Tao's parents are first-generation immigrants from Hong Kong to Australia. Tao's father, Billy Tao (Chinese: ; pinyin: Táo Xiàngguó), was a pediatrician who was born in Shanghai, China, and earned his medical degree (MBBS) from the University of Hong Kong in 1969. Tao's mother, Grace (Chinese: ; Jyutping: Loeng4 Wai6-laan4), is from Hong Kong; she received a first-class honours degree in astrophysics and mathematics at the University of Hong Kong. She was a secondary school teacher of mathematics and physics in Hong Kong. Billy and Grace met as students at the University of Hong Kong. They then emigrated from Hong Kong to Australia in 1972.
A child prodigy, Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He is one of only two children in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just eight years old; Tao scored a 760.Julian Stanley, Director of the Study of Mathematically Precocious Youth stated that he had the greatest mathematical reasoning ability he had found in years of intensive searching. Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal, respectively. He remains the youngest winner of each of the three medals in the Olympiad's history, winning a gold medal at the age of 13 in 1988.
Within the field of mathematics, Tao is known for his collaboration with British mathematician Ben J. Green of Oxford University; together they proved the Green-Tao theorem. Known for his collaborative mindset, by 2006, Tao had worked with over 30 others in his discoveries, reaching 68 co-authors by October 2015.
In a book review, the British mathematician Timothy Gowers remarked on Tao's accomplishments:
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 of 3-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 David 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.
Tao has won numerous mathematician honours and awards over the years.
In 2004, Dr. Tao, along with Ben Green, a mathematician now at the University of Cambridge in England, solved a problem related to the Twin Prime Conjecture by looking at prime number progressions--series of numbers equally spaced. (For example, 3, 7 and 11 constitute a progression of prime numbers with a spacing of 4; the next number in the sequence, 15, is not prime.) Dr. Tao and Dr. Green proved that it is always possible to find, somewhere in the infinity of integers, a progression of prime numbers of equal spacing and any length.
Such is Tao's reputation that mathematicians now compete to interest him in their problems, and he is becoming a kind of Mr Fix-it for frustrated researchers. "If you're stuck on a problem, then one way out is to interest Terence Tao," says Charles Fefferman [professor of mathematics at Princeton University].
In April 2008, Tao received the Alan T. Waterman Award, which recognizes an early career scientist for outstanding contributions in their field. In addition to a medal, Waterman awardees also receive a $500,000 grant for advanced research.
In December 2008, he was named the Lars Onsager lecturer of 2008, for "his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics". He was presented the Onsager Medal, and held his Lars Onsager lecture entitled "Structure and randomness in the prime numbers" at NTNU, Norway.
Tao was also elected a Fellow of the American Academy of Arts and Sciences in 2009.
In the 1950s, Eugene Wigner initiated the study of random matrices and their eigenvalues. Wigner studied the case of hermitian and symmetric matrices, proving a "semicircle law" for their eigenvalues. In 2010, Tao and Van Vu made a major contribution to the study of non-symmetric random matrices. They showed that if n is large and the entries of a n × n matrix A are selected randomly according to any fixed probability distribution of average 0 and standard deviation 1, then the eigenvalues of A will tend to be uniformly scattered across the disk of radius n1/2 around the origin; this can be made precise using the language of measure theory. This gave a proof of the long-conjectured circular law, which had previously been proved in weaker formulations by many other authors. In Tao and Vu's formulation, the circular law becomes an immediate consequence of a "universality principle" stating that the distribution of the eigenvalues can only depend on the average and standard deviation of the given component-by-component probability distribution, thereby providing a reduction of the general circular law to a calculation for specially-chosen probability distributions.
