Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the
University of California, Los Angeles
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the Californ ...
(UCLA), where he holds the James and Carol Collins chair. His research includes topics in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
,
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s,
algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in alg ...
,
arithmetic combinatorics,
geometric combinatorics Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets ...
,
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
,
compressed sensing and
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
.
Tao was born to ethnic Chinese immigrant parents and raised in
Adelaide
Adelaide ( ) is the capital city of South Australia, the state's largest city and the fifth-most populous city in Australia. "Adelaide" may refer to either Greater Adelaide (including the Adelaide Hills) or the Adelaide city centre. The dem ...
. Tao won the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
in 2006 and won the
Royal Medal
The Royal Medal, also known as The Queen's Medal and The King's Medal (depending on the gender of the monarch at the time of the award), is a silver-gilt medal, of which three are awarded each year by the Royal Society, two for "the most important ...
and
Breakthrough Prize in Mathematics
The Breakthrough Prize in Mathematics is an annual award of the Breakthrough Prize series announced in 2013.
It is funded by Yuri Milner and Mark Zuckerberg and others. The annual award comes with a cash gift of $3 million. The Breakthrough Pri ...
in 2014. He is also a
2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers. He is widely regarded as one of the greatest living mathematicians and has been referred to as the "
Mozart
Wolfgang Amadeus Mozart (27 January 17565 December 1791), baptised as Joannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the Classical period. Despite his short life, his rapid pace of composition r ...
of mathematics".
Life and career
Family
Tao's parents are first-generation
immigrants from
Hong Kong
Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China (abbr. Hong Kong SAR or HKSAR), is a List of cities in China, city and Special administrative regions of China, special ...
to
Australia
Australia, officially the Commonwealth of Australia, is a sovereign country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. With an area of , Australia is the largest country by ...
.
['']Wen Wei Po
''Wen Wei Po'' is a pro-Beijing state-owned newspaper based in Hong Kong. The newspaper was established in Hong Kong on 9 September 1948, after its Shanghai edition was launched in 1938.
Its head office is in the Hing Wai Centre () in Aber ...
'', Page A4, 24 August 2006. Tao's father, Billy Tao (), was a Chinese
paediatrician who was born in
Shanghai
Shanghai (; , , Standard Chinese, Standard Mandarin pronunciation: ) is one of the four Direct-administered municipalities of China, direct-administered municipalities of the China, People's Republic of China (PRC). The city is located on the ...
and earned his
medical degree (
MBBS
Bachelor of Medicine, Bachelor of Surgery ( la, Medicinae Baccalaureus, Baccalaureus Chirurgiae; abbreviated most commonly MBBS), is the primary medical degree awarded by medical schools in countries that follow the tradition of the United Kin ...
) from the
University of Hong Kong
The University of Hong Kong (HKU) (Chinese: 香港大學) is a public research university in Hong Kong
Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China (abbr. Hon ...
in 1969. Tao's mother, Grace Leong (), was born in Hong Kong; she received a first-class honours degree in
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
and mathematics at the University of Hong Kong.
[Terence Tao: the Mozart of maths](_blank)
7 March 2015, Stephanie Wood, The Sydney Morning Herald
''The Sydney Morning Herald'' (''SMH'') is a daily compact newspaper published in Sydney, New South Wales, Australia, and owned by Nine. Founded in 1831 as the ''Sydney Herald'', the ''Herald'' is the oldest continuously published newspaper ...
. 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 also has two brothers, who are living in Australia. Both formerly represented the country at the
International Mathematical Olympiad
The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre- university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, excep ...
.
[Nigel makes Waves: Google's bid to overthrow email](_blank)
Asher Moses, Sydney Morning Herald
''The Sydney Morning Herald'' (''SMH'') is a daily compact newspaper published in Sydney, New South Wales, Australia, and owned by Nine. Founded in 1831 as the ''Sydney Herald'', the ''Herald'' is the oldest continuously published newspaper ...
