Terence Tao

Q: What is the Green-Tao theorem?

Proved by Ben Green and Terence Tao in 2004: the primes contain arbitrarily long arithmetic progressions. That is, for any number k, there exist k primes in arithmetic progression (equally spaced). A deep result combining analytic number theory and ergodic theory.

Q: How prolific is Tao really?

Extremely. By his late forties he had published over 300 papers across almost every area of analysis and number theory. He is sometimes called the 'Mozart of mathematics' for his combination of precocity, range, and productivity.

Q: What did Tao prove about the Collatz conjecture?

In 2019 he proved that for almost all starting numbers, the Collatz sequence eventually becomes very small — one of the strongest results ever obtained on this notoriously resistant problem. The full conjecture remains open.

Terence Chi-Shen Tao (born 1975) is an Australian-American mathematician who has become one of the defining figures of 21st-century mathematics. A Fields Medalist at 31, he has contributed to analysis, number theory, combinatorics, partial differential equations, and several other fields — and done so at a pace that keeps him, year after year, at the top of the discipline.

Life

Tao was born in Adelaide, Australia, to immigrant parents from Hong Kong. He was an extreme prodigy: at 7 he was studying at high-school level; at 10, 11, and 12 he won medals in the International Mathematical Olympiad, becoming the youngest-ever medalist. He entered university at 9, got his master’s at 16, and his PhD from Princeton at 21.

He joined UCLA in 1996 at age 24 and has been there ever since. He is now Professor of Mathematics and holds the James and Carol Collins Chair.

Contributions

The Green–Tao theorem

In 2004, Tao and Ben Green proved a stunning result in analytic number theory: the primes contain arbitrarily long arithmetic progressions. That is, for any integer $k$ , there exist primes $p_1 < p_2 < \cdots < p_k$ that form an arithmetic progression (equally spaced).

The proof combined deep techniques from ergodic theory (via Furstenberg’s work) with combinatorial methods (via Szemerédi’s theorem). It was a spectacular technical achievement and opened a new frontier in additive combinatorics.

Compressed sensing and signal processing

With Emmanuel Candès, Tao developed mathematical foundations for compressed sensing — the theory that allows accurate reconstruction of signals from far fewer measurements than classical theory required. The results have had major applications in MRI imaging, where they can reduce scan times dramatically.

Collatz

In 2019, Tao published a paper showing that for almost every starting number, the Collatz sequence eventually becomes very small. Specifically, he proved that the minimum value of the Collatz iterates, normalized appropriately, is bounded by any function that tends to infinity — for a density-1 set of starting values. It is one of the strongest results ever obtained on the Collatz Conjecture, though it does not prove the full statement.

Work across fields

Tao is remarkable for his breadth. He publishes important work in:

Additive combinatorics (Green–Tao theorem and beyond)
Harmonic analysis (restriction theorems, dispersive equations)
Partial differential equations (Navier-Stokes, wave equations, Schrödinger equations)
Random matrix theory
Elementary number theory (including recent work on prime gaps)

As of 2025, he has authored or co-authored more than 300 papers and over 20 books.

Recognition

Tao received the Fields Medal in 2006, at age 31 — “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.” He has also received the Breakthrough Prize in Mathematics (2014, with three million dollars), the Crafoord Prize (2012), the MacArthur Fellowship (2006, the “genius grant”), and the King Faisal International Prize (2010).

He is a member of the US National Academy of Sciences and the Royal Society.

Public mathematics

Tao runs one of the most-read mathematics blogs in the world — What’s new — where he posts expositions, problem discussions, and informal thoughts. Many of his blog posts have grown into books. He has become, effectively, the most accessible and visible public face of elite pure mathematics in the 21st century.

He has also been thoughtful about the role of AI in mathematics, writing publicly about both the promise and the limits of large language models as mathematical collaborators.

Legacy

Tao is still mid-career. His legacy continues to accumulate. But even now, the combination of speed, range, depth, and public generosity has made him the most-cited mathematician of his generation — and the one most young mathematicians measure themselves against.

Known for

Green–Tao theorem on arithmetic progressions of primes
Fields Medal (2006)
Extensive work across analysis, number theory, and partial differential equations

Frequently asked

What is the Green-Tao theorem?

Proved by Ben Green and Terence Tao in 2004: the primes contain arbitrarily long arithmetic progressions. That is, for any number k, there exist k primes in arithmetic progression (equally spaced). A deep result combining analytic number theory and ergodic theory.

How prolific is Tao really?

Extremely. By his late forties he had published over 300 papers across almost every area of analysis and number theory. He is sometimes called the 'Mozart of mathematics' for his combination of precocity, range, and productivity.

What did Tao prove about the Collatz conjecture?

In 2019 he proved that for almost all starting numbers, the Collatz sequence eventually becomes very small — one of the strongest results ever obtained on this notoriously resistant problem. The full conjecture remains open.