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{{Short description|Australian-American mathematician (born 1975)}}
{{Use dmy dates|date=February 2022}}
{{Use Australian English|date=May 2011}}
{{Infobox scientist
| name              = Terence Tao
| honorific_suffix  = [[Fellow of the Australian Academy of Science|FAA]] [[Fellow of the Royal Society|FRS]]
| image             = Terence Tao, PCAST Member (cropped).jpg
| caption           = 
| birth_name        = 
| birth_date        = {{birth date and age|df=yes|1975|7|17}}
| birth_place       = [[Adelaide]], South Australia, Australia
| spouse            = Laura Tao
| children          = 2
| citizenship       = {{hlist|Australia|United States<ref name="Vitae Tao">{{cite web|url=https://www.math.ucla.edu/~tao/preprints/cv.html|title=Vitae and Bibliography for Terence Tao |date=12 October 2009|access-date=21 January 2010}}</ref>}}
| fields            = [[Harmonic analysis]]
| alma_mater        = {{plainlist}}
* [[Flinders University]] ([[Bachelor of Science|BSc]], [[Master of Science|MSc]])
* [[Princeton University]] ([[PhD]])
{{endplainlist}}
| thesis_title      = Three Regularity Results in Harmonic Analysis<ref name="Vitae Tao" />
| thesis_year       = 1996
| doctoral_advisor  = [[Elias M. Stein]]
| doctoral_students = [[Monica Vișan]]
| known_for         = [[Partial differential equations]], [[analytic number theory]], [[random matrices]], [[compressed sensing]], [[combinatorics]], [[dynamical systems]]
| website           = {{ubl|{{URL|https://www.math.ucla.edu/~tao/}}|{{URL|https://terrytao.wordpress.com}}}}{{URL|https://mathstodon.xyz/@tao}}
| work_institutions = [[University of California, Los Angeles]]
| prizes            = {{nowrap|[[Fields Medal]]&nbsp;(2006)}}{{collapsible list
 |framestyle    = border:none; padding:0;
 |liststyle     = list-style-type:disc; text-align:left;
 |title         = {{nobold|''List''}}
   |[[Salem Prize]] (2000)|[[Bôcher Memorial Prize]] (2002)|[[Clay Research Award]] (2003)|[[Australian Mathematical Society Medal]] (2005)|[[Ostrowski Prize]] (2005)
   |[[Levi L. Conant Prize]] (2005)|[[MacArthur Award]] (2006)|[[SASTRA Ramanujan Prize]] (2006)|[[Sloan Fellowship]] (2006)|[[Fellow of the Royal Society]] (2007)
   |[[Alan T. Waterman Award]] (2008)|[[Onsager Medal]] (2008)| [[King Faisal International Prize]] (2010)<ref>[http://www.kff.com/ King Faisal Foundation] – retrieved 11 January 2010.</ref>|[[Nemmers Prize in Mathematics]] (2010)
   |[[Pólya Prize (SIAM)|Pólya Prize]] (2010)<ref>{{cite web|url=http://www.siam.org/prizes/sponsored/polya.php| archive-url=https://ghostarchive.org/archive/20211023/http://www.siam.org/prizes/sponsored/polya.php| archive-date=23 October 2021|title=SIAM: George Pólya Prize|access-date=5 September 2015}}{{cbignore}}</ref>|[[Crafoord Prize]] (2012)|[[Simons Investigator]] (2012)|[[Breakthrough Prize in Mathematics]] (2014)
   |[[Royal Medal]] (2014)|[[PROSE Awards|PROSE Award]] (2015)|[[Riemann Prize]] (2019)|[[Princess of Asturias Award]] (2020)|[[Bolyai Prize]] (2020)|[[IEEE Jack S. Kilby Signal Processing Medal]] (2021)|[[Global Australian of the Year Award]] (2022) | [[Grande Médaille]] (2023)| [[Best Paper Award]] (2023)| [[ Alexanderson Award]] (2023)}}
}}
{{Infobox Chinese
| t = 陶哲軒
| s = 陶哲轩
| p = Táo Zhéxuān
| suz = Dau Tseh-shie
| y = Tòuh Jit-hīn
| j = Tou4 Zit3-hin1
| ci = {{IPAc-yue|t|ou|4|-|z|it|3|.|h|in|1}}
}}
'''Terence Chi-Shen Tao''' {{post-nominals|country=AUS|FAA|FRS}} ({{zh|t=陶哲軒}}; born 17 July 1975) is an Australian-American mathematician, [[Fields Medal|Fields medalist]], and professor of mathematics at the [[University of California, Los Angeles]] (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences. His research includes topics in [[harmonic analysis]], [[partial differential equation]]s, [[algebraic combinatorics]], [[arithmetic combinatorics]], [[geometric combinatorics]], [[probability theory]], [[compressed sensing]] and [[analytic number theory]].<ref>{{Cite web | url=https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/| archive-url=https://ghostarchive.org/archive/20211023/https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/| archive-date=23 October 2021 | title=Mathematician Proves Huge Result on 'Dangerous' Problem| date=11 December 2019}}{{cbignore}}</ref>

Tao was born to Chinese immigrant parents and raised in [[Adelaide]]. Tao won the [[Fields Medal]] in 2006 and won the [[Royal Medal]] and [[Breakthrough Prize in Mathematics]] in 2014, and is a [[MacArthur Fellows Program|2006 MacArthur Fellow]]. Tao has been the author or co-author of over three hundred research papers, and is widely regarded as one of the greatest living mathematicians.<ref>{{Cite web |title=Search &#124; arXiv e-print repository |url=https://arxiv.org/search/?query=%22Terence+Tao%22&searchtype=author&source=header}}</ref><ref name="singmind">{{Cite news|last=Cook|first=Gareth|date=24 July 2015|title=The Singular Mind of Terry Tao (Published 2015)|work=The New York Times|url=https://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html|access-date=15 February 2021|issn=0362-4331}}</ref><ref>{{Cite web |first1=Dana|last1=Mackenzie|url=https://www.smithsonianmag.com/innovation/primed-for-success-166127128/| title=Primed for Success| date=2 October 2007}}{{cbignore}}</ref><ref>{{Cite web | url=https://bidenwhitehouse.archives.gov/pcast/members/terence-tao/| title=PRESIDENT'S COUNCIL OF ADVISORS ON SCIENCE AND TECHNOLOGY: Terence Tao, PhD| date=2021}}{{cbignore}}</ref><ref>{{Cite web | url=https://newsroom.ucla.edu/releases/Terence-Tao-Mozart-of-Math-7252| title=Terence Tao, 'Mozart of Math,' is first UCLA math prof to win Fields Medal| date=8 August 2006}}{{cbignore}}</ref><ref name="Terence Tao: the Mozart of maths" />

==Life and career==
===Family===
Tao's parents are first generation [[immigrants]] from [[Hong Kong]] to [[Australia]].<ref name="ReferenceA">''[[Wen Wei Po]]'', Page A4, 24 August 2006.</ref> Tao's father, Billy Tao,{{Efn|[[Chinese language|Chinese]]: 陶象國; [[pinyin]]: ''Táo Xiàngguó''|name=|group=a}} was a Chinese [[pediatrics|paediatrician]] who was born in [[Shanghai]] and earned his [[medical degree]] ([[MBBS]]) from the [[University of Hong Kong]] in 1969.<ref>[https://www.healthshare.com.au/profile/professional/135326-dr-billy-tao/ Dr Billy Tao], Healthshare.</ref> Tao's mother, Grace Leong,{{Efn|[[Chinese language|Chinese]]: 梁蕙蘭; [[Jyutping]]: ''Loeng⁴ Wai⁶-laan⁴''|name=|group=b}} was born in Hong Kong; she received a first-class honours degree in [[mathematics]] and [[physics]] at the [[University of Hong Kong]].<ref name="Terence Tao: the Mozart of maths">[https://www.smh.com.au/good-weekend/terence-tao-the-mozart-of-maths-20150216-13fwcv.html Terence Tao: the Mozart of maths], 7 March 2015, Stephanie Wood, [[The Sydney Morning Herald]].</ref> She was a secondary school teacher of mathematics and physics in Hong Kong.<ref>''[[Oriental Daily]]'', Page A29, 24 August 2006.</ref> Billy and Grace met as students at the University of Hong Kong.<ref>[http://www-history.mcs.st-andrews.ac.uk/Biographies/Tao.html Terence Chi-Shen Tao], MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland.</ref> They then emigrated from Hong Kong to Australia in 1972.<ref name="ReferenceA"/><ref name="Terence Tao: the Mozart of maths"/>