, 2 October 2009 Tao speaks Cantonese but cannot write Chinese. Tao is married to Laura Tao, a Chinese-American woman who is an electrical engineer at
NASA
The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research.
NASA was established in 1958, succeedin ...
's
Jet Propulsion Laboratory
The Jet Propulsion Laboratory (JPL) is a Federally funded research and development centers, federally funded research and development center and NASA field center in the City of La Cañada Flintridge, California, La Cañada Flintridge, California ...
.
They live with their son and daughter in
Los Angeles
Los Angeles ( ; es, Los Ángeles, link=no , ), often referred to by its initials L.A., is the largest city in the state of California and the second most populous city in the United States after New York City, as well as one of the world ...
, California.
Childhood
A
child prodigy
A child prodigy is defined in psychology research literature as a person under the age of ten who produces meaningful output in some domain at the level of an adult expert. The term is also applied more broadly to young people who are extraor ...
, 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
The Julian C. Stanley Study of Exceptional Talent (SET) is an outgrowth of the Study of Mathematically Precocious Youth (SMPY) at Johns Hopkins University. Founded in 1971 by Professor Julian Stanley, SMPY pioneered the concept of above-grade-leve ...
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
The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre- university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, excep ...
, 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, having won the gold medal at the age of 13 in 1988.
Career
At age 14, Tao attended the
Research Science Institute
The Research Science Institute (RSI) is an international summer research program for high school students. RSI is sponsored by the Center for Excellence in Education (CEE) and hosted by MIT in Cambridge, Massachusetts. RSI brings together the top S ...
, 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
Flinders University is a public research university based in Adelaide, South Australia, with a footprint extending across 11 locations in South Australia and the Northern Territory. Founded in 1966, it was named in honour of British navigator M ...
under the direction of Garth Gaudry.
[It's prime time as numbers man Tao tops his Field](_blank)
Stephen Cauchi, 23 August 2006. Retrieved 31 August 2006. In 1992, he won a Postgraduate
Fulbright Scholarship
The Fulbright Program, including the Fulbright–Hays Program, is one of several United States Cultural Exchange Programs with the goal of improving intercultural relations, cultural diplomacy, and intercultural competence between the people ...
to undertake research in mathematics at
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
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
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the Californ ...
. 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 is known for his collaborative mindset; by 2006, Tao had worked with over 30 others in his discoveries,
reaching 68 co-authors by October 2015.
Tao has had a particularly extensive collaboration with British mathematician
Ben J. Green
Ben Joseph Green FRS (born 27 February 1977) is a British mathematician, specialising in combinatorics and number theory. He is the Waynflete Professor of Pure Mathematics at the University of Oxford.
Early life and education
Ben Green was ...
; 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 number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. ''
The New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...
'' 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 famous
Navier–Stokes existence and smoothness Millennium Problem
[ ]
* his 2015 resolution of the
Erdős discrepancy problem, which used entropy estimates within
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
* his 2019 progress on the
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
, 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 3 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 de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function ''H''(''λ'', ''z''), where ''λ'' is a real paramete ...
, the nonpositivity of which is equivalent to the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
, 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.
Recognition
British mathematician and Fields medalist
Timothy Gowers remarked on Tao's breadth of knowledge:
An article by ''
New Scientist
''New Scientist'' is a magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organisation publish ...
'' writes of his ability:
Tao has won numerous mathematician honours and awards over the years. He is a
Fellow of the Royal Society
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathemati ...
, the
Australian Academy of Science (Corresponding Member), the
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...
(Foreign member), the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
, the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
, and the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
. In 2006 he received the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
; he was the first Australian, the first
UCLA
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California ...
faculty member, and one of the youngest mathematicians to receive the award.
He was also awarded the
MacArthur Fellowship. He has been featured in ''
The New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...
