Terence Tao: Difference between revisions
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{{Short description| |
{{Short description|Australian-American mathematician (born 1975)}} |
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'''Terence Chi-Shen Tao''' {{post-nominals|country=AUS|FAA|FRS}} ({{zh|陶哲軒}}; born 17 July 1975) is an Australian mathematician |
'''Terence Chi-Shen Tao''' {{post-nominals|country=AUS|FAA|FRS}} ({{zh|t=陶哲軒}}; born 17 July 1975) is an Australian and American mathematician who is a 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> |
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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 |
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, for which he is widely regarded as one of the greatest living mathematicians.<ref>{{Cite web |title=Search | 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 | 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://www.whitehouse.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" /> |
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==Life and career== |
==Life and career== |
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===Childhood=== |
===Childhood=== |
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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> 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> |
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> 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> |
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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}}</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> |
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===Career=== |
===Career=== |
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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=http://www.cbc.ca/technology/story/2006/08/22/math-fields.html| 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://www.whitehouse.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> |
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=http://www.cbc.ca/technology/story/2006/08/22/math-fields.html| 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://www.whitehouse.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> |
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As of 2022, Tao |
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> |
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==Research contributions== |
==Research contributions== |
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A technical tour de force 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> |
A technical tour de force 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> |
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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 |
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. |
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===Harmonic analysis=== |
===Harmonic analysis=== |
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===Random matrices=== |
===Random matrices=== |
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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 [[ |
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. |
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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 |
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 |
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::''Global regularity of wave maps II. Small energy in two dimensions.'' [[Communications in Mathematical Physics|Comm. Math. Phys.]] 2244 (2001), no. 2, 443–544. |
::''Global regularity of wave maps II. Small energy in two dimensions.'' [[Communications in Mathematical Physics|Comm. Math. Phys.]] 2244 (2001), no. 2, 443–544. |
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: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." |
: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." |
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* 2003 – [[Clay Research Award]] for:<ref> |
* 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> |
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::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 |
::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 |
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* 2005 – [[Australian Mathematical Society Medal]] |
* 2005 – [[Australian Mathematical Society Medal]] |
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::"For numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations and analytic number theory." |
::"For numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations and analytic number theory." |
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* 2014 – [[Royal Medal]] |
* 2014 – [[Royal Medal]] |
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* 2015 – PROSE award in the category of "Mathematics" for:<ref> |
* 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> |
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::"Hilbert's Fifth Problem and Related Topics" {{ISBN|978-1-4704-1564-8}} |
::"Hilbert's Fifth Problem and Related Topics" {{ISBN|978-1-4704-1564-8}} |
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* 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 – [[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> |
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* 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> |
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* 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|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 – [[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|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]] |
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* 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> |
* 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> |