Tao also has two brothers, Trevor and Nigel, who are currently living in Australia. Both formerly represented Australia at the [[International Mathematical Olympiad]].<ref name="NigelIMO">{{cite web | title=International Mathematical Olympiad | website=
	International Mathematical Olympiad - Nigel Tao
 | url=https://www.imo-official.org/participant_r.aspx?id=783 | language=en  | access-date=5 October 2024}}</ref><ref name="TrevorIMO">{{cite web | title=International Mathematical Olympiad | website=
	International Mathematical Olympiad - Trevor Tao
 | url=https://www.imo-official.org/participant_r.aspx?id=3893 | language=en | access-date=5 October 2024}}</ref> Furthermore, Trevor Tao has been representing Australia internationally in chess and holds the title of Chess International Master.<ref>{{cite web | url=https://ratings.fide.com/profile/3201961 | title=Tao, Trevor }}</ref> 

Tao speaks Cantonese but cannot write Chinese. Tao is married to Laura Tao, an electrical engineer at [[NASA]]'s [[Jet Propulsion Laboratory]].<ref name="Terence Tao: the Mozart of maths" /><ref>{{Cite web |url=http://www.smithsonianmag.com/science-nature/tao.html |title=History, Travel, Arts, Science, People, Places – Smithsonian |url-status=dead |archive-url=https://archive.today/20120910212718/http://www.smithsonianmag.com/science-nature/tao.html |archive-date=10 September 2012 |access-date=5 September 2015}}</ref> They live in [[Los Angeles]], California, and have two children.<ref>{{Cite web |last=Wood |first=Stephanie |date=2015-03-04 |title=Terence Tao: the Mozart of maths |url=https://www.smh.com.au/lifestyle/terence-tao-the-mozart-of-maths-20150216-13fwcv.html |access-date=2023-02-13 |website=The Sydney Morning Herald |language=en}}</ref>

===Childhood===
A [[child prodigy]],<ref>{{Citation |last=Clements |first=M. A. (Ken) |title=Terence Tao |jstor=3482178 |journal=Educational Studies in Mathematics |volume=15 |issue=3 |pages=213–238 |year=1984 |doi=10.1007/BF00312075|s2cid=189827772 }}.</ref> Terence Tao skipped 5 grades.<ref>{{cite web | url=http://www.asiapacific-mathnews.com/01/0101/0023_0026.html | title=Asia Pacific Math Newsletter }}</ref><ref>{{cite web | url=https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking/chapters/terence-tao-s-journey | title=Terence Tao's Journey }}</ref>  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.<ref>{{Cite web|url=https://www.davidsongifted.org/resource-library/|title=Resources for Gifted Children & their Families|website=Davidson Institute}}</ref> [[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.<ref name="singmind" /><ref>{{Cite web| url=http://www.davidsongifted.org/Search-Database/entry/A10116| archive-url=https://ghostarchive.org/archive/20211023/http://www.davidsongifted.org/Search-Database/entry/A10116| archive-date=23 October 2021 |title=Radical Acceleration in Australia: Terence Tao |website=www.davidsongifted.org}}{{cbignore}}</ref>

Tao was the [[List of International Mathematical Olympiad participants|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, having won the gold medal at the age of 13 in 1988.<ref>{{Cite web|url=https://www.imo-official.org/year_info.aspx?year=1988|title=International Mathematical Olympiad|website=www.imo-official.org}}</ref>

===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.<ref name="theage.com.au">[http://www.theage.com.au/articles/2006/08/22/1156012542775.html It's prime time as numbers man Tao tops his Field] Stephen Cauchi, 23 August 2006. Retrieved 31 August 2006.</ref> 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.<ref name="theage.com.au" /> 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.<ref name="theage.com.au" />

He is known for his collaborative mindset; by 2006, Tao had worked with over 30 others in his discoveries,<ref name="singmind" /> 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 progression]]s of [[prime number]]s. ''[[The New York Times]]'' described it this way:<ref>{{cite news | url=https://www.nytimes.com/2007/03/13/science/13prof.html| archive-url=https://ghostarchive.org/archive/20211023/https://www.nytimes.com/2007/03/13/science/13prof.html| archive-date=23 October 2021 | title=Journeys to the Distant Fields of Prime | author=Kenneth Chang | date=13 March 2007 | work=The New York Times}}{{cbignore}}</ref><ref>{{cite news | url =https://www.nytimes.com/2007/03/13/pageoneplus/14corrections.ART-010.html| archive-url=https://ghostarchive.org/archive/20211023/https://www.nytimes.com/2007/03/13/pageoneplus/14corrections.ART-010.html| archive-date=23 October 2021 | title = Corrections: For the Record| date=13 March 2007| work =The New York Times}}{{cbignore}}</ref>

{{blockquote|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. (For example, 3, 7 and 11 constitute a progression of prime numbers with a spacing of 4; the next number in the sequence, 15, is not prime.) 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.}}

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 Prize Problems|Millennium Problem]]<ref name="quantanavier">{{cite web| url = https://www.quantamagazine.org/a-fluid-new-path-in-grand-math-challenge-20140224/| title = Quanta Magazine| date = 24 February 2014}}</ref>
* his 2015 resolution of the [[Erdős discrepancy problem]], which used entropy estimates within [[analytic number theory]]<ref>{{cite web|url=https://www.quantamagazine.org/terence-taos-answer-to-the-erdos-discrepancy-problem-20151001/| archive-url=https://web.archive.org/web/20190226123018/https://www.quantamagazine.org/terence-taos-answer-to-the-erdos-discrepancy-problem-20151001| archive-date=26 February 2019|title=Terence Tao's Answer to the Erdős Discrepancy Problem|website=Quanta Magazine|date=October 2015}}{{cbignore}}</ref>
* his 2019 progress on the [[Collatz conjecture]], in which he proved the probabilistic claim that almost all Collatz orbits attain almost bounded values.<ref>{{cite arXiv |last1=Tao |first1=Terence |title=Almost all orbits of the Collatz map attain almost bounded values |year=2019 |class=math.PR |eprint=1909.03562 }}</ref>
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 nonpositivity of which is equivalent to the [[Riemann hypothesis]], is nonnegative.<ref>{{cite journal|arxiv=1801.05914|date=6 April 2020|title=The De Bruijn-Newman constant is non-negative|last2=Tao|first2=Terence|last1=Rodgers|first1=Brad|journal=Forum of Mathematics, Pi|volume=8|pages=e6|doi=10.1017/fmp.2020.6
|doi-access=free}}.</ref> 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 of a polynomial|degree]].<ref>{{cite journal
| last1=Tao | first1=Terence 
| title=Sendov's conjecture for sufficiently high degree polynomials
| date=2022
| arxiv=2012.04125
| journal=Acta Mathematica
| volume=229
| issue=2
| pages=347–392
| doi=10.4310/ACTA.2022.v229.n2.a3 | doi-access=free}}</ref>

===Recognition===
[[File:Terence Tao.jpg|thumb|Tao at ICM 2006.]]

Tao has won numerous mathematician honours and awards over the years.<ref>{{cite web |url=https://www.math.ucla.edu/~tao/preprints/cv.html |title=Vitae |publisher= UCLA |access-date=5 September 2015}}</ref> 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]],<ref>{{Cite web|title=APS Member History|url=https://search.amphilsoc.org/memhist/search?creator=Terence+Tao&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced|access-date=19 March 2021|website=search.amphilsoc.org}}</ref> and the [[American Mathematical Society]].<ref>[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 25 August 2013.</ref> 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.<ref name="2006 Fields Medals awarded">{{cite journal| title = 2006 Fields Medals awarded | journal = [[Notices of the American Mathematical Society]]| volume = 53|issue = 9|date=October 2006 | url = http://www.ams.org/notices/200609/comm-prize-fields.pdf| archive-url=https://web.archive.org/web/20061102135505/http://www.ams.org/notices/200609/comm-prize-fields.pdf| archive-date=2 November 2006| pages= 1037–1044}}</ref><ref name="cbc.ca">{{cite news|url=https://www.cbc.ca/news/science/reclusive-russian-turns-down-math-world-s-highest-honour-1.622524| archive-url=https://ghostarchive.org/archive/20211023/http://www.cbc.ca/technology/story/2006/08/22/math-fields.html| archive-date=23 October 2021|title=Reclusive Russian turns down math world's highest honour|publisher=[[Canadian Broadcasting Corporation]] (CBC)|date=22 August 2006|access-date=26 August 2006}}{{cbignore}}</ref> He 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.<ref>{{cite web |url=https://www.math.ucla.edu/~tao/media.html| archive-url=https://ghostarchive.org/archive/20211023/https://www.math.ucla.edu/~tao/media.html| archive-date=23 October 2021 |title=Media information |publisher= UCLA |access-date=5 September 2015}}</ref> In 2014, Tao received a CTY Distinguished Alumni Honor from [[Center for Talented Youth|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.<ref>{{cite web|url=https://bidenwhitehouse.archives.gov/briefing-room/statements-releases/2021/09/22/president-biden-announces-members-of-presidents-council-of-advisors-on-science-and-technology/| archive-url=https://ghostarchive.org/archive/20211023/https://www.whitehouse.gov/briefing-room/statements-releases/2021/09/22/president-biden-announces-members-of-presidents-council-of-advisors-on-science-and-technology/| archive-date=23 October 2021|title=President Biden Announces Members of President's Council of Advisors on Science and Technology|website=White House|date=22 September 2021}}</ref> 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]].<ref>{{Cite news|date=21 September 2021|title=Terence Tao, il matematico con il QI più alto al mondo: "Non so cantare e ho fallito un paio di esami""|work=[[Huffington Post]] Italy|url=https://www.huffingtonpost.it/entry/terence-tao-il-matematico-con-il-qi-piu-alto-al-mondo-non-so-cantare-e-ho-fallito-un-paio-di-esami_it_614992f4e4b00171833fea6a| archive-url=https://ghostarchive.org/archive/20211023/https://www.huffingtonpost.it/entry/terence-tao-il-matematico-con-il-qi-piu-alto-al-mondo-non-so-cantare-e-ho-fallito-un-paio-di-esami_it_614992f4e4b00171833fea6a| archive-date=23 October 2021}}</ref> Tao was a finalist to become [[Australian of the Year]] in 2007.<ref>National Australia Day Committee, [https://web.archive.org/web/20221005030637/https://cms.australianoftheyear.org.au/recipients/professor-terence-tao Terence Tao – Australian of the Year]. Retrieved 3 February 2023.</ref>