'',
CNN, ''
USA Today
''USA Today'' (stylized in all uppercase) is an American daily middle-market newspaper and news broadcasting company. Founded by Al Neuharth on September 15, 1982, the newspaper operates from Gannett's corporate headquarters in Tysons, Virgini ...
'', ''
Popular Science
''Popular Science'' (also known as ''PopSci'') is an American digital magazine carrying popular science content, which refers to articles for the general reader on science and technology subjects. ''Popular Science'' has won over 58 awards, incl ...
'', and many other media outlets. 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 that are 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. In 2021, Tao was awarded the
Riemann Prize Week as recipient of the inaugural Riemann Prize 2019 by the Riemann International School of Mathematics at the
University of Insubria
The University of Insubria ( it, Università degli Studi dell'Insubria) is an Italian university located in Como and Varese, with secondary locations in Busto Arsizio and Saronno. It was founded in 1998, it has been named after the area where it i ...
. Tao was a finalist to become
Australian of the Year
The Australian of the Year is a national award conferred on an Australian citizen by the National Australia Day Council, a not-for-profit Australian Governmentowned social enterprise. Similar awards are also conferred at the State and Territ ...
in 2007.
As of 2022, Tao has published over three hundred articles, along with sixteen books. He has an
Erdős number of 2. He is a
highly cited researcher
Clarivate Plc is a British-American publicly traded analytics company that operates a collection of subscription-based services, in the areas of bibliometrics and scientometrics; business / market intelligence, and competitive profiling for ...
.
Research contributions
Dispersive partial differential equations
From 2001 to 2010, Tao was part of a well-known collaboration with
James Colliander, Markus Keel,
Gigliola Staffilani
Gigliola Staffilani (born March 24, 1966) is an Italian-American mathematician who works as the Abby Rockefeller Mauze Professor of Mathematics at the Massachusetts Institute of Technology. , and Hideo Takaoka. They found a number of novel results, many to do with the
well-posedness of
weak solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precise ...
s, for
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
s,
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. 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.
A technical tour de force 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. 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 In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that
describes one-dimensional internal waves in deep water.
It was introduced by and .
The Benjamin–Ono equation is
:u_t+uu_x+Hu_=0
where ''H'' i ...
; Alexandru Ionescu and Kenig later obtained improved results with Tao's methods.
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
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...
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 2022) conjectural.
Harmonic analysis
Bent Fuglede introduced the
Fuglede conjecture in the 1970s, positing a
tile
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or ...
-based characterisation of those Euclidean domains for which a Fourier ensemble provides a basis of Tao resolved the conjecture in the negative for dimensions larger than 5, based upon the construction of an elementary counterexample to an analogous problem in the setting of
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
s.
With Camil Muscalu and
Christoph Thiele
Christoph Thiele (born 1968 in Bielefeld) is a German mathematician working in the field of harmonic analysis. After completing his undergraduate studies at TU Darmstadt and Bielefeld University, his Ph.D. was obtained in 1995 at Yale under the ...
, Tao considered certain multilinear
singular integral operators with the multiplier allowed to degenerate on a hyperplane, identifying conditions which ensure operator continuity relative to spaces. This unified and extended earlier notable results of
Ronald Coifman
Ronald Raphael Coifman is the Sterling professor of Mathematics at Yale University. Coifman earned a doctorate from the University of Geneva in 1965, supervised by Jovan Karamata.
Coifman is a member of the American Academy of Arts and Sciences, t ...
,
Carlos Kenig,
Michael Lacey,
Yves Meyer
Yves F. Meyer (; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017.
Biography
Born in Paris to a Jewish family, Yves Meyer ...
,
Elias Stein, and Thiele, among others. Similar problems were analyzed by Tao in 2001 in the context of Bourgain spaces, rather than the usual spaces. Such estimates are used in establishing well-posedness results for dispersive partial differential equations, following famous earlier work of
Jean Bourgain, Kenig,
Gustavo Ponce, and
Luis Vega, among others.