In 2011, Tao and Vu established a "four moment theorem," which applies to random hermitian matrices whose components are independently distributed, each with average 0 and standard deviation 1, and which are exponentially unlikely to be large (as for a Gaussian distribution). If one considers two such random matrices which agree on the average value of any quadratic polynomial in the diagonal entries and on the average value of any quartic polynomial in the off-diagonal entries, then Tao and Vu show that the expected value of a large number of functions of the eigenvalues will also coincide, up to an error which is uniformly controllable by the size of the matrix and which becomes arbitrarily small as the size of the matrix increases. Similar results were obtained around the same time by László Erdös, Horng-Tzer Yau, and Jun Yin.
Analytic number theory and arithmetic combinatorics
In 2004, Tao, together with Jean Bourgain and Nets Katz, studied the additive and multiplicative structure of subsets of finite fields of prime order. It is well known that there are no nontrivial subrings of such a field. Bourgain, Katz, and Tao provided a quantitative formulation of this fact, showing that for any subset of such a field, the number of sums and products of elements of the subset must be quantitatively large, as compared to the size of the field and the size of the subset itself. Improvements of their result were later given by Bourgain, Alexey Glibichuk, and Sergei Konyagin.
In work published in 2008 but known since 2004, Tao and Ben Green proved the existence of arbitrarily long arithmetic progressions in the prime numbers; this result is generally referred to as the Green-Tao theorem. The source of Green and Tao's arithmetic progressions is Endre Szemerédi's seminal 1975 theorem on existence of arithmetic progressions in certain sets of integers. Green and Tao showed that one can use a "transference principle" to extend the validity of Szemerédi's theorem to further sets of integers. The Green-Tao theorem then arises as a special case, although it is not trivial to show that the prime numbers satisfy the conditions of Green and Tao's extension of the Szemerédi theorem.
In 2010, Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions. Given a k × n matrix A and a k × 1 matrix v whose components are all integers, Green and Tao give conditions on when there exist infinitely many n × 1 matrices x such that all components of Ax + v are prime numbers. The proof of Green and Tao was incomplete, as it was conditioned upon unproven conjectures. Those conjectures were proved in later work of Green, Tao, and Tamar Ziegler.
In 2012, in joint work with longtime co-author Ben Green, proofs were announced for the Dirac-Motzkin conjecture and the "orchard-planting problem" (which asks for the maximum number of lines through exactly 3 points in a set of n points in the plane, not all on a line). That same year, Tao published the first monograph on the topic of higher order Fourier analysis.
In 2014, Tao received a CTY Distinguished Alumni Honor from Johns Hopkins Center for Gifted and Talented Youth in front of 963 attendees in 8th and 9th grade that are in the same program from which Tao graduated. That year, Tao presented work on a possible attack on the Navier-Stokes existence and smoothness Millennium Problem, by establishing finite time blowup for an averaged three-dimensional Navier-Stokes equation.
That year he also, jointly with several co-authors, proved several results on short and long prime gaps.
Global regularity of wave maps I. Small critical Sobolev norm in high dimensions. Internat. Math. Res. Notices (2001), no. 6, 299-328.
Global regularity of wave maps II. Small energy in two dimensions. Comm. Math. Phys. 2244 (2001), no. 2, 443-544.
in addition to "his remarkable series of papers, written in collaboration with J. Colliander, M. Keel, G. Staffilani, and H. Takaoka, on global regularity in optimal Sobolev spaces for KdV and other equations, as well as his many deep contributions to Strichartz and bilinear estimates."
Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp. ISBN0-8218-4143-2
Structure and randomness. Pages from year one of a mathematical blog. American Mathematical Society, Providence, RI, 2008. xii+298 pp. ISBN978-0-8218-4695-7
Poincaré's legacies, pages from year two of a mathematical blog. Part I. American Mathematical Society, Providence, RI, 2009. x+293 pp. ISBN978-0-8218-4883-8
Poincaré's legacies, pages from year two of a mathematical blog. Part II. American Mathematical Society, Providence, RI, 2009. x+292 pp. ISBN978-0-8218-4885-2
Research articles on dispersive partial differential equations
Keel, Markus; Tao, Terence. Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955-980.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness for Schrödinger equations with derivative. SIAM J. Math. Anal. 33 (2001), no. 3, 649-669.
