Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician and professor of mathematics at the University of California, Los Angeles (UCLA). He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory. Among his best-known results is the Green–Tao theorem, proved with Ben Green, which states that the prime numbers contain arbitrarily long arithmetic progressions.
Tao was born to Chinese immigrant parents and raised in Adelaide, South Australia. After earning his doctorate in mathematics from Princeton University and joining the faculty at UCLA, he went on to be a researcher, known for the diversity of his own interests and his collaborations with others. Tao has won many prizes for his work, including the Fields Medal in 2006 and the Royal Medal and Breakthrough Prize in Mathematics in 2014. He is a 2006 MacArthur Fellow.
Tao's published research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing, and analytic number theory. He is regarded by many as "the finest mathematician of his generation".
Early life and career Family Tao's father, Billy Tao, was a Chinese paediatrician who was born in Shanghai and received a medical degree (MBBS) from the University of Hong Kong in 1969.Dr Billy Tao, Healthshare. Tao's mother, Grace Leong, was born in Hong Kong; she received a first-class honours bachelor's degree majoring in mathematics and physics from the University of Hong Kong. She was a secondary school teacher of mathematics and physics in Hong Kong.Oriental Daily, Page A29, 24 August 2006. Billy and Grace met as students at the University of Hong Kong.Terence Chi-Shen Tao, MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland. They then emigrated from Hong Kong to Australia in 1972.Wen Wei Po, Page A4, 24 August 2006.
Tao also has two brothers, Trevor and Nigel, who are currently living in Australia. Both formerly represented Australia at the International Mathematical Olympiad.
Childhood A child prodigy,. Terence Tao skipped five grades. Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He is one of only three 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 Tao 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. Tao remains the youngest winner of each of the three medals in the Olympiad's history.
Career At age 14, Tao attended the Research Science Institute, a summer program for secondary students. In 1991, he received his bachelor's and master's degrees at the age of 16 from Flinders University under the direction of Garth Gaudry.It's prime time as numbers man Tao tops his Field Stephen Cauchi, 23 August 2006. Retrieved 31 August 2006. In 1992, he won a postgraduate Fulbright Scholarship to undertake research in mathematics at Princeton University in the United States. From 1992 to 1996, Tao was a graduate student at Princeton University under the direction of Elias Stein, receiving his PhD at the age of 21. In 1996, he joined the faculty of the University of California, Los Angeles. In 1999, when he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution.
He became known for fruitful collaborations and working in multiple specializations; by 2006, Tao had co-authored publications with over 30 other researchers, reaching 68 co-authors by October 2015.
Tao has had a particularly extensive collaboration with British mathematician Ben J. Green; together they proved the Green–Tao theorem, which is well known among both amateur and professional mathematicians. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:
Many other results of Tao have received mainstream attention in the scientific press, including: his establishment of finite time blowup for a modification of the Navier–Stokes existence and smoothness Millennium Problem his 2015 resolution of the Erdős discrepancy problem, which used entropy estimates within analytic number theory his 2019 progress on the Collatz conjecture, in which he proved the probabilistic claim that almost all Collatz orbits attain almost bounded values. Tao has also resolved or made progress on a number of conjectures. In 2012, Green and Tao announced proofs of the conjectured "orchard-planting problem," which asks for the maximum number of lines through exactly three points in a set of n points in the plane, not all on a line. In 2018, with Brad Rodgers, Tao showed that the de Bruijn–Newman constant, the nonpositivity of which is equivalent to the Riemann hypothesis, is nonnegative.. In 2020, Tao proved Sendov's conjecture, concerning the locations of the roots and critical points of a complex polynomial, in the special case of polynomials with sufficiently high degree. In 2024 and 2025, Tao solved problems 121, 442, 135, 685, 69, and 1102 from the Erdős problems website.
Recognition
Tao has won numerous awards and mathematician honours over the years. He was appointed a Companion of the Order of Australia in the 2026 King's Birthday Honours for "eminent service to the mathematical sciences, to the global mathematics community, and to tertiary education and academia". He is a Fellow of the Royal Society, the Australian Academy of Science (Corresponding Member), the National Academy of Sciences (Foreign member), the American Academy of Arts and Sciences, the American Philosophical Society, and the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 25 August 2013. In 2006 he received the Fields Medal; he was the first Australian, the first UCLA faculty member, and one of the youngest mathematicians to receive the award. He was also awarded the MacArthur Fellowship. In 2014, Tao was selected as one of the inaugural Breakthrough Prize laureates; he felt himself unqualified and unsuccessfully argued that the prize money be distributed among more researchers instead. After receiving the prize money, he used part of it to establish fellowships for students. In 2019, Tao was chosen as the inaugural winner of the Riemann Prize by the Riemann International School of Mathematics at the University of Insubria.
