both my kids surpass me in this.
Let us end with a quote from [8]:-
Tao says that his insights arrive, when they do, after much hard work. He gets ideas from reading, from other mathematicians, from taking long walks. Sometimes, one idea reminds him of a similar problem he saw somewhere else that might prove useful now. Most paths lead nowhere, but he learns something even in the cul de sacs. "Once you solve a problem," he continues, "you tend to remember only the short path that got you from A to B. You forget all the dead ends. It's a bit of a shame. It gives the wrong impression that people who are good at mathematics only choose the right steps. But there is a lot of trial and error and really terribly embarrassing ideas. Sometimes there is a 'eureka' moment, but it's more of a hitting-your-head moment: 'Of course, why was I so dense?'

References (show)

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  81. T Tao, Higher order Fourier analysis (American Mathematical Society, Providence, RI, 2012).
  82. T Tao, Compactness and contradiction (American Mathematical Society, Providence, RI, 2013).
  83. T Tao, Hilbert's fifth problem and related topics (American Mathematical Society, Providence, RI, 2014).
  84. T Tao, Expansion in finite simple groups of Lie type (American Mathematical Society, Providence, RI, 2015).
  85. T Tao, Does one have to be a genius to do maths?, Terry Tao's blog (23 July 2023).
  86. T Tao, Embracing change and resetting expectations, Microsoft Unlocked (12 June 2023).
  87. T Tao, What is good mathematics?, Bulletin of the American Mathematical Society 44 (4) (2007), 623-634.
  88. T Tao, Garth Gaudry, Terry Tao's blog (18 October 2012).
  89. T Tao, 'Hooked for life' on math, CNN (25 April 2010).
  90. T Tao, Elias Stein, Terry Tao's blog (24 December 2018).
  91. T Tao and T Ziegler, Infinite partial sumsets in the primes, arXiv:2301.10303v3 [math.NT].
  92. Terence Tao, 2002 Bôcher Prize, Notices of the American Mathematical Society 49 (4) (2002), 473-474.
  93. Terence Tao Named 2021 USIA Award Winner for Mathematics, United Sigma Intelligence Association (2021).
  94. Terence Tao Ph.D., The White House (22 September 2021).
  95. Terence Tao, Ph.D., Professor, University of California, Los Angeles, Simons Foundation.
  96. Terence Tao, Breakthrough Prize in Mathematics (2015).
  97. Terence Tao to receive inaugural Riemann Prize, Riemann International School of Mathematics (2019).
  98. Terence Tao quotes, Brainy Quote.
  99. Terence Tao, Great Immigrants 2019, Carnegie Corporation of New York.
  100. Terence Tao a Bolyai János Nemzetközi Matematikai Díj 2020-as kitüntetettje, Hungarian Academy of Sciences (3 May 2021).
  101. Terence Tao, Lauréat de la Grande Médaille 2022, The Academy of Sciences (21 March 2023).
  102. Terence Tao, The Royal Society.
  103. Terence Tao, Scholar, Institute for Advanced Study (2023).
  104. Terence Tao, American Academy of Arts & Sciences (September 2023).
  105. The 2003 Clay Research Awards, The Year 2003, Clay Mathematics Institute, 3.
  106. P V Tremmel, Major Math and Science Awards Announced: Leading economist and 'Mozart of math' receive Nemmers prizes, Northwestern University (12 April 2010).
  107. N Tzvetkov, Terence Tao et les équations aux dérivées partielles: un bref aperçu, Gazette des Mathématiciens (112) (2007), 23-25.
  108. L Vaserstein, Review: Additive combinatorics, by Terence Tao and Van H Vu, SIAM Review 52 (1) (2010), 215-216.
  109. D L Vestal, Review: Additive combinatorics, by Terence Tao and Van H Vu, Mathematical Association of America (6 June 2007).
  110. Vitae and Bibliography for Terence Tao, University of California, Los Angeles (16 October 2020).
  111. Who I am, University of California, Los Angeles (23 October 2021).
  112. S Wolpert, Terence Tao selected by Royal Society to receive 2014 Royal Medal, University of California, Los Angeles (6 August 2014).
  113. S Wolpert, Terence Tao, 'Mozart of Math,' is first UCLA math prof to win Fields Medal, University of California, Los Angeles (22 August 2006).
  114. S Wood, Terence Tao: the Mozart of maths, The Sydney Morning Herald (5 March 2015).
  115. World's greatest mathematician named 2022 Global Australian of the Year, Global Australian Awards (2022).
  116. First-Hand: IEEE Award Recipient Series: Terence Chi-Shen Tao, Engineering and Technology History Wiki (20 April 2021).

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Written by J J O'Connor and E F Robertson
Last Update December 2023