As of 2022, Tao had published over three hundred articles, along with sixteen books.<ref>{{cite web|url=https://mathscinet.ams.org/mathscinet/MRAuthorID/361755|title=Terence C. Tao|website=[[MathSciNet]]|publisher=[[American Mathematical Society]]|access-date=November 24, 2022}}</ref> He has an [[Erdős number]] of 2.<ref>{{cite web |url=https://www.math.ucla.edu/~tao/whoami.html| archive-url=https://ghostarchive.org/archive/20211023/https://www.math.ucla.edu/~tao/whoami.html| archive-date=23 October 2021 |title=Who am I? |publisher= UCLA |access-date=5 September 2015}}</ref> He is a [[Highly Cited Researcher|highly cited researcher]].<ref>{{Cite web|title=Terence Tao's Publons profile|url=https://publons.com/researcher/2356887/terence-tao/| archive-url=https://ghostarchive.org/archive/20211023/https://publons.com/researcher/2356887/terence-tao/| archive-date=23 October 2021|access-date=6 February 2021|website=publons.com|language=en}}</ref><ref>{{Cite web|title=Highly Cited Researchers|url=https://publons.com/awards/highly-cited/2019/| archive-url=https://ghostarchive.org/archive/20211023/https://publons.com/awards/highly-cited/2019/| archive-date=23 October 2021|access-date=6 February 2021|website=publons.com}}</ref>

An article by ''[[New Scientist]]''<ref>''[[New Scientist|NewScientist.com]]'', [https://www.newscientist.com/article/dn9813-prestigious-fields-medals-for-mathematics-awarded.html Prestigious Fields Medals for mathematics awarded], 22 August 2006.</ref> writes of his ability:

{{blockquote|Such is Tao's 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 you're stuck on a problem, then one way out is to interest Terence Tao," says [[Charles Fefferman]] [professor of mathematics at Princeton University].<ref name="2006 Fields Medals awarded"/>}}

British mathematician and Fields medalist [[Timothy Gowers]] remarked on Tao's breadth of knowledge:<ref>Mathematical Reviews MR2523047, Review by Timothy Gowers of 
Terence Tao's ''Poincaré's legacies, part I'', http://mathscinet</ref>

{{blockquote|Tao's 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 Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later.}}

==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-posed problem|well-posedness]] of [[weak solution]]s, for [[Schrödinger equation]]s, [[Korteweg–De Vries equation|KdV equations]], and KdV-type equations.{{ran|C+03}}[[File:Paul Erdos with Terence Tao.jpg|left|thumb|Tao at the age of 10 with mathematician [[Paul Erdős]] in 1985]][[F. Michael Christ|Michael Christ]], Colliander, and Tao developed methods of [[Carlos Kenig]], [[Gustavo Ponce]], and [[Luis Vega (mathematician)|Luis Vega]] to establish ill-posedness of certain Schrödinger and KdV equations for Sobolev data of sufficiently low exponents.{{ran|CCT03}}<ref>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.</ref> 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.{{ran|BT06}}

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.{{ran|C+08}} 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.{{ran|TVZ07}} [[Rowan Killip]], Tao, and Vișan later made notable progress on the two-dimensional problem in radial symmetry.{{ran|KTV09}}

An article by Tao in 2001 considered the [[wave maps equation]] with two-dimensional domain and spherical range.{{ran|T01a}} He built upon earlier innovations of [[Daniel Tataru]], who considered wave maps valued in [[Minkowski space]].<ref>Tataru, Daniel. On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37–77.</ref> 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.{{ran|T04a}}<ref>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.</ref>

In 2016, Tao constructed a variant of the [[Navier–Stokes equations]] which possess solutions exhibiting irregular behavior in finite time.{{ran|T16}} 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|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.<ref>{{cite book|mr=3469428|last1=Lemarié-Rieusset|first1=Pierre Gilles|title=The Navier–Stokes problem in the 21st century|publisher=[[CRC Press]]|location=Boca Raton, FL|year=2016|isbn=978-1-4665-6621-7|doi=10.1201/b19556|zbl=1342.76029|s2cid=126089972 }}</ref> 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.<ref name="singmind" /><ref name="quantanavier" /> However, such results remain (as of 2024) conjectural.

===Harmonic analysis===
[[Bent Fuglede]] introduced the [[Fuglede's conjecture|Fuglede conjecture]] in the 1970s, positing a [[Tessellation|tile]]-based characterisation of those Euclidean domains for which a Fourier ensemble provides a basis of {{math|''L''<sup>2</sup>.}}<ref>Fuglede, Bent. Commuting self-adjoint partial differential operators and a group theoretic problem. J. Functional Analysis 16 (1974), 101–121.</ref> 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]]s.{{ran|T04b}}

With Camil Muscalu and [[Christoph Thiele]], Tao considered certain multilinear [[singular integral]] operators with the multiplier allowed to degenerate on a hyperplane, identifying conditions which ensure operator continuity relative to {{math|''L''<sup>''p''</sup>}} spaces.{{ran|MTT02}} This unified and extended earlier notable results of [[Ronald Coifman]], [[Carlos Kenig]], [[Michael Lacey (mathematician)|Michael Lacey]], [[Yves Meyer]], [[Elias Stein]], and Thiele, among others.<ref>Coifman, R. R.; Meyer, Yves On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315–331.</ref><ref>Coifman, R.; Meyer, Y. Commutateurs d'intégrales singulières et opérateurs multilinéaires. (French) Ann. Inst. Fourier (Grenoble) 28 (1978), no. 3, xi, 177–202.</ref><ref>Coifman, Ronald R.; Meyer, Yves Au delà des opérateurs pseudo-différentiels. Astérisque, 57. Société Mathématique de France, Paris, 1978. i+185 pp.</ref><ref>Lacey, Michael; Thiele, Christoph. {{math|''L''<sup>''p''</sup>}} estimates on the bilinear Hilbert transform for {{math|2<''p''<∞}}. Ann. of Math. (2) 146 (1997), no. 3, 693–724.</ref><ref>Lacey, Michael; Thiele, Christoph On Calderón's conjecture. Ann. of Math. (2) 149 (1999), no. 2, 475–496.</ref><ref>Kenig, Carlos E.; Stein, Elias M. Multilinear estimates and fractional integration. Math. Res. Lett. 6 (1999), no. 1, 1–15.</ref> Similar problems were analysed by Tao in 2001 in the context of Bourgain spaces, rather than the usual {{math|''L''<sup>''p''</sup>}} spaces.{{ran|T01b}} 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 (mathematician)|Luis Vega]], among others.<ref>Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996), no. 2, 573–603.</ref><ref>Ginibre, Jean. Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain). Séminaire Bourbaki, Vol. 1994/95. Astérisque No. 237 (1996), Exp. No. 796, 4, 163–187.</ref>

A number of Tao's results deal with "restriction" phenomena in Fourier analysis, which have been widely studied since the time of the articles of [[Charles Fefferman]], [[Robert Strichartz]], and Peter Tomas in the 1970s.<ref>Fefferman, Charles. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.</ref><ref>Tomas, Peter A. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477–478.</ref><ref>Strichartz, Robert S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705–714.</ref> Here one studies the operation which restricts input functions on Euclidean space to a [[submanifold]] and outputs the product of the [[Fourier transform]]s of the corresponding measures. It is of major interest to identify exponents such that this operation is continuous relative to {{math|''L''<sup>''p''</sup>}} spaces. Such multilinear problems originated in the 1990s, including in notable work of [[Jean Bourgain]], [[Sergiu Klainerman]], and [[Matei Machedon]].<ref>Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3 (1993), no. 2, 107–156</ref><ref>Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation. Geom. Funct. Anal. 3 (1993), no. 3, 209–262.</ref><ref>Klainerman, S.; Machedon, M. Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268.</ref> In collaboration with Ana Vargas and [[Luis Vega (mathematician)|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]] instead of the Fourier transform.{{ran|TVV98}} In 2003, Tao adapted ideas developed by [[Thomas Wolff]] for bilinear restriction to conical sets into the setting of restriction to quadratic hypersurfaces.{{ran|T03}}<ref>Wolff, Thomas. A sharp bilinear cone restriction estimate. Ann. of Math. (2) 153 (2001), no. 3, 661–698.</ref> The multilinear setting for these problems was further developed by Tao in collaboration with [[Jonathan Bennett (mathematician)|Jonathan Bennett]] and Anthony Carbery; their work was extensively used by Bourgain and [[Larry Guth]] in deriving estimates for general [[oscillatory integral|oscillatory integral operators]].{{ran|BCT06}}<ref>Bourgain, Jean; Guth, Larry. Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295.</ref>