A number of Tao's results deal with "restriction" phenomena in Fourier analysis, which have been widely studied since seminal articles of
Charles Fefferman,
Robert Strichartz, and Peter Tomas in the 1970s. Here one studies the operation which restricts input functions on Euclidean space to a
submanifold and outputs the product of the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s of the corresponding measures. It is of major interest to identify exponents such that this operation is continuous relative to spaces. Such multilinear problems originated in the 1990s, including in notable work of
Jean Bourgain,
Sergiu Klainerman, and
Matei Machedon
Matei Machedon (born 10 February 1960 in Romania) is a Romanian-American mathematician, specializing in partial differential equations and mathematical physics.
Machedon graduated from the University of Chicago with B.A./M.S. in 1982. He received ...
. In collaboration with Ana Vargas and
Luis Vega, Tao made some foundational contributions to the study of the bilinear restriction problem, establishing new exponents and drawing connections to the linear restriction problem. They also found analogous results for the bilinear Kakeya problem which is based upon the
X-ray transform
In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon trans ...
instead of the Fourier transform. In 2003, Tao adapted ideas developed by
Thomas Wolff
Thomas Hartwig Wolff (July 14, 1954, New York City – July 31, 2000, Kern County) was a noted mathematician, working primarily in the fields of harmonic analysis, complex analysis, and partial differential equations. As an undergraduate at Harv ...
for bilinear restriction to conical sets into the setting of restriction to quadratic hypersurfaces. The multilinear setting for these problems was further developed by Tao in collaboration with
Jonathan Bennett and Anthony Carbery; their work was extensively used by Bourgain and
Larry Guth in deriving estimates for general
oscillatory integral operators.
Compressed sensing and statistics
In collaboration with
Emmanuel Candes and Justin Romberg, Tao has made notable contributions to the field of
compressed sensing. In mathematical terms, most of their results identify settings in which a convex optimisation problem correctly computes the solution of an optimisation problem which seems to lack a computationally tractable structure. These problems are of the nature of finding the solution of an underdetermined linear system with the minimal possible number of nonzero entries, referred to as "sparsity". Around the same time,
David Donoho considered similar problems from the alternative perspective of high-dimensional geometry.
Motivated by striking numerical experiments, Candes, Romberg, and Tao first studied the case where the matrix is given by the discrete Fourier transform. Candes and Tao abstracted the problem and introduced the notion of a "restricted linear isometry," which is a matrix that is quantitatively close to an isometry when restricted to certain subspaces. They showed that it is sufficient for either exact or optimally approximate recovery of sufficiently sparse solutions. Their proofs, which involved the theory of convex duality, were markedly simplified in collaboration with Romberg, to use only linear algebra and elementary ideas of harmonic analysis. These ideas and results were later improved by Candes. Candes and Tao also considered relaxations of the sparsity condition, such as power-law decay of coefficients. They complemented these results by drawing on a large corpus of past results in random matrix theory to show that, according to the Gaussian ensemble, a large number of matrices satisfy the restricted isometry property.
In 2009, Candes and Benjamin Recht considered an analogous problem for recovering a matrix from knowledge of only a few of its entries and the information that the matrix is of low rank. They formulated the problem in terms of convex optimisation, studying minimisation of the nuclear norm. Candes and Tao, in 2010, developed further results and techniques for the same problem. Improved results were later found by Recht. Similar problems and results have also been considered by a number of other authors.
In 2007, Candes and Tao introduced a novel statistical estimator for linear regression, which they called the "Dantzig selector." They proved a number of results on its success as an estimator and model selector, roughly in parallel to their earlier work on compressed sensing. A number of other authors have since studied the Dantzig selector, comparing it to similar objects such as the
statistical lasso introduced in the 1990s.