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* 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> |
* 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> |
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* 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 – 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> |
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* 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> |
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* 2023 - [[Grande Médaille]] |
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{{refend}} |
{{refend}} |
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==Major publications== |
==Major publications== |
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{{refbegin}} |
{{refbegin}} |
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=== Textbooks === |
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* {{cite book|mr=2265113|last1=Tao|first1=Terence|author-mask=1|title=Solving mathematical problems. A personal perspective|publisher=[[Oxford University Press]]|location=Oxford|year=2006|isbn=978-0-19-920560-8|zbl=1098.00006|edition=Second edition of 1992 original}} |
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* {{ |
* {{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}} |
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* {{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}} |
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* {{ |
* {{Cite book |last=Tao |first=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://www.maa.org/press/maa-reviews/additive-combinatorics-0 |website=Mathematical Association of America}}</ref> |
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⚫ | |||
* {{ |
* {{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}} |
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* {{ |
* {{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}} |
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* {{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}} |
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* {{cite book|mr=2760403|last1=Tao|first1=Terence|author-mask=1|url=http://terrytao.files.wordpress.com/2012/12/gsm-117-tao3-epsilon1.pdf|title=An epsilon of room, I: real analysis. Pages from year three of a mathematical blog|zbl=1216.46002|series=[[Graduate Studies in Mathematics]]|volume=117|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2010|isbn=978-0-8218-5278-1|doi=10.1090/gsm/117}}<ref>{{cite web|author=Stenger, Allen|date=March 4, 2011|title=Review of ''A Epsilon of Room, I: Real Analysis: Pages from year three of a mathematical blog'' by Terence Tao|website=MAA Reviews, Mathematical Association of America|url=https://www.maa.org/press/maa-reviews/an-epsilon-of-room-i-real-analysis-pages-from-year-three-of-a-mathematical-blog}}</ref> |
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* {{ |
* {{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 ''A Epsilon of Room, I: Real Analysis: Pages from year three of a mathematical blog'' by Terence Tao |url=https://www.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> |
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⚫ | |||
* {{ |
* {{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://www.maa.org/press/maa-reviews/an-introduction-to-measure-theory |website=Mathematical Association of America}}</ref> |
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* {{cite book|mr=2906465|last1=Tao|first1=Terence|author-mask=1|url=http://terrytao.files.wordpress.com/2011/08/matrix-book.pdf|zbl=1256.15020|title=Topics in random matrix theory|series=[[Graduate Studies in Mathematics]]|volume=132|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2012|isbn=978-0-8218-7430-1|doi=10.1090/gsm/132}} |
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* {{ |
* {{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}} |
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* {{ |
* {{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}} |
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* {{ |
* {{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}} |
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* {{ |
* {{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}} |
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* {{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}} |
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* {{cite book|mr=3237440|last1=Tao|first1=Terence|author-mask=1|url=http://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/|zbl=1298.22001|title=Hilbert's fifth problem and related topics|series=[[Graduate Studies in Mathematics]]|volume=153|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2014|isbn=978-1-4704-1564-8|doi=10.1090/gsm/153}} |
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* {{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}} |
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* {{ |
* {{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> |
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'''Research articles.''' Tao is the author of over 300 articles. The following, among the most cited, are surveyed below. |
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* {{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}} |
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* {{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}} |
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=== Research articles === |
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{{Refbegin|30em}} |
{{Refbegin|30em}} |
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{{longitem|{{rma|KT98|tw=4em|{{ |
{{longitem|{{rma|KT98|tw=4em|{{Cite journal |last=Keel |first=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}}}}}} |
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{{longitem|{{rma|TVV98|tw=4em|{{ |
{{longitem|{{rma|TVV98|tw=4em|{{Cite journal |last=Tao |first=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}}}}}} |