===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.<ref>Donoho, David L. Compressed sensing. IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306.</ref>

Motivated by striking numerical experiments, Candes, Romberg, and Tao first studied the case where the matrix is given by the discrete Fourier transform.{{ran|CRT06a}} 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.{{ran|CT05}} 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.{{ran|CRT06b}} These ideas and results were later improved by Candes.<ref>Candès, Emmanuel J. The restricted isometry property and its implications for compressed sensing. C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 589–592.</ref> Candes and Tao also considered relaxations of the sparsity condition, such as power-law decay of coefficients.{{ran|CT06}} 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.{{ran|CT06}}

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.{{ran|CT07}} A number of other authors have since studied the Dantzig selector, comparing it to similar objects such as the [[Lasso (statistics)|statistical lasso]] introduced in the 1990s.<ref>Bickel, Peter J.; Ritov, Ya'acov; Tsybakov, Alexandre B. Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 (2009), no. 4, 1705–1732.</ref> [[Trevor Hastie]], [[Robert Tibshirani]], and [[Jerome H. Friedman]] conclude that it is "somewhat unsatisfactory" in a number of cases.<ref>Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome The elements of statistical learning. Data mining, inference, and prediction. Second edition. Springer Series in Statistics. Springer, New York, 2009. xxii+745 pp. {{ISBN|978-0-387-84857-0}}</ref> Nonetheless, it remains of significant interest in the statistical literature.

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.<ref>Candès, Emmanuel J.; Recht, Benjamin Exact matrix completion via convex optimization. Found. Comput. Math. 9 (2009), no. 6, 717–772.</ref> 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.{{ran|CT10}} Improved results were later found by Recht.<ref>Recht, Benjamin A simpler approach to matrix completion. J. Mach. Learn. Res. 12 (2011), 3413–3430.</ref> Similar problems and results have also been considered by a number of other authors.<ref>Keshavan, Raghunandan H.; Montanari, Andrea; Oh, Sewoong Matrix completion from a few entries. IEEE Trans. Inform. Theory 56 (2010), no. 6, 2980–2998.</ref><ref>Recht, Benjamin; Fazel, Maryam; Parrilo, Pablo A. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 (2010), no. 3, 471–501.</ref><ref>Candès, Emmanuel J.; Plan, Yaniv Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEEE Trans. Inform. Theory 57 (2011), no. 4, 2342–2359.</ref><ref>Koltchinskii, Vladimir; Lounici, Karim; Tsybakov, Alexandre B. Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist. 39 (2011), no. 5, 2302–2329.</ref><ref>Gross, David Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theory 57 (2011), no. 3, 1548–1566.</ref>

===Random matrices===
In the 1950s, [[Eugene Wigner]] initiated the study of [[random matrices]] and their eigenvalues.<ref>Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. [[Annals of Mathematics]] (2) 62 (1955), 548–564.</ref><ref>Wigner, Eugene P. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958), 325–327.</ref> Wigner studied the case of [[Hermitian matrix|hermitian]] and [[symmetric matrices]], proving a "semicircle law" for their eigenvalues. In 2010, Tao and [[Van H. Vu|Van Vu]] made a major contribution to the study of non-symmetric random matrices. They showed that if {{mvar|n}} is large and the entries of a {{math|''n'' × ''n''}} matrix {{mvar|A}} are selected randomly according to any fixed [[probability distribution]] of [[expected value|expectation]] 0 and [[standard deviation]] 1, then the eigenvalues of {{mvar|A}} will tend to be uniformly scattered across the disk of radius {{math|''n''<sup>1/2</sup>}} around the origin; this can be made precise using the language of [[measure theory]].{{ran|TV10}} 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 (mathematics)|moment]] 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]]). 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.{{ran|TV11}} Similar results were obtained around the same time by László Erdös, [[Horng-Tzer Yau]], and Jun Yin.<ref>{{cite journal
| last1=Erdős | first1=László
| last2=Yau | first2=Horng-Tzer | authorlink2=Horng-Tzer Yau
| last3=Yin | first3=Jun
| title=Rigidity of eigenvalues of generalized Wigner matrices
| journal=[[Advances in Mathematics]]
| volume=229
| date=2012
| issue=3
| pages=1435–1515
| doi=10.1016/j.aim.2011.12.010 | doi-access=free| arxiv=1007.4652
}}</ref><ref>{{cite journal
| last1=Erdős | first1=László
| last2=Yau | first2=Horng-Tzer | authorlink2=Horng-Tzer Yau
| last3=Yin | first3=Jun
| title=Bulk universality for generalized Wigner matrices
| journal=Probability Theory and Related Fields
| volume=154
| date=2012
| issue=1–2
| pages=341–407
| doi=10.1007/s00440-011-0390-3| s2cid=253977494
| doi-access=free
| arxiv=1001.3453
}}</ref>

===Analytic number theory and arithmetic combinatorics===
[[File:Professor Tao with his Devoted Students.jpg|thumb|Tao (second from left) with UCLA undergraduate students in 2021]]
In 2004, Tao, together with [[Jean Bourgain]] and [[Nets Katz]], studied the additive and multiplicative structure of subsets of [[finite fields]] of prime order.{{ran|BKT04}} It is well known that there are no nontrivial [[subring]]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]].<ref>Bourgain, J. More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (2005), no. 1, 1–32.</ref><ref>Bourgain, J.; Glibichuk, A.A.; Konyagin, S.V. Estimates for the number of sums and products and for exponential sums in fields of prime order. J. London Math. Soc. (2) 73 (2006), no. 2, 380–398.</ref>

Tao and [[Ben Green (mathematician)|Ben Green]] proved the existence of arbitrarily long [[arithmetic progression]]s in the [[prime number]]s; this result is generally referred to as the [[Green–Tao theorem]], and is among Tao's most well-known results.{{ran|GT08}} The source of Green and Tao's arithmetic progressions is [[Endre Szemerédi]]'s [[Szemerédi's theorem|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 [[Dirichlet's theorem on arithmetic progressions|theorem on arithmetic progressions]]. Given a {{math|''k'' × ''n''}} matrix {{mvar|A}} and a {{math|''k'' × 1}} matrix {{mvar|v}} whose components are all integers, Green and Tao give conditions on when there exist infinitely many {{math|''n'' × 1}} matrices {{mvar|x}} such that all components of {{math|''Ax'' + ''v''}} are prime numbers.{{ran|GT10}} 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]].{{ran|GTZ12}}

