Terence Tao
| Terence Tao | |
|---|---|
| |
| Institutions | University of California, Los Angeles |
| Nationality | Australian-American |
| Awards | Fields Medal (2006), MacArthur Fellowship, Breakthrough Prize in Mathematics, Crafoord Prize |
| Birth date | Template:Birth date and age |
| Doctoral advisor | Elias M. Stein |
| Fields | Mathematics |
| Birth place | Adelaide, South Australia |
| Known for | Green–Tao theorem, Additive combinatorics, Harmonic analysis, Partial differential equations |
| Alma mater | Princeton University (PhD, 1996) |
| Wikidata | Q295981 |
Terence Tao (born 1975) is an Australian–American mathematician whose work has transformed many areas of mathematics. A professor at UCLA (where he holds the James and Carol Collins Chair), Tao received the Fields Medal in 2006 for breakthroughs in partial differential equations, combinatorics, harmonic analysis, and additive number theory. His colleagues praise his talent – one UCLA professor quipped “Terry is like Mozart; mathematics just flows out of him” – and he is widely regarded as one of the greatest living mathematicians.
Early Life and Education
Tao was born in Adelaide, Australia in 1975 to ethnic Chinese immigrant parents. He showed extraordinary ability from childhood: he learned calculus by age 7 and excelled at math competitions. Notably, at 11 he began taking university-level math courses, and by age 13 he became the youngest-ever gold medalist at the International Mathematical Olympiad. At 14 he enrolled full-time at Flinders University (Australia), and at 15 he wrote his first book, Solving Mathematical Problems, aimed at helping students and teachers tackle advanced puzzles. He earned his B.Sc. with first-class honors in 1991 (at age 16).
In 1993 he went to Princeton University for graduate work under Elias Stein. Tao completed his Ph.D. in 1996 at age 20 with a thesis in harmonic analysis. The same year he joined UCLA’s faculty. Remarkably, he was promoted to full professor at UCLA by age 24, making him one of the youngest full professors ever. He later became a U.S. citizen and today holds the endowed James and Carol Collins Chair in UCLA’s College of Letters and Sciences. Over his career he has published hundreds of papers and expanded UCLA’s reputation in analysis and number theory.
Research Contributions
Tao’s research spans an unusually broad range of fields. Some of his most celebrated contributions include:
- Harmonic Analysis and PDE: Tao made foundational advances in Fourier analysis and wave equations. He proved sharp restriction estimates (studying how Fourier transforms behave on curved surfaces) and resolved long-standing Kakeya-type problems about minimal-size sets that contain a unit segment in every direction. He also solved important nonlinear wave and dispersive equations: for example, he established global existence (no blowup) for certain KdV- and wave-map equations. In the theory of fluid mechanics, he tackled the famous 3D Navier–Stokes equations (which model incompressible fluid flow). In 2014 he constructed an “averaged” Navier–Stokes model obeying the usual energy law but allowing a finite-time blowup. This result shows why the true Navier–Stokes problem is so hard: even when all known analytical estimates hold, a solution can still blow up. In short, Tao’s work in analysis has clarified the fine line between well-behaved and singular solutions of many equations.
- Number Theory and Additive Combinatorics: Tao revolutionized the study of prime numbers and related combinatorics. In 2004 he and Ben Green proved the Green–Tao theorem, showing that the prime numbers contain arithmetic progressions of every finite length. In other words, somewhere in the primes one can find sequences like (length 3), and in fact arbitrarily long ones; there are infinitely many progressions of length 100, 1,000, etc. Their proof cleverly combined combinatorial ideas (Szemerédi’s theorem on arithmetic progressions) with number theory techniques. This breakthrough earned prizes (e.g. Ostrowski, Nemmers) and inspired much further research on primes.