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{{longitem|{{rma|KT99|tw=4em|{{ |
{{longitem|{{rma|KT99|tw=4em|{{Cite journal |last=Knutson |first=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}}}}}} |
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{{longitem|{{rma|C+01|tw=4em|{{ |
{{longitem|{{rma|C+01|tw=4em|{{Cite journal |last=Colliander |first=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}}}}}} |
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{{longitem|{{rma|T01a|tw=4em|{{ |
{{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}}}}}} |
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{{longitem|{{rma|T01b|tw=4em|{{ |
{{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}}}}}} |
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{{longitem|{{rma|C+02a|tw=4em|{{ |
{{longitem|{{rma|C+02a|tw=4em|{{Cite journal |last=Colliander |first=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}}}}}} |
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{{longitem|{{rma|C+02b|tw=4em|{{ |
{{longitem|{{rma|C+02b|tw=4em|{{Cite journal |last=Colliander |first=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}}}}}} |
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{{longitem|{{rma|MTT02|tw=4em|{{ |
{{longitem|{{rma|MTT02|tw=4em|{{Cite journal |last=Muscalu |first=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}}}}}} |
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{{longitem|{{rma|CCT03|tw=4em|{{ |
{{longitem|{{rma|CCT03|tw=4em|{{Cite journal |last=Christ |first=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}}}}}} |
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{{longitem|{{rma|C+03|tw=4em|{{ |
{{longitem|{{rma|C+03|tw=4em|{{Cite journal |last=Colliander |first=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}}}}}} |
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{{longitem|{{rma|T03|tw=4em|{{ |
{{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}}}}}} |
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{{longitem|{{rma|BKT04|tw=4em|{{ |
{{longitem|{{rma|BKT04|tw=4em|{{Cite journal |last=Bourgain |first=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}}}}}} |
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{{longitem|{{rma|C+04|tw=4em|{{ |
{{longitem|{{rma|C+04|tw=4em|{{Cite journal |last=Colliander |first=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}}}}}} |
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{{longitem|{{rma|KTW04|tw=4em|{{ |
{{longitem|{{rma|KTW04|tw=4em|{{Cite journal |last=Knutson |first=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}}}}}} |
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{{longitem|{{rma|T04a|tw=4em|{{ |
{{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}}}}}} |
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{{longitem|{{rma|T04b|tw=4em|{{ |
{{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}}}}}} |
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{{longitem|{{rma|CT05|tw=4em|{{ |
{{longitem|{{rma|CT05|tw=4em|{{Cite journal |last=Candes |first=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}}}}}} |
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{{longitem|{{rma|BT06|tw=4em|{{ |
{{longitem|{{rma|BT06|tw=4em|{{Cite journal |last=Bejenaru |first=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}}}}}} |
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{{longitem|{{rma|BCT06|tw=4em|{{ |
{{longitem|{{rma|BCT06|tw=4em|{{Cite journal |last=Bennett |first=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}}}}}} |
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{{longitem|{{rma|CRT06a|tw=4em|{{ |
{{longitem|{{rma|CRT06a|tw=4em|{{Cite journal |last=Candès |first=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}}}}}} |
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{{longitem|{{rma|CRT06b|tw=4em|{{ |
{{longitem|{{rma|CRT06b|tw=4em|{{Cite journal |last=Candès |first=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}}}}}} |
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{{longitem|{{rma|CT06|tw=4em|{{ |
{{longitem|{{rma|CT06|tw=4em|{{Cite journal |last=Candes |first=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}}}}}} |
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{{longitem|{{rma|CT07|tw=4em|{{ |
{{longitem|{{rma|CT07|tw=4em|{{Cite journal |last=Candes |first=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}}}}}} |
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{{longitem|{{rma|TVZ07|tw=4em|{{ |
{{longitem|{{rma|TVZ07|tw=4em|{{Cite journal |last=Tao |first=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}}}}}} |
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{{longitem|{{rma|C+08|tw=4em|{{ |
{{longitem|{{rma|C+08|tw=4em|{{Cite journal |last=Colliander |first=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}}}}}} |
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{{longitem|{{rma|GT08|tw=4em| {{ |
{{longitem|{{rma|GT08|tw=4em| {{Cite journal |last=Green |first=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}}}}}} |
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{{longitem|{{rma|KTV09|tw=4em|{{ |
{{longitem|{{rma|KTV09|tw=4em|{{Cite journal |last=Killip |first=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}}}}}} |
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{{longitem|{{rma|CT10|tw=4em|{{ |
{{longitem|{{rma|CT10|tw=4em|{{Cite journal |last=Candès |first=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}}}}}} |
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{{longitem|{{rma|GT10|tw=4em|{{ |
{{longitem|{{rma|GT10|tw=4em|{{Cite journal |last=Green |first=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}}}}}} |
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{{longitem|{{rma|TV10|tw=4em|{{ |
{{longitem|{{rma|TV10|tw=4em|{{Cite journal |last=Tao |first=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}}}}}} |
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{{longitem|{{rma|TV11|tw=4em|{{ |