==Notable awards==
Terence Tao has won numerous awards for his work. He won the [[Fields Medal]], the highest award of mathematics, in 2006.
{{refbegin|30em}}
* 1999 – [[Packard Fellowship]]
* 2000 – [[Salem Prize]] for:<ref>[http://www.ams.org/notices/200008/people.pdf Mathematics People.] Notices of the AMS</ref>
::"his work in {{math|''L<sup>p</sup>''}} harmonic analysis and on related questions in [[geometric measure theory]] and [[partial differential equation]]s."
* 2002 – [[Bôcher Memorial Prize]] for:<ref>{{cite journal|journal=[[Notices of the American Mathematical Society]]|date=April 2002|volume=49|issue=4|title=2002 Bôcher Prize|pages=472–475|url=https://www.ams.org/notices/200204/comm-bocherprz.pdf}}</ref>
::''Global regularity of wave maps I. Small critical Sobolev norm in high dimensions.'' [[International Mathematics Research Notices|Internat. Math. Res. Notices]] (2001), no. 6, 299–328.
::''Global regularity of wave maps II. Small energy in two dimensions.'' [[Communications in Mathematical Physics|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:<ref>{{Cite web|url=https://www.claymath.org/news/research-fellow-james-maynard/|title=Research Fellow: James Maynard|website=Clay Mathematics Institute}}</ref>
::his restriction theorems in [[Fourier analysis]], his work on [[wave maps equation|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]] (with [[Ben Green (mathematician)|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]] for:
::"his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory"
* 2006 – [[MacArthur Award]]
* 2006 – [[SASTRA Ramanujan Prize]]<ref name="Alladi SASTRA">{{cite journal | last=Alladi | first=Krishnaswami | title=Ramanujan's legacy: the work of the SASTRA prize winners | journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences | publisher=The Royal Society | volume=378 | issue=2163 | date=2019-12-09 | issn=1364-503X | doi=10.1098/rsta.2018.0438 | page=20180438| pmid=31813370 | s2cid=198231874 | doi-access=free }}</ref>
* 2006 – [[Sloan Fellowship]]
* 2007 – [[Fellow of the Royal Society]]<ref>[http://royalsociety.org/WorkArea/DownloadAsset.aspx?id=4294969200 Fellows and Foreign Members of the Royal Society], retrieved 9 June 2010.</ref>
* 2008 – [[Alan T. Waterman Award]] for:<ref>[[National Science Foundation]], [https://www.nsf.gov/od/waterman/waterman.jsp Alan T. Waterman Award]. Retrieved 18 April 2008.</ref>
::"his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis"
* 2008 – [[Onsager Medal]]<ref>{{cite web|title=The Lars Onsager Lecture and Professorship – IMF|url=http://www.ntnu.no/imf/onsager|url-status=dead|archive-url=https://web.archive.org/web/20090203103723/http://www.ntnu.no/imf/onsager|archive-date=3 February 2009|access-date=13 January 2009}}</ref> 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 [[Norwegian University of Science and Technology|NTNU]], Norway.<ref>{{YouTube|L61HNEWuUbE|NTNU's Onsager Lecture, by Terence Tao}}</ref>
* 2009 – Inducted into the [[American Academy of Arts and Sciences]]<ref>{{Cite web|title=Alphabetical Index of Active AAAS Members|url=https://www.amacad.org/multimedia/pdfs/alphalist.pdf| archive-url=https://web.archive.org/web/20131005011917/https://www.amacad.org/multimedia/pdfs/alphalist.pdf| archive-date=5 October 2013|publisher=[[American Academy of Arts and Sciences|amacad.org]]|access-date=21 November 2013}}<br />His 2009 induction ceremony is [https://www.youtube.com/watch?v=Hxe2n_tejyQ here].</ref>
* 2010 – [[King Faisal International Prize]]<ref>{{cite journal|journal=[[Notices of the American Mathematical Society]]|title=Bombieri and Tao Receive King Faisal Prize|volume=57|issue=5|pages=642–643|date=May 2010|url=https://www.ams.org/notices/201005/rtx100500642p.pdf}}</ref>
* 2010 – [[Nemmers Prize in Mathematics]]<ref>{{cite web|url=http://www.northwestern.edu/newscenter/stories/2010/04/nemmers.html| archive-url=https://web.archive.org/web/20100416204215/http://www.northwestern.edu/newscenter/stories/2010/04/nemmers.html| archive-date=16 April 2010|title=Major Math and Science Awards Announced: Northwestern University News|access-date=5 September 2015}}</ref>
* 2010 – [[Pólya Prize (SIAM)|Polya Prize]] (with [[Emmanuel Candès]])
* 2012 – [[Crafoord Prize]]<ref>{{cite web |title=The Crafoord Prize in Mathematics 2012 and The Crafoord Prize in Astronomy 2012|author=<!--Staff writer(s); no by-line.--> |date=19 January 2012 |url=http://www.crafoordprize.se/press/arkivpressreleases/thecrafoordprizeinmathematics2012andthecrafoordprizeinastronomy2012.5.6018c17913483dc064280001363.html| archive-url=https://ghostarchive.org/archive/20211023/http://www.crafoordprize.se/press/arkivpressreleases/thecrafoordprizeinmathematics2012andthecrafoordprizeinastronomy2012.5.6018c17913483dc064280001363.html| archive-date=23 October 2021 |publisher= Royal Swedish Academy of Sciences|access-date=13 November 2014}}{{cbignore}}</ref><ref>{{cite web|url=http://chronicle.com/blogs/ticker/4-scholars-win-crafoord-prizes-in-astronomy-and-math/39807| archive-url=https://ghostarchive.org/archive/20211023/http://chronicle.com/blogs/ticker/4-scholars-win-crafoord-prizes-in-astronomy-and-math/39807| archive-date=23 October 2021|title=4 Scholars Win Crafoord Prizes in Astronomy and Math – The Ticker – Blogs – The Chronicle of Higher Education| date=19 January 2012|access-date=5 September 2015}}{{cbignore}}</ref>
* 2012 – [[Simons Investigator]]<ref>{{cite web|title=Simons Investigators Awardees|url=https://www.simonsfoundation.org/mathematics-physical-sciences/simons-investigators/simons-investigators-awardees/| archive-url=https://ghostarchive.org/archive/20211023/https://www.simonsfoundation.org/mathematics-physical-sciences/simons-investigators/simons-investigators-awardees/| archive-date=23 October 2021|work=Simons Foundation|access-date=9 September 2017}}{{cbignore}}</ref>
* 2014 – [[Breakthrough Prize in Mathematics]]
::"For numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations and analytic number theory."
* 2014 – [[Royal Medal]]
* 2015 – PROSE award in the category of "Mathematics" for:<ref>{{Cite web|url=https://proseawards.com/winners/2015-award-winners/|title=2015 Award Winners|website=PROSE Awards}}</ref>
::"Hilbert's Fifth Problem and Related Topics" {{ISBN|978-1-4704-1564-8}}
* 2019 – [[Riemann Prize]]<ref>{{cite web|url=https://www.rism.it/index.php/news/riemann-prize-2019| archive-url=https://web.archive.org/web/20191220093858/https://www.rism.it/index.php/news/riemann-prize-2019| archive-date=20 December 2019|title=Riemann Prize laureate 2019: Terence Tao|access-date=23 November 2019}}</ref>
* 2019 – [[The Carnegie Corporation of New York]] honored Tao with 2019 [[Great Immigrants Award|Great Immigrant Award]].<ref>{{Cite web|title=Terence Tao|url=https://www.carnegie.org/awards/honoree/terence-tao/ |access-date=2024-06-27|publisher=Carnegie Corporation of New York|language=en}}</ref>
* 2020 – [[Princess of Asturias Award]] for Technical and Scientific Research,<ref>{{cite web|title=Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès, Princess of Asturias Award for Technical and Scientific Research 2020|work=The Princess of Asturias Foundation |publisher=Princess of Asturias Foundation|url=https://www.fpa.es/en/princess-of-asturias-awards/laureates/2020-yves-meyer-ingrid-daubechies-terence-tao-and-emmanuel-candes.html| archive-url=https://web.archive.org/web/20200626135512/https://www.fpa.es/en/princess-of-asturias-awards/laureates/2020-yves-meyer-ingrid-daubechies-terence-tao-and-emmanuel-candes.html| archive-date=26 June 2020|access-date=23 June 2020}}</ref> with [[Emmanuel Candès]], for their work on [[compressed sensing]]
* 2020 – [[Bolyai Prize]]<ref>{{cite web|url=https://www.math.ucla.edu/~tao/preprints/cv.html|title=Vitae and Bibliography for Terence Tao|publisher=UCLA|access-date=13 November 2020}}</ref>
* 2021 – [[IEEE Jack S. Kilby Signal Processing Medal]]<ref name="IEEE Awards">{{cite web | title=IEEE Awards | website=IEEE Awards | date=2022-06-27 | url=https://corporate-awards.ieee.org/corporate-awards/ | access-date=2022-09-10}}</ref>
* 2022 – Global Australian of the Year (Advance Global Australians; Advance.org)<ref name="2022-09-08_A">[https://advance.org/app/uploads/2022/09/2022-Global-Australian-Awards_Winner_Media-Release_8-Sept.pdf World’s greatest mathematician named 2022 Global Australian of the Year], Advance.org, media release 2022-09-08, accessed 2022-09-14</ref><ref name="2022-09-07_FR">[https://www.afr.com/work-and-careers/leaders/why-this-maths-genius-refuses-to-work-for-a-hedge-fund-20220902-p5bew6 Why this maths genius refuses to work for a hedge fund], Tess Bennett, [[Australian Financial Review]], 2022-09-07, accessed 2022-09-14</ref>
* 2022 – [[Grande Médaille]]<ref>{{cite web|title=Séance de remise de la Grande médaille de l'Académie des sciences 2022 à Terence Tao|url=https://www.academie-sciences.fr/fr/Colloques-conferences-et-debats/ceremonie-grande-medaille-2022-terence-tao.html|date=21 March 2023|publisher=French Academy of Sciences|access-date=2024-07-10}}</ref>
* 2023 –  [[Alexanderson Award]] (with [[Kaisa Matomäki]], [[Maksym Radziwiłł]], Joni Teräväinen, and [[Tamar Ziegler]]) for:<ref>{{cite web|title=Alexanderson Award 2023|url=https://aimath.org/alexanderson-award-2023/|date=|publisher=American Institute of Mathematics|access-date=2024-10-07}}</ref>
::''Higher uniformity of bounded multiplicative functions in short intervals on average.'' [[Annals of Mathematics]], Second Series (2023), 197(2): 739–857.
{{refend}}