Tao also organized large-scale collaborations on primes. After Yitang Zhang proved in 2013 that infinitely many prime pairs differ by at most 70 million, Tao helped lead the Polymath 8 project: by crowdsourcing work online, the team quickly lowered the proven gap bound to below 10,000. (This effort involved dozens of researchers sharing ideas on Tao’s blog and forums.) In additive combinatorics more broadly, Tao co-authored a leading text Additive Combinatorics and established deep “sum–product” results showing that sets with additive structure force multiplicative structure and vice versa. In 2015, using both classical proofs and computer assistance, he resolved the 80-year-old Erdős Discrepancy Problem: he showed that any infinite sequence of +1s and –1s must have arbitrarily large imbalance (discrepancy) in some arithmetic progression. This creative mix of analysis and collaboration finally settled a question Paul Erdős posed long ago.
- Other Areas: Tao’s work also touches geometry, algebra, and applied mathematics. Along with his expository skill, he has solved problems in linear algebra and computer science. For example, with Allen Knutson he proved Horn’s conjecture about the possible eigenvalues of sums of Hermitian matrices. With Emmanuel Candès he developed the mathematical theory of compressed sensing – showing how sparse signals can be recovered from very few measurements – which earned them the 2010 George Pólya Prize. He has also worked on random matrix theory, numerical analysis, and ergodic theory, often moving between seemingly unrelated fields. (In total he has authored over 300 research papers and about 17 books, covering everything from graduate textbooks to research monographs.)
In all these areas, Tao combines “sheer technical power” with creative insight. His style emphasizes clarity and structure: as he once wrote about problem-solving, a good solution “should be relatively short, understandable, and hopefully have a touch of elegance. It should be fun to discover”. This philosophy underlies both his research and his expository work.
Public Engagement and Educational Outreach
Tao is also committed to teaching and public communication of mathematics. Unusually for a research mathematician, he began outreach very early. At 15 he wrote Solving Mathematical Problems: A Personal Perspective, a book intended to train high-school teachers and students in creative problem-solving strategies. Throughout his career he has written many expository pieces and textbooks on analysis, combinatorics, and more, aiming to explain ideas clearly. He maintains a popular blog “What’s New”, where he posts lecture notes, research expositions, and even answers to reader questions. For example, after a reader asked about the Collatz conjecture on his blog, Tao applied analytic ideas to prove a strong partial result on Collatz dynamics, demonstrating how he welcomes community input.
Tao has given numerous public lectures and taught online courses. His writing and speaking style is noted for being accessible and enthusiastic. Princeton’s alumni magazine described his book Solving Mathematical Problems (1990) as “engaging” and full of useful strategies for young learners. In total he has published about 17 books and over 300 papers, ranging from Olympiad guides to cutting-edge research monographs. His educational influence extends beyond professional math: many students and even non-mathematicians have encountered his clear explanations of complex topics via videos, interviews, and his blog posts.
Recognition and Impact
Tao’s towering contributions have been recognized by nearly every top honor in mathematics. In addition to the 2006 Fields Medal, he won a MacArthur “genius grant” in 2006 and the Breakthrough Prize in Mathematics in 2014. He shared the 2012 Crafoord Prize (often called a “Nobel Prize for mathematics”) with Jean Bourgain for their work in analysis, PDE, number theory, and combinatorics. He received the 2010 King Faisal Prize and the 2014 Royal Medal of the Royal Society, among others, and holds fellowships in prestigious societies (e.g. U.S. National Academy of Sciences, Royal Society).
These honors reflect Tao’s lasting impact. Experts note that he has made “deep contributions” that often bridge disparate fields. For example, his blend of ideas has led to new insights on prime distribution, data reconstruction, and geometric analysis. Several sources describe him as one of the most prolific and versatile mathematicians alive. As one citation puts it, “Terry Tao is one of the most universal, penetrating and prolific mathematicians in the world”. His work continues to inspire both specialists and students, shaping modern approaches in analytic number theory, combinatorics, and beyond.
In summary, Terence Tao’s career exemplifies a rare combination of creativity, technical mastery, and clarity. He has solved and advanced many famous problems, often by bringing together techniques from different areas. At the same time he makes mathematics engaging through his writing and teaching. This breadth and influence – from proving theorems about primes to writing books for learners – underline why Tao is regarded as a leading figure in contemporary mathematics.