Trevor Hastie,
Robert Tibshirani
Robert Tibshirani (born July 10, 1956) is a professor in the Departments of Statistics and Biomedical Data Science at Stanford University. He was a professor at the University of Toronto from 1985 to 1998. In his work, he develops statistical ...
, and
Jerome H. Friedman
Jerome Harold Friedman (born December 29, 1939) is an American statistician, consultant and Professor of Statistics at Stanford University, known for his contributions in the field of statistics and data mining. conclude that it is "somewhat unsatisfactory" in a number of cases. Nonetheless it remains of significant interest in the statistical literature.
Random matrices
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 is large and the entries of a matrix are selected randomly according to any fixed
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
0 and
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
1, then the eigenvalues of will tend to be uniformly scattered across the disk of radius around the origin; this can be made precise using the language of
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
. 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 depend only 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
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
). 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
László () is a Hungarian male given name and surname after the King-Knight Saint Ladislaus I of Hungary (1077–1095). It derives from Ladislav, a variant of Vladislav. Other versions are Lessl or Laszly. The name has a history of being frequen ...
,
Horng-Tzer Yau
Horng-Tzer Yau (; born 1959 in Taiwan) is a Taiwanese-American mathematician. He received his B.Sc. in 1981 from National Taiwan University and his Ph.D. in 1987 from Princeton University. Yau joined the faculty of NYU in 1988, and became a full ...
, 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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of prime order. It is well known that there are no nontrivial
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
s 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
Sergei Vladimirovich Konyagin (russian: Серге́й Владимирович Конягин; born 25 April 1957) is a Russian mathematician. He is a professor of mathematics at the Moscow State University.
Konyagin participated in the Internat ...
.
Tao and
Ben Green proved the existence of arbitrarily long
arithmetic progressions in the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s; this result is generally referred to as the
Green–Tao theorem, and is among Tao's most well-known results. 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 matrix and a matrix whose components are all integers, Green and Tao give conditions on when there exist infinitely many matrices such that all components of 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.
Notable awards
* 1992 –
Fulbright Scholarship
The Fulbright Program, including the Fulbright–Hays Program, is one of several United States Cultural Exchange Programs with the goal of improving intercultural relations, cultural diplomacy, and intercultural competence between the people ...
* 1999 –
Packard Fellowship
The David and Lucile Packard Foundation is a private foundation that provides grants to not-for-profit organizations. It was created in 1964 by David Packard (co-founder of HP) and his wife Lucile Salter Packard. Following David Packard's death ...
* 2000 –
Salem Prize for:
::"his work in harmonic analysis and on related questions in
geometric measure theory and
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s."
* 2002 –
Bôcher Memorial Prize for:
::''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."
* 2003 –
Clay Research Award for:
::his restriction theorems in
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
, his work on
wave maps, his global existence theorems for KdV-type equations, and for his solution with
Allen Knutson of Horn's conjecture
* 2005 –
Australian Mathematical Society Medal
* 2005 –
Ostrowski Prize
The Ostrowski Prize is a mathematics award given every odd year for outstanding mathematical achievement judged by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands. Al ...
(with
Ben Green) for:
::"their exceptional achievements in the area of analytic and combinatorial number theory"
* 2005 –
Levi L.Conant Prize (with
Allen Knutson) for:
::their expository article "Honeycombs and Sums of Hermitian Matrices" (Notices of the AMS. 48 (2001), 175–186.)
* 2006 –
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
for:
::"his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory"
* 2006 –
MacArthur Award
The MacArthur Fellows Program, also known as the MacArthur Fellowship and commonly but unofficially known as the "Genius Grant", is a prize awarded annually by the John D. and Catherine T. MacArthur Foundation typically to between 20 and 30 indi ...
* 2006 –
SASTRA Ramanujan Prize
* 2006 –
Sloan Fellowship
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States.
...
* 2007 –
Fellow of the Royal Society
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathemati ...