==Major publications==
{{refbegin}}

=== Textbooks ===
* {{Cite book |last=Tao |first=Terence |title=Solving mathematical problems. A personal perspective |publisher=[[Oxford University Press]] |year=2006 |isbn=978-0-19-920560-8 |edition=Second edition of 1992 original |mr=2265113 |zbl=1098.00006 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Nonlinear dispersive equations. Local and global analysis |publisher=[[American Mathematical Society]] |year=2006 |isbn=0-8218-4143-2 |series=CBMS Regional Conference Series in Mathematics |volume=106 |location=Providence, RI |doi=10.1090/cbms/106 |mr=2233925 |zbl=1106.35001 |author-mask=1}}
* {{Cite book |last1=Tao |first1=Terence |title=Additive combinatorics |last2=Vu |first2=Van H. |author-link2=Van H. Vu |publisher=[[Cambridge University Press]] |year=2006 |isbn=978-0-521-85386-6 |series=Cambridge Studies in Advanced Mathematics |volume=105 |mr=2289012 |zbl=1127.11002 |author-mask=1}}<ref>{{Cite journal |last=Green |first=Ben |author-link=Ben Green (mathematician) |year=2009 |title=''Review: ''Additive combinatorics'' by Terence C. Tao and Van H. Vu'' |url=http://www.ams.org/journals/bull/2009-46-03/S0273-0979-09-01231-2/S0273-0979-09-01231-2.pdf |journal=Bull. Amer. Math. Soc. (N.S.) |volume=46 |issue=3 |pages=489–497 |doi=10.1090/s0273-0979-09-01231-2 |archive-url=https://web.archive.org/web/20120311114011/http://www.ams.org/journals/bull/2009-46-03/S0273-0979-09-01231-2/S0273-0979-09-01231-2.pdf |archive-date=11 March 2012 |doi-access=free}}</ref><ref>{{Cite web |last=Vestal |first=Donald L. |date=June 6, 2007 |title=Review of ''Additive Combinatorics'' by Terence Tao and Van H. Vu |url=https://old.maa.org/press/maa-reviews/additive-combinatorics-0 |website=Mathematical Association of America}}</ref>
* {{Cite book |last=Tao |first=Terence |title=Structure and randomness. Pages from year one of a mathematical blog |publisher=American Mathematical Society |year=2008 |isbn=978-0-8218-4695-7 |location=Providence, RI |doi=10.1090/mbk/059 |mr=2459552 |zbl=1245.00024 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Poincaré's legacies, pages from year two of a mathematical blog. Part I |publisher=American Mathematical Society |year=2009 |isbn=978-0-8218-4883-8 |location=Providence, RI |doi=10.1090/mbk/066 |mr=2523047 |zbl=1171.00003 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Poincaré's legacies, pages from year two of a mathematical blog. Part II |publisher=American Mathematical Society |year=2009 |isbn=978-0-8218-4885-2 |location=Providence, RI |doi=10.1090/mbk/067 |mr=2541289 |zbl=1175.00010 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=An epsilon of room, I: real analysis. Pages from year three of a mathematical blog |publisher=American Mathematical Society |year=2010 |isbn=978-0-8218-5278-1 |series=Graduate Studies in Mathematics |volume=117 |location=Providence, RI |doi=10.1090/gsm/117 |mr=2760403 |zbl=1216.46002 |author-mask=1}}<ref>{{Cite web |last=Stenger |first=Allen |date=March 4, 2011 |title=Review of ''An Epsilon of Room, I: Real Analysis: Pages from year three of a mathematical blog'' by Terence Tao |url=https://old.maa.org/press/maa-reviews/an-epsilon-of-room-i-real-analysis-pages-from-year-three-of-a-mathematical-blog |website=Mathematical Association of America}}</ref>
* {{Cite book |last=Tao |first=Terence |title=An epsilon of room, II. Pages from year three of a mathematical blog |publisher=[American Mathematical Society |year=2010 |isbn=978-0-8218-5280-4 |location=Providence, RI |doi=10.1090/mbk/077 |mr=2780010 |zbl=1218.00001 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=An introduction to measure theory |publisher=American Mathematical Society |year=2011 |isbn=978-0-8218-6919-2 |series=Graduate Studies in Mathematics |volume=126 |location=Providence, RI |doi=10.1090/gsm/126 |mr=2827917 |zbl=1231.28001 |author-mask=1}}<ref>{{Cite web |last=Poplicher |first=Mihaela |date=April 14, 2012 |title=Review of ''An Introduction to Measure Theory'' by Terence Tao |url=https://old.maa.org/press/maa-reviews/an-introduction-to-measure-theory |website=Mathematical Association of America}}</ref>
* {{Cite book |last=Tao |first=Terence |title=Topics in random matrix theory |publisher=American Mathematical Society |year=2012 |isbn=978-0-8218-7430-1 |series=Graduate Studies in Mathematics |volume=132 |location=Providence, RI |doi=10.1090/gsm/132 |mr=2906465 |zbl=1256.15020 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Higher order Fourier analysis |publisher=American Mathematical Society |year=2012 |isbn=978-0-8218-8986-2 |series=Graduate Studies in Mathematics |volume=142 |location=Providence, RI |doi=10.1090/gsm/142 |mr=2931680 |zbl=1277.11010 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Compactness and contradiction |publisher=American Mathematical Society |year=2013 |isbn=978-0-8218-9492-7 |location=Providence, RI |doi=10.1090/mbk/081 |mr=3026767 |zbl=1276.00007 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Analysis. I |publisher=Hindustan Book Agency |year=2014 |isbn=978-93-80250-64-9 |edition=3rd |series=Texts and Readings in Mathematics |volume=37 |location=New Delhi |mr=3309891 |zbl=1300.26002 |author-mask=1 |orig-date=2006}}
* {{Cite book |last=Tao |first=Terence |title=Analysis. II |publisher=Hindustan Book Agency |year=2014 |isbn=978-93-80250-65-6 |edition=3rd |series=Texts and Readings in Mathematics |volume=38 |location=New Delhi |mr=3310023 |zbl=1300.26003 |author-mask=1 |orig-date=2006}}
* {{Cite book |last=Tao |first=Terence |title=Hilbert's fifth problem and related topics |publisher=American Mathematical Society |year=2014 |isbn=978-1-4704-1564-8 |series=Graduate Studies in Mathematics |volume=153 |location=Providence, RI |doi=10.1090/gsm/153 |mr=3237440 |zbl=1298.22001 |author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Expansion in finite simple groups of Lie type |publisher=American Mathematical Society |year=2015 |isbn=978-1-4704-2196-0 |series=Graduate Studies in Mathematics |volume=164 |location=Providence, RI |doi=10.1090/gsm/164 |mr=3309986 |zbl=1336.20015 |author-mask=1 |s2cid=118288443}}<ref>{{Cite journal |last=Lubotzky |first=Alexander |author-link=Alexander Lubotzky |date=25 January 2018 |title=Review of ''Expansion in finite simple groups of Lie type'' by Terence Tao |journal=Bull. Amer. Math. Soc. (N.S.) |pages=1 |doi=10.1090/bull/1610 |postscript=; review published electronically |doi-access=free}}</ref>
* {{Cite book |last=Tao |first=Terence |title=Analysis I |publisher=Springer Singapore |year=2022 |isbn=978-981-19-7261-4 |edition=4th |series=Texts and Readings in Mathematics |volume=37|author-mask=1}}
* {{Cite book |last=Tao |first=Terence |title=Analysis II |year=2022 |publisher=Springer Singapore |isbn=978-981-19-7284-3 |edition=4th |series=Texts and Readings in Mathematics |volume=38|author-mask=1}}

