## ⓘ Terence Tao

Terence Chi-Shen Tao is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles.

Tao was a recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of 275 research papers.

Tao is the second mathematician of Han Chinese descent to win the Fields medal after Shing-Tung Yau, and the first Australian citizen to win the medal.

### * 1.1. Personal life * Family

Taos parents are first generation immigrants from Hong Kong to Australia. Taos father, Dr. Billy Tao Chinese: 陶象國 ; pinyin: Tao Xiàngguo, was a pediatrician who was born in Shanghai, China and earned his medical degree MBBS from the University of Hong Kong in 1969. Taos mother, Grace Chinese: 梁蕙蘭, English: Leung Wai-lan, is from Hong Kong; she received a first-class honours degree in physics 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.

Tao has two brothers, Nigel and Trevor, living in Australia. Both formerly represented Australia at the International Mathematical Olympiad.

Taos wife, Laura, is an electrical engineer at NASAs Jet Propulsion Laboratory. They live with their son William and daughter Madeleine in Los Angeles, California.

### * 1.2. Personal life * Childhood

A child prodigy, Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He and Lenhard Ng are the 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 nine 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. He remains the youngest winner of each of the three medals in the Olympiads history, winning the gold medal at the age of 12 in 1988. He was said to possibly be the brightest of his age in the whole world.

At age 14, Tao attended the Research Science Institute. When he was 15, he published his first assistant paper. In 1991, he received his bachelors and masters degrees at the age of 16 from Flinders University under the direction of Garth Gaudry. In 1992, he won a Postgraduate Fulbright Scholarship to undertake research in mathematics topology 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.

## 2. Research and awards

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 Taos accomplishments:

Taos 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 Taos knowledge, and if you do then you may well find that the gaps have been filled a year later.

Tao has won numerous honours and awards over the years.

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, and the American Mathematical Society. In 2006 he received the Fields Medal "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory", and was also awarded the MacArthur Fellowship. He has been featured in The New York Times, CNN, USA Today, Popular Science, and many other media outlets.

As of 2019, Tao has published nearly 350 research papers and 17 books. He has an Erdos number of 2.

In 2018, Tao proved bounding the de Bruijn–Newman constant. In 2019, Tao proved for the Collatz Conjecture using probability that almost all Collatz orbits attain almost bounded values.

## 3. Green–Tao theorem and compressed sensing

In 2004, Ben Green and Tao released a preprint proving what is now known as the Green–Tao theorem. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:

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. 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.

For this and other work Tao was awarded the Australian Mathematical Society Medal of 2004. He was awarded a Fields Medal in August 2006 at the 25th International Congress of Mathematicians in Madrid. He was the first Australian, the first UCLA faculty member, and one of the youngest mathematicians to receive the award.

An article by New Scientist writes of his ability:

Such is Taos 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 youre stuck on a problem, then one way out is to interest Terence Tao," says Charles Fefferman.

Tao was a finalist to become Australian of the Year in 2007. He is a corresponding member of the Australian Academy of Science, and in 2007 was elected as a Fellow of the Royal Society. In the same year Tao also published Taos inequality, an extension to the Szemeredi regularity lemma in the field of information theory.

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 2010, he received the King Faisal International Prize jointly with Enrico Bombieri. Also in 2010, he was awarded the Nemmers Prize in Mathematics and the Polya Prize SIAM jointly with Emmanuel Candes for their work on Compressed Sensing.

## 4. Random matrices, Hardy–Littlewood prime tuples conjecture, approximate groups

In 2007, Tao and Van H. Vu solved the circular law conjecture.

In 2010, joint work with Ben Green culminated in the proof of the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.

Tao also made contributions to the study of the Erdos–Straus conjecture in 2011, by showing that the number of solutions to the Erdos–Straus equation increases polylogarithmically as n tends to infinity.

In 2012, he and Jean Bourgain received the Crafoord Prize in Mathematics from the Royal Swedish Academy of Sciences. Also, in 2012, he was selected as a Simons Investigator. He proved that every odd integer greater than 1 is the sum of at most five primes.

## 5. Other work

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.

In September 2015, Tao announced a proof of the Erdos discrepancy problem, using for the first time entropy-estimates within analytic number theory.

## 6. Notable awards

He received the Salem Prize in 2000, the Bocher Memorial Prize in 2002, and the Clay Research Award in 2003, for his contributions to analysis including work on the Kakeya conjecture and wave maps. In 2005, he received the American Mathematical Societys Levi L. Conant Prize with Allen Knutson for a proof of the Horn conjecture, and in 2006 he was awarded the SASTRA Ramanujan Prize.

- Levi L.Conant Prize 2005
- Johns Hopkins CTY Distinguished Alumnus 2014
- Salem Prize 2000
- Nemmers Prize in Mathematics 2010
- Polya Prize 2010
- Ostrowski Prize 2005
- Fellow of the Royal Society 2007
- Onsager Medal 2008
- MacArthur Award 2006
- SASTRA Ramanujan Prize 2006
- Australian Mathematical Society Medal 2005
- ISAAC award2005
- Bocher Memorial Prize 2002
- Inaugural recipient of the Center for Excellence in Educations Joseph I. Lieberman Award 2013
- Clay Research Award 2003
- PROSE award 2015
- Riemann Prize 2019
- Fields Medal 2006
- Inducted into the American Academy of Arts and Sciences 2009
- Royal Medal 2014
- Fulbright Scholarship 1992
- Breakthrough Prize in Mathematics 2015, awarded in 2014
- Convocation Award 2008
- Alan T. Waterman Award 2008
- Simons Investigator 2012
- Crafoord Prize 2012
- King Faisal International Prize 2010
- Sloan Fellowship 2006

## 7. Publications

- Compactness and Contradiction, American Mathematical Society, 2013 online version
- Analysis, Vols I and II, Hindustan Book Agency, 2006
- Poincares legacies: pages from year two of a mathematical blog, Vols. I and II, American Mathematical Society, 2009
- Expansion in Finite Simple Groups of Lie Type, American Mathematical Society, 2015 online version
- Higher-order Fourier Analysis, American Mathematical Society, 2012 online version
- An Introduction to Measure Theory. American Mathematical Society, 2011, online version
- Nonlinear dispersive equations: local and global analysis, CBMS regional series in mathematics, 2006.
- Topics in Random Matrix Theory, American Mathematical Society, 2012 online version
- An Epsilon of Room, I: Real Analysis: pages from year three of a mathematical blog, American Mathematical Society, 2011
- Solving Mathematical Problems: A Personal Perspective, Oxford University Press, 2006
- Hilberts Fifth Problem and Related Topics, American Mathematical Society, 2014 online version
- An Epsilon of Room, II: pages from year three of a mathematical blog, American Mathematical Society, 2011
- Structure and Randomness: pages from year one of a mathematical blog, American Mathematical Society. 2008
- Additive Combinatorics, with Van H. Vu, Cambridge University Press, 2006