Tao, Terence. Global regularity of wave maps. II. Small energy in two dimensions. Comm. Math. Phys. 224 (2001), no. 2, 443-544.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. A refined global well-posedness result for Schrödinger equations with derivative. SIAM J. Math. Anal. 34 (2002), no. 1, 64-86.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation. Math. Res. Lett. 9 (2002), no. 5-6, 659-682.
Christ, Michael; Colliander, James; Tao, Terrence. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 (2003), no. 6, 1235-1293.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on R and T. J. Amer. Math. Soc. 16 (2003), no. 3, 705-749.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211 (2004), no. 1, 173-218.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3. Comm. Pure Appl. Math. 57 (2004), no. 8, 987-1014.
Tao, Terence. Global well-posedness of the Benjamin-Ono equation in H1(R). J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 27-49.
Bejenaru, Ioan; Tao, Terence. Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. J. Funct. Anal. 233 (2006), no. 1, 228-259.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3. Ann. of Math. (2) 167 (2008), no. 3, 767-865.
Killip, Rowan; Tao, Terence; Visan, Monica. The cubic nonlinear Schrödinger equation in two dimensions with radial data. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203-1258.
Research articles on compressed sensing
Candes, Emmanuel J.; Tao, Terence. Decoding by linear programming. IEEE Trans. Inform. Theory 51 (2005), no. 12, 4203-4215.
Candes, Emmanuel J.; Tao, Terence. Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52 (2006), no. 12, 5406-5425.
Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (2006), no. 8, 1207-1223.
Candès, Emmanuel J.; Romberg, Justin; Tao, Terence. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 (2006), no. 2, 489-509.
Candes, Emmanuel; Tao, Terence. The Dantzig selector: statistical estimation when p is much larger than n. Ann. Statist. 35 (2007), no. 6, 2313-2351.
Candès, Emmanuel J.; Tao, Terence. The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inform. Theory 56 (2010), no. 5, 2053-2080.
Research articles on analytic number theory
Bourgain, J.; Katz, N.; Tao, T. A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14 (2004), no. 1, 27-57.
Green, Ben; Tao, Terence. The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 (2008), no. 2, 481-547.
Green, Benjamin; Tao, Terence. Linear equations in primes. Ann. of Math. (2) 171 (2010), no. 3, 1753-1850.
Research articles on random matrices
Tao, Terence; Vu, Van. Random matrices: universality of ESDs and the circular law. With an appendix by Manjunath Krishnapur. Ann. Probab. 38 (2010), no. 5, 2023-2065.
Tao, Terence; Vu, Van. Random matrices: universality of local eigenvalue statistics. Acta Math. 206 (2011), no. 1, 127-204.
Research articles on harmonic analysis
Tao, Terence; Vargas, Ana; Vega, Luis. A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc. 11 (1998), no. 4, 967-1000.
Tao, Terence. Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123 (2001), no. 5, 839-908.
Muscalu, Camil; Tao, Terence; Thiele, Christoph. Multi-linear operators given by singular multipliers. J. Amer. Math. Soc. 15 (2002), no. 2, 469-496.
Tao, T. A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13 (2003), no. 6, 1359-1384.
Tao, Terence. Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11 (2004), no. 2-3, 251-258.
Bennett, Jonathan; Carbery, Anthony; Tao, Terence. On the multilinear restriction and Kakeya conjectures. Acta Math. 196 (2006), no. 2, 261-302.
Research articles on algebraic combinatorics
Knutson, Allen; Tao, Terence. The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090.
Knutson, Allen; Tao, Terence; Woodward, Christopher. The honeycomb model of GLn(C) tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone. J. Amer. Math. Soc. 17 (2004), no. 1, 19-48.