=== Research articles ===
{{Refbegin|30em}}
{{longitem|{{rma|KT98|tw=4em|{{Cite journal |last1=Keel |first1=Markus |last2=Tao |first2=Terence |year=1998 |title=Endpoint Strichartz estimates |journal=[[American Journal of Mathematics]] |volume=120 |issue=5 |pages=955–980 |citeseerx=10.1.1.599.1892 |doi=10.1353/ajm.1998.0039 |jstor=25098630 |mr=1646048 |s2cid=13012479 |zbl=0922.35028}}}}}}
{{longitem|{{rma|TVV98|tw=4em|{{Cite journal |last1=Tao |first1=Terence |last2=Vargas |first2=Ana |last3=Vega |first3=Luis |author-link3=Luis Vega (mathematician) |year=1998 |title=A bilinear approach to the restriction and Kakeya conjectures |journal=[[Journal of the American Mathematical Society]] |volume=11 |issue=4 |pages=967–1000 |doi=10.1090/S0894-0347-98-00278-1 |mr=1625056 |zbl=0924.42008 |doi-access=free}}}}}}
{{longitem|{{rma|KT99|tw=4em|{{Cite journal |last1=Knutson |first1=Allen |author-link=Allen Knutson |last2=Tao |first2=Terence |year=1999 |title=The honeycomb model of <math>GL_n(\mathbb{C})</math> tensor products. I. Proof of the saturation conjecture |journal=Journal of the American Mathematical Society |volume=12 |issue=4 |pages=1055–1090 |doi=10.1090/S0894-0347-99-00299-4 |mr=1671451 |zbl=0944.05097 |doi-access=free}}}}}}
{{longitem|{{rma|C+01|tw=4em|{{Cite journal |last1=Colliander |first1=J. |author-link=James Colliander |last2=Keel |first2=M. |last3=Staffilani |first3=G. |author-link3=Gigliola Staffilani |last4=Takaoka |first4=H. |last5=Tao |first5=T. |year=2001 |title=Global well-posedness for Schrödinger equations with derivative |journal=[[SIAM Journal on Mathematical Analysis]] |volume=33 |issue=3 |pages=649–669 |arxiv=math/0101263 |doi=10.1137/S0036141001384387 |mr=1871414 |zbl=1002.35113}}}}}}
{{longitem|{{rma|T01a|tw=4em|{{Cite journal |last=Tao |first=Terence |year=2001 |title=Global regularity of wave maps. II. Small energy in two dimensions |journal=[[Communications in Mathematical Physics]] |volume=224 |issue=2 |pages=443–544 |arxiv=math/0011173 |bibcode=2001CMaPh.224..443T |doi=10.1007/PL00005588 |mr=1869874 |s2cid=119634411 |zbl=1020.35046}} {{erratum|https://terrytao.wordpress.com/2016/08/26/an-erratum-to-global-regularity-of-wave-maps-ii-small-energy-in-two-dimensions|checked=yes}}}}}}
{{longitem|{{rma|T01b|tw=4em|{{Cite journal |last=Tao |first=Terence |year=2001 |title=Multilinear weighted convolution of L<sup>2</sup>-functions, and applications to nonlinear dispersive equations |journal=[[American Journal of Mathematics]] |volume=123 |issue=5 |pages=839–908 |arxiv=math/0005001 |doi=10.1353/ajm.2001.0035 |jstor=25099087 |mr=1854113 |s2cid=984131 |zbl=0998.42005}}}}}}
{{longitem|{{rma|C+02a|tw=4em|{{Cite journal |last1=Colliander |first1=J. |author-link=James Colliander |last2=Keel |first2=M. |last3=Staffilani |first3=G. |author-link3=Gigliola Staffilani |last4=Takaoka |first4=H. |last5=Tao |first5=T. |year=2002 |title=A refined global well-posedness result for Schrödinger equations with derivative |journal=[[SIAM Journal on Mathematical Analysis]] |volume=34 |issue=1 |pages=64–86 |arxiv=math/0110026 |doi=10.1137/S0036141001394541 |mr=1950826 |s2cid=9007785 |zbl=1034.35120}}}}}}
{{longitem|{{rma|C+02b|tw=4em|{{Cite journal |last1=Colliander |first1=J. |author-link=James Colliander |last2=Keel |first2=M. |last3=Staffilani |first3=G. |author-link3=Gigliola Staffilani |last4=Takaoka |first4=H. |last5=Tao |first5=T. |year=2002 |title=Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation |journal=Mathematical Research Letters |volume=9 |issue=5–6 |pages=659–682 |arxiv=math/0203218 |doi=10.4310/MRL.2002.v9.n5.a9 |mr=1906069 |zbl=1152.35491 |doi-access=free}}}}}}
{{longitem|{{rma|MTT02|tw=4em|{{Cite journal |last1=Muscalu |first1=Camil |last2=Tao |first2=Terence |last3=Thiele |first3=Christoph |author-link3=Christoph Thiele |year=2002 |title=Multi-linear operators given by singular multipliers |journal=[[Journal of the American Mathematical Society]] |volume=15 |issue=2 |pages=469–496 |doi=10.1090/S0894-0347-01-00379-4 |mr=1887641 |zbl=0994.42015 |doi-access=free}}}}}}
{{longitem|{{rma|CCT03|tw=4em|{{Cite journal |last1=Christ |first1=Michael |author-link=F. Michael Christ |last2=Colliander |first2=James |author-link2=James Colliander |last3=Tao |first3=Terrence |year=2003 |title=Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations |journal=[[American Journal of Mathematics]] |volume=125 |issue=6 |pages=1235–1293 |arxiv=math/0203044 |doi=10.1353/ajm.2003.0040 |mr=2018661 |s2cid=11001499 |zbl=1048.35101}}}}}}
{{longitem|{{rma|C+03|tw=4em|{{Cite journal |last1=Colliander |first1=J. |author-link=James Colliander |last2=Keel |first2=M. |last3=Staffilani |first3=G. |author-link3=Gigliola Staffilani |last4=Takaoka |first4=H. |last5=Tao |first5=T. |year=2003 |title=Sharp global well-posedness for KdV and modified KdV on <math>\mathbb{R}</math> and <math>\mathbb{T}</math> |journal=[[Journal of the American Mathematical Society]] |volume=16 |issue=3 |pages=705–749 |doi=10.1090/S0894-0347-03-00421-1 |mr=1969209 |zbl=1025.35025 |doi-access=free}}}}}}
{{longitem|{{rma|T03|tw=4em|{{Cite journal |last=Tao |first=T. |year=2003 |title=A sharp bilinear restrictions estimate for paraboloids |journal=[[Geometric and Functional Analysis]] |volume=13 |issue=6 |pages=1359–1384 |arxiv=math/0210084 |doi=10.1007/s00039-003-0449-0 |mr=2033842 |s2cid=15873489 |zbl=1068.42011}}}}}}
{{longitem|{{rma|BKT04|tw=4em|{{Cite journal |last1=Bourgain |first1=J. |author-link=Jean Bourgain |last2=Katz |first2=N. |author-link2=Nets Katz |last3=Tao |first3=T. |year=2004 |title=A sum-product estimate in finite fields, and applications |journal=[[Geometric and Functional Analysis]] |volume=14 |issue=1 |pages=27–57 |arxiv=math/0301343 |doi=10.1007/s00039-004-0451-1 |mr=2053599 |s2cid=14097626 |zbl=1145.11306}}}}}}
{{longitem|{{rma|C+04|tw=4em|{{Cite journal |last1=Colliander |first1=J. |author-link=James Colliander |last2=Keel |first2=M. |last3=Staffilani |first3=G. |author-link3=Gigliola Staffilani |last4=Takaoka |first4=H. |last5=Tao |first5=T. |year=2004 |title=Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on {{math|ℝ<sup>3</sup>}} |journal=[[Communications on Pure and Applied Mathematics]] |volume=57 |issue=8 |pages=987–1014 |arxiv=math/0301260 |doi=10.1002/cpa.20029 |mr=2053757 |s2cid=16423475 |zbl=1060.35131}}}}}}
{{longitem|{{rma|KTW04|tw=4em|{{Cite journal |last1=Knutson |first1=Allen |author-link=Allen Knutson |last2=Tao |first2=Terence |last3=Woodward |first3=Christopher |year=2004 |title=The honeycomb model of <math>GL_n(\mathbb{C})</math> tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone |journal=[[Journal of the American Mathematical Society]] |volume=17 |issue=1 |pages=19–48 |doi=10.1090/S0894-0347-03-00441-7 |mr=2015329 |zbl=1043.05111 |doi-access=free}}}}}}
{{longitem|{{rma|T04a|tw=4em|{{Cite journal |last=Tao |first=Terence |year=2004 |title=Global well-posedness of the Benjamin–Ono equation in {{math|''H''<sup>1</sup>(ℝ)}} |journal=[[Journal of Hyperbolic Differential Equations]] |volume=1 |issue=1 |pages=27–49 |arxiv=math/0307289 |doi=10.1142/S0219891604000032 |mr=2052470 |zbl=1055.35104}}}}}}
{{longitem|{{rma|T04b|tw=4em|{{Cite journal |last=Tao |first=Terence |year=2004 |title=Fuglede's conjecture is false in 5 and higher dimensions |journal=Mathematical Research Letters |volume=11 |issue=2–3 |pages=251–258 |arxiv=math/0306134 |doi=10.4310/MRL.2004.v11.n2.a8 |mr=2067470 |zbl=1092.42014 |doi-access=free}}}}}}
{{longitem|{{rma|CT05|tw=4em|{{Cite journal |last1=Candes |first1=Emmanuel J. |author-link=Emmanuel Candes |last2=Tao |first2=Terence |year=2005 |title=Decoding by linear programming |journal=[[IEEE Transactions on Information Theory]] |volume=51 |issue=12 |pages=4203–4215 |arxiv=math/0502327 |doi=10.1109/TIT.2005.858979 |mr=2243152 |s2cid=12605120 |zbl=1264.94121}}}}}}