{{longitem|{{rma|TV11|tw=4em|{{Cite journal |last=Tao |first=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}}}}}} |
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{{longitem|{{rma|GTZ12|tw=4em|{{ |
{{longitem|{{rma|GTZ12|tw=4em|{{Cite journal |last=Green |first=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}}}}}} |
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{{longitem|{{rma|T16|tw=4em|{{ |
{{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}}}}}} |
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{{Refend}} |
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==See also== |
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* [[Erdős discrepancy problem]] |
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* [[Inscribed square problem]] |
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== References == |
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Revision as of 00:29, 11 July 2024
Terence Tao | |||||||||||||||||||
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Simplified Chinese | 陶哲轩 | ||||||||||||||||||
|
Terence Chi-Shen Tao FAA FRS (Chinese: 陶哲軒; born 17 July 1975) is an Australian and American mathematician who is a 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 equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.[4]
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 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers, for which he is widely regarded as one of the greatest living mathematicians.[5][6][7][8][9][10]
Life and career
Family
Tao's parents are first-generation immigrants from Hong Kong to Australia.[11] Tao's father, Billy Tao,[a] was a Chinese paediatrician who was born in Shanghai and earned his medical degree (MBBS) from the University of Hong Kong in 1969.[12] Tao's mother, Grace Leong,[b] was born in Hong Kong; she received a first-class honours degree in mathematics and physics at the University of Hong Kong.[10] She was a secondary school teacher of mathematics and physics in Hong Kong.[13] Billy and Grace met as students at the University of Hong Kong.[14] They then emigrated from Hong Kong to Australia in 1972.[11][10]
Tao also has two brothers, Trevor and Nigel, who are currently living in Australia. Both formerly represented the states at the International Mathematical Olympiad.[15] Furthermore, Trevor Tao has been representing Australia internationally in chess and holds the title of Chess International Master.[16] Tao speaks Cantonese but cannot write Chinese. Tao is married to Laura Tao, an electrical engineer at NASA's Jet Propulsion Laboratory.[10][17] They live in Los Angeles, California, and have two children.[18]
Childhood
A child prodigy,[19] 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.[20] 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.[6][21]
Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal, 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.[22]
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.[23] 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.[23] 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.[23]
He is known for his collaborative mindset; by 2006, Tao had worked with over 30 others in his discoveries,[6] 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 progressions of prime numbers. The New York Times described it this way:[24][25]
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 Problem[26]
- his 2015 resolution of the Erdős discrepancy problem, which used entropy estimates within analytic number theory[27]
- his 2019 progress on the Collatz conjecture, in which he proved the probabilistic claim that almost all Collatz orbits attain almost bounded values.[28]
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.[29] 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.[30]
Recognition
British mathematician and Fields medalist Timothy Gowers remarked on Tao's breadth of knowledge:[31]
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.
An article by New Scientist[32] writes of his ability:
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].[33]
Tao has won numerous mathematician honours and awards over the years.[34] 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,[35] and the American Mathematical Society.[36] 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.[33][37] 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.[38] 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.[39] 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.[40] Tao was a finalist to become Australian of the Year in 2007.[41]
As of 2022, Tao had published over three hundred articles, along with sixteen books.[42] He has an Erdős number of 2.[43] He is a highly cited researcher.[44][45]
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, and Hideo Takaoka. They found a number of novel results, many to do with the well-posedness of weak solutions, for Schrödinger equations, KdV equations, and KdV-type equations.[C+03]
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.[CCT03][46] 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.[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.[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.[TVZ07] Rowan Killip, Tao, and Vișan later made notable progress on the two-dimensional problem in radial symmetry.[KTV09]
A technical tour de force by Tao in 2001 considered the wave maps equation with two-dimensional domain and spherical range.[T01a] He built upon earlier innovations of Daniel Tataru, who considered wave maps valued in Minkowski space.[47] 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.[T04a][48]
In 2016, Tao constructed a variant of the Navier–Stokes equations which possess solutions exhibiting irregular behavior in finite time.[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 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.[49] 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.[6][26] However, such results remain (as of 2024) conjectural.