He has been featured in The New York Times, CNN, USA Today, Popular Science, and many other media outlets. Tao was a finalist to become Australian of the Year in 2007.National Australia Day Committee, Terence Tao – Australian of the Year. Retrieved 3 February 2023. In 2014, Tao received a CTY Distinguished Alumni Honor from Johns Hopkins Center for Gifted and Talented Youth in front of 979 attendees in 8th and 9th grade who were in the same program from which Tao graduated. In 2021, President Joe Biden announced Tao had been selected as one of 30 members of his President's Council of Advisors on Science and Technology, a body bringing together America's most distinguished leaders in science and technology.
As of 2026, Tao had published five hundred sixty articles, along with nineteen books. He has an Erdős number of 2 and is a highly cited researcher.
Per the New York Times, "many regard Tao as the finest mathematician of his generation." An article by New ScientistNewScientist.com, Prestigious Fields Medals for mathematics awarded, 22 August 2006. writes of his ability:
British mathematician and Fields medalist Timothy Gowers remarked on Tao's breadth of knowledge:Mathematical Reviews, Review by Timothy Gowers of Terence Tao's Poincaré's legacies, part I,
Research contributions Dispersive partial differential equations From 2001 to 2010, Tao was part of a collaboration with James Colliander, Markus Keel, Gigliola Staffilani, and Hideo Takaoka. They found a number of novel results, many to do with the well-posedness of weak solutions, for Schrödinger equations, KdV equations, and KdV-type equations.Michael Christ, Colliander, and Tao developed methods of Carlos Kenig, Gustavo Ponce, and Luis Vega to establish ill-posedness of certain Schrödinger and KdV equations for Sobolev data of sufficiently low exponents.Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106 (2001), no. 3, 617–633. In many cases these results were sharp enough to perfectly complement well-posedness results for sufficiently large exponents as due to Bourgain, Colliander−Keel−Staffilani−Takaoka−Tao, and others. Further such notable results for Schrödinger equations were found by Tao in collaboration with Ioan Bejenaru.
A particularly notable result of the Colliander−Keel−Staffilani−Takaoka−Tao collaboration established the long-time existence and scattering theory of a power-law Schrödinger equation in three dimensions. Their methods, which made use of the scale-invariance of the simple power law, were extended by Tao in collaboration with Monica Vișan and Xiaoyi Zhang to deal with nonlinearities in which the scale-invariance is broken. Rowan Killip, Tao, and Vișan later made notable progress on the two-dimensional problem in radial symmetry.
An article by Tao in 2001 considered the wave maps equation with two-dimensional domain and spherical range. He built upon earlier innovations of Daniel Tataru, who considered wave maps valued in Minkowski space.Tataru, Daniel. On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37–77. Tao proved the global well-posedness of solutions with sufficiently small initial data. The fundamental difficulty is that Tao considers smallness relative to the critical Sobolev norm, which typically requires sophisticated techniques. Tao later adapted some of his work on wave maps to the setting of the Benjamin–Ono equation; Alexandru Ionescu and Kenig later obtained improved results with Tao's methods.Ionescu, Alexandru D.; Kenig, Carlos E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Amer. Math. Soc. 20 (2007), no. 3, 753–798.
In 2016, Tao constructed a variant of the Navier–Stokes equations which possess solutions exhibiting irregular behavior in finite time. Due to structural similarities between Tao's system and the Navier–Stokes equations themselves, it follows that any positive resolution of the Navier–Stokes existence and smoothness problem must take into account the specific nonlinear structure of the equations. In particular, certain previously proposed resolutions of the problem could not be legitimate. Tao speculated that the Navier–Stokes equations might be able to simulate a Turing complete system, and that as a consequence it might be possible to (negatively) resolve the existence and smoothness problem using a modification of his results. However, such results remain (as of 2025) conjectural.
Harmonic analysis