{{longitem|{{rma|BT06|tw=4em|{{Cite journal |last1=Bejenaru |first1=Ioan |last2=Tao |first2=Terence |year=2006 |title=Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation |journal=[[Journal of Functional Analysis]] |volume=233 |issue=1 |pages=228–259 |arxiv=math/0508210 |doi=10.1016/j.jfa.2005.08.004 |mr=2204680 |zbl=1090.35162 |doi-access=free}}}}}}
{{longitem|{{rma|BCT06|tw=4em|{{Cite journal |last1=Bennett |first1=Jonathan |author-link=Jonathan Bennett (mathematician) |last2=Carbery |first2=Anthony |last3=Tao, Terence |year=2006 |title=On the multilinear restriction and Kakeya conjectures |journal=[[Acta Mathematica]] |volume=196 |issue=2 |pages=261–302 |arxiv=math/0509262 |doi=10.1007/s11511-006-0006-4 |mr=2275834 |zbl=1203.42019 |doi-access=free}}}}}}
{{longitem|{{rma|CRT06a|tw=4em|{{Cite journal |last1=Candès |first1=Emmanuel J. |author-link=Emmanuel Candes |last2=Romberg |first2=Justin K. |last3=Tao, Terence |year=2006 |title=Stable signal recovery from incomplete and inaccurate measurements |journal=[[Communications on Pure and Applied Mathematics]] |volume=59 |issue=8 |pages=1207–1223 |arxiv=math/0503066 |doi=10.1002/cpa.20124 |mr=2230846 |s2cid=119159284 |zbl=1098.94009}}}}}}
{{longitem|{{rma|CRT06b|tw=4em|{{Cite journal |last1=Candès |first1=Emmanuel J. |author-link=Emmanuel Candes |last2=Romberg |first2=Justin |last3=Tao, Terence |year=2006 |title=Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information |journal=[[IEEE Transactions on Information Theory]] |volume=52 |issue=2 |pages=489–509 |arxiv=math/0409186 |doi=10.1109/TIT.2005.862083 |mr=2236170 |s2cid=7033413 |zbl=1231.94017}}}}}}
{{longitem|{{rma|CT06|tw=4em|{{Cite journal |last1=Candes |first1=Emmanuel J. |author-link=Emmanuel Candes |last2=Tao |first2=Terence |year=2006 |title=Near-optimal signal recovery from random projections: universal encoding strategies? |journal=[[IEEE Transactions on Information Theory]] |volume=52 |issue=12 |pages=5406–5425 |arxiv=math/0410542 |doi=10.1109/TIT.2006.885507 |mr=2300700 |s2cid=1431305 |zbl=1309.94033}}}}}}
{{longitem|{{rma|CT07|tw=4em|{{Cite journal |last1=Candes |first1=Emmanuel |author-link=Emmanuel Candes |last2=Tao |first2=Terence |year=2007 |title=The Dantzig selector: statistical estimation when p is much larger than n |journal=[[Annals of Statistics]] |volume=35 |issue=6 |pages=2313–2351 |arxiv=math/0506081 |doi=10.1214/009053606000001523 |mr=2382644 |zbl=1139.62019 |doi-access=free}}}}}}
{{longitem|{{rma|TVZ07|tw=4em|{{Cite journal |last1=Tao |first1=Terence |last2=Visan |first2=Monica |author-link2=Monica Vișan |last3=Zhang |first3=Xiaoyi |year=2007 |title=The nonlinear Schrödinger equation with combined power-type nonlinearities |journal=Communications in Partial Differential Equations |volume=32 |issue=7–9 |pages=1281–1343 |arxiv=math/0511070 |doi=10.1080/03605300701588805 |mr=2354495 |s2cid=15109526 |zbl=1187.35245}}}}}}
{{longitem|{{rma|C+08|tw=4em|{{Cite journal |last1=Colliander |first1=J. |author-link=James Colliander |last2=Keel |first2=M. |last3=Staffilani |first3=G. |author-link3=Gigliola Staffilani |last4=Takaoka |first4=H. |last5=Tao |first5=T. |year=2008 |title=Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in {{math|ℝ<sup>3</sup>}} |journal=[[Annals of Mathematics]] |series=Second Series |volume=167 |issue=3 |pages=767–865 |doi=10.4007/annals.2008.167.767 |mr=2415387 |zbl=1178.35345 |doi-access=free}}}}}}
{{longitem|{{rma|GT08|tw=4em| {{Cite journal |last1=Green |first1=Ben |author-link=Ben Green (mathematician) |last2=Tao |first2=Terence |year=2008 |title=The primes contain arbitrarily long arithmetic progressions |journal=[[Annals of Mathematics]] |series=Second Series |volume=167 |issue=2 |pages=481–547 |arxiv=math/0404188 |doi=10.4007/annals.2008.167.481 |mr=2415379 |zbl=1191.11025 |doi-access=free}}}}}}
{{longitem|{{rma|KTV09|tw=4em|{{Cite journal |last1=Killip |first1=Rowan |last2=Tao |first2=Terence |last3=Visan |first3=Monica |author-link3=Monica Vișan |year=2009 |title=The cubic nonlinear Schrödinger equation in two dimensions with radial data |journal=[[Journal of the European Mathematical Society]] |volume=11 |issue=6 |pages=1203–1258 |arxiv=0707.3188 |doi=10.4171/JEMS/180 |mr=2557134 |zbl=1187.35237 |doi-access=free}}}}}}
{{longitem|{{rma|CT10|tw=4em|{{Cite journal |last1=Candès |first1=Emmanuel J. |author-link=Emmanuel Candes |last2=Tao |first2=Terence |year=2010 |title=The power of convex relaxation: near-optimal matrix completion |journal=[[IEEE Transactions on Information Theory]] |volume=56 |issue=5 |pages=2053–2080 |arxiv=0903.1476 |doi=10.1109/TIT.2010.2044061 |mr=2723472 |s2cid=1255437 |zbl=1366.15021}}}}}}
{{longitem|{{rma|GT10|tw=4em|{{Cite journal |last1=Green |first1=Ben |author-link=Ben Green (mathematician) |last2=Tao |first2=Terence |year=2008 |title=The primes contain arbitrarily long arithmetic progressions |journal=[[Annals of Mathematics]] |series=Second Series |volume=167 |issue=2 |pages=481–547 |arxiv=math/0404188 |doi=10.4007/annals.2008.167.481 |mr=2415379 |zbl=1191.11025 |doi-access=free}}}}}}
{{longitem|{{rma|TV10|tw=4em|{{Cite journal |last1=Tao |first1=Terence |last2=Vu |first2=Van |author-link2=Van Vu |year=2010 |others=With an appendix by Manjunath Krishnapur |title=Random matrices: universality of ESDs and the circular law |journal=[[Annals of Probability]] |volume=38 |issue=5 |pages=2023–2065 |arxiv=0807.4898 |doi=10.1214/10-AOP534 |mr=2722794 |zbl=1203.15025 |doi-access=free}}}}}}
{{longitem|{{rma|TV11|tw=4em|{{Cite journal |last1=Tao |first1=Terence |last2=Vu |first2=Van |author-link2=Van Vu |year=2011 |title=Random matrices: universality of local eigenvalue statistics. |journal=[[Acta Mathematica]] |volume=206 |issue=1 |pages=127–204 |arxiv=0908.1982 |doi=10.1007/s11511-011-0061-3 |mr=2784665 |zbl=1217.15043 |doi-access=free}}}}}}
{{longitem|{{rma|GTZ12|tw=4em|{{Cite journal |last1=Green |first1=Ben |author-link=Ben Green (mathematician) |last2=Tao |first2=Terence |last3=Ziegler |first3=Tamar |author-link3=Tamar Ziegler |year=2012 |title=An inverse theorem for the Gowers U<sup>s+1</sup>[N]-norm |journal=[[Annals of Mathematics]] |series=Second Series |volume=176 |issue=2 |pages=1231–1372 |arxiv=1006.0205 |doi=10.4007/annals.2012.176.2.11 |mr=2950773 |zbl=1282.11007 |doi-access=free}}}}}}
{{longitem|{{rma|T16|tw=4em|{{Cite journal |last=Tao |first=Terence |year=2016 |title=Finite time blowup for an averaged three-dimensional Navier–Stokes equation |journal=[[Journal of the American Mathematical Society]] |volume=29 |issue=3 |pages=601–674 |arxiv=1402.0290 |doi=10.1090/jams/838 |mr=3486169 |zbl=1342.35227 |doi-access=free}}}}}}
{{Refend}}
{{refend}}

==Notes==
{{notelist}}

==See also==
* [[Cramer conjecture]]
* [[Erdős discrepancy problem]]
* [[Goldbach's weak conjecture]]
* [[Inscribed square problem]]

== References ==
{{Reflist}}

==External links==
{{Wikiquote}}
{{commons category|Terence Tao}}
* [https://www.math.ucla.edu/~tao Terence Tao's home page]
* [http://terrytao.wordpress.com/ Tao's research blog]
* [https://mathoverflow.net/users/766/terry-tao Tao's MathOverflow page]
* {{MacTutor Biography|id=Tao}}
* {{MathGenealogy |id=43967}}
* Terence Tao's entry in the [http://www.numericana.com/fame/index.htm#tao Numericana Hall of Fame]
* {{IMO results}}

{{FRS 2007}}
{{Fields medalists}}
{{Breakthrough Prize laureates}}
{{Princess of Asturias Award for Technical and Scientific Research}}
{{Recipients of SASTRA Ramanujan Prize winners|state=collapsed}}
{{Authority control}}

{{DEFAULTSORT:Tao, Terence Chi-Shen}}

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