Harmonic analysis
Bent Fuglede introduced the Fuglede conjecture in the 1970s, positing a tile-based characterisation of those Euclidean domains for which a Fourier ensemble provides a basis of L2.[50] 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 groups.[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 Lp spaces.[MTT02] This unified and extended earlier notable results of Ronald Coifman, Carlos Kenig, Michael Lacey, Yves Meyer, Elias Stein, and Thiele, among others.[51][52][53][54][55][56] Similar problems were analysed by Tao in 2001 in the context of Bourgain spaces, rather than the usual Lp spaces.[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, among others.[57][58]
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.[59][60][61] Here one studies the operation which restricts input functions on Euclidean space to a submanifold and outputs the product of the Fourier transforms of the corresponding measures. It is of major interest to identify exponents such that this operation is continuous relative to Lp spaces. Such multilinear problems originated in the 1990s, including in notable work of Jean Bourgain, Sergiu Klainerman, and Matei Machedon.[62][63][64] 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 instead of the Fourier transform.[TVV98] In 2003, Tao adapted ideas developed by Thomas Wolff for bilinear restriction to conical sets into the setting of restriction to quadratic hypersurfaces.[T03][65] 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.[BCT06][66]
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.[67]
Motivated by striking numerical experiments, Candes, Romberg, and Tao first studied the case where the matrix is given by the discrete Fourier transform.[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.[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.[CRT06b] These ideas and results were later improved by Candes.[68] Candes and Tao also considered relaxations of the sparsity condition, such as power-law decay of coefficients.[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.[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.[CT07] 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.[69] Trevor Hastie, Robert Tibshirani, and Jerome H. Friedman conclude that it is "somewhat unsatisfactory" in a number of cases.[70] 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.[71] 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.[CT10] Improved results were later found by Recht.[72] Similar problems and results have also been considered by a number of other authors.[73][74][75][76][77]
Random matrices
In the 1950s, Eugene Wigner initiated the study of random matrices and their eigenvalues.[78][79] 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 n is large and the entries of a n × n matrix A are selected randomly according to any fixed probability distribution of expectation 0 and standard deviation 1, then the eigenvalues of A will tend to be uniformly scattered across the disk of radius n1/2 around the origin; this can be made precise using the language of measure theory.[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 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.[TV11] Similar results were obtained around the same time by László Erdös, Horng-Tzer Yau, and Jun Yin.[80][81]
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 of prime order.[BKT04] It is well known that there are no nontrivial subrings 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.[82][83]
Tao and Ben Green proved the existence of arbitrarily long arithmetic progressions in the prime numbers; this result is generally referred to as the Green–Tao theorem, and is among Tao's most well-known results.[GT08] 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 k × n matrix A and a k × 1 matrix v whose components are all integers, Green and Tao give conditions on when there exist infinitely many n × 1 matrices x such that all components of Ax + v are prime numbers.[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.[GTZ12]
Notable awards
- 1999 – Packard Fellowship
- 2000 – Salem Prize for:[84]
- "his work in Lp harmonic analysis and on related questions in geometric measure theory and partial differential equations."