* 2008 –
Alan T. Waterman Award for:
::"his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis"
* 2008 –
Onsager Medal for:
::"his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics". His Lars Onsager lecture was entitled "Structure and randomness in the prime numbers" at
NTNU, Norway.
* 2009 – Inducted into the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
* 2010 –
King Faisal International Prize
* 2010 –
Nemmers Prize in Mathematics
* 2010 –
Polya Prize (with
Emmanuel Candès)
* 2012 –
Crafoord Prize
* 2012 –
Simons Investigator
The Simons Foundation is a private foundation established in 1994 by Marilyn and Jim Simons with offices in New York City. As one of the largest charitable organizations in the US with assets of over $5 billion in 2022, the foundation's mission ...
* 2014 –
Breakthrough Prize in Mathematics
The Breakthrough Prize in Mathematics is an annual award of the Breakthrough Prize series announced in 2013.
It is funded by Yuri Milner and Mark Zuckerberg and others. The annual award comes with a cash gift of $3 million. The Breakthrough Pri ...
::"For numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations and analytic number theory."
* 2014 –
Royal Medal
The Royal Medal, also known as The Queen's Medal and The King's Medal (depending on the gender of the monarch at the time of the award), is a silver-gilt medal, of which three are awarded each year by the Royal Society, two for "the most important ...
* 2015 – PROSE award in the category of "Mathematics" for:
::"Hilbert's Fifth Problem and Related Topics"
* 2019 –
Riemann Prize
* 2020 –
Princess of Asturias Award for Technical and Scientific Research, with
Emmanuel Candès, for their work on
compressed sensing
* 2020 –
Bolyai Prize
* 2021 –
IEEE Jack S. Kilby Signal Processing Medal
The IEEE Jack S. Kilby Signal Processing Medal is presented "for outstanding achievements in signal processing" theory, technology or commerce. The recipients of this award will receive a gold medal, together with a replica in bronze, a certific ...
* 2021 –
USIA Award
* 2022 –
Education & Research award finalist
* 2022 - Global Australian of the Year (Advance Global Australians; Advance.org)
[World’s greatest mathematician named 2022 Global Australian of the Year](_blank)
Advance.org, media release 2022-09-08, accessed 2022-09-14[Why this maths genius refuses to work for a hedge fund](_blank)
Tess Bennett, Australian Financial Review
''The Australian Financial Review'' (abbreviated to the ''AFR'') is an Australian business-focused, compact daily newspaper covering the current business and economic affairs of Australia and the world. The newspaper is based in Sydney, New Sou ...
, 2022-09-07, accessed 2022-09-14
* 2022 -
Research.com Mathematics in United States Leader Award
Major publications
Textbooks
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Research articles. Tao is the author of over 300 articles. The following, among the most cited, are surveyed above.
See also
*
Erdős discrepancy problem
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Inscribed square problem
*
Goldbach's weak conjecture
*
Cramer conjecture
References
External links
Terence Tao's home pageTao's research blogTao's MathOverflow page*
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* Terence Tao's entry in th
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* (See
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
.)
* (See
Sendov's conjecture.)
*
* (See
Singmaster's conjecture.)
{{DEFAULTSORT:Tao, Terence Chi-Shen
1975 births
20th-century American mathematicians
21st-century American mathematicians
Additive combinatorialists
American bloggers
American people of Hong Kong descent
American people of Chinese descent
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Australian emigrants to the United States
Australian mathematicians
Australian people of Hong Kong descent
Australian people of Chinese descent
Clay Research Award recipients
Educators from California
Fellows of the American Academy of Arts and Sciences
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Fellows of the Australian Academy of Science
Fellows of the Royal Society
Fields Medalists
Flinders University alumni
Foreign associates of the National Academy of Sciences
Harmonic analysis
International Mathematical Olympiad participants
Living people
MacArthur Fellows
Mathematical analysts
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Number theorists
PDE theorists
Scientists from Adelaide
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