- 2002 – Bôcher Memorial Prize for:[85]
- 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:[86]
- his restriction theorems in Fourier analysis, 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 (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 for:
- "his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory"
- 2006 – MacArthur Award
- 2006 – SASTRA Ramanujan Prize[87]
- 2006 – Sloan Fellowship
- 2007 – Fellow of the Royal Society[88]
- 2008 – Alan T. Waterman Award for:[89]
- "his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis"
- 2008 – Onsager Medal[90] for:
- 2009 – Inducted into the American Academy of Arts and Sciences[92]
- 2010 – King Faisal International Prize[93]
- 2010 – Nemmers Prize in Mathematics[94]
- 2010 – Polya Prize (with Emmanuel Candès)
- 2012 – Crafoord Prize[95][96]
- 2012 – Simons Investigator[97]
- 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:[98]
- "Hilbert's Fifth Problem and Related Topics" ISBN 978-1-4704-1564-8
- 2019 – Riemann Prize[99]
- 2019 – The Carnegie Corporation of New York honored Tao with 2019 Great Immigrant Award.[100]
- 2020 – Princess of Asturias Award for Technical and Scientific Research,[101] with Emmanuel Candès, for their work on compressed sensing
- 2020 – Bolyai Prize[102]
- 2021 – IEEE Jack S. Kilby Signal Processing Medal[103]
- 2022 – Global Australian of the Year (Advance Global Australians; Advance.org)[104][105]
- 2022 – Grande Médaille[106]
Major publications
Textbooks
- — (2006). Solving mathematical problems. A personal perspective (Second edition of 1992 original ed.). Oxford University Press. ISBN 978-0-19-920560-8. MR 2265113. Zbl 1098.00006.
- — (2006). Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics. Vol. 106. Providence, RI: American Mathematical Society. doi:10.1090/cbms/106. ISBN 0-8218-4143-2. MR 2233925. Zbl 1106.35001.
- —; Vu, Van H. (2006). Additive combinatorics. Cambridge Studies in Advanced Mathematics. Vol. 105. Cambridge University Press. ISBN 978-0-521-85386-6. MR 2289012. Zbl 1127.11002.[107][108]
- — (2008). Structure and randomness. Pages from year one of a mathematical blog. Providence, RI: American Mathematical Society. doi:10.1090/mbk/059. ISBN 978-0-8218-4695-7. MR 2459552. Zbl 1245.00024.
- — (2009). Poincaré's legacies, pages from year two of a mathematical blog. Part I. Providence, RI: American Mathematical Society. doi:10.1090/mbk/066. ISBN 978-0-8218-4883-8. MR 2523047. Zbl 1171.00003.
- — (2009). Poincaré's legacies, pages from year two of a mathematical blog. Part II. Providence, RI: American Mathematical Society. doi:10.1090/mbk/067. ISBN 978-0-8218-4885-2. MR 2541289. Zbl 1175.00010.
- — (2010). An epsilon of room, I: real analysis. Pages from year three of a mathematical blog. Graduate Studies in Mathematics. Vol. 117. Providence, RI: American Mathematical Society. doi:10.1090/gsm/117. ISBN 978-0-8218-5278-1. MR 2760403. Zbl 1216.46002.[109]
- — (2010). An epsilon of room, II. Pages from year three of a mathematical blog. Providence, RI: [American Mathematical Society. doi:10.1090/mbk/077. ISBN 978-0-8218-5280-4. MR 2780010. Zbl 1218.00001.
- — (2011). An introduction to measure theory. Graduate Studies in Mathematics. Vol. 126. Providence, RI: American Mathematical Society. doi:10.1090/gsm/126. ISBN 978-0-8218-6919-2. MR 2827917. Zbl 1231.28001.[110]
- — (2012). Topics in random matrix theory. Graduate Studies in Mathematics. Vol. 132. Providence, RI: American Mathematical Society. doi:10.1090/gsm/132. ISBN 978-0-8218-7430-1. MR 2906465. Zbl 1256.15020.
- — (2012). Higher order Fourier analysis. Graduate Studies in Mathematics. Vol. 142. Providence, RI: American Mathematical Society. doi:10.1090/gsm/142. ISBN 978-0-8218-8986-2. MR 2931680. Zbl 1277.11010.
- — (2013). Compactness and contradiction. Providence, RI: American Mathematical Society. doi:10.1090/mbk/081. ISBN 978-0-8218-9492-7. MR 3026767. Zbl 1276.00007.
- — (2014) [2006]. Analysis. I. Texts and Readings in Mathematics. Vol. 37 (3rd ed.). New Delhi: Hindustan Book Agency. ISBN 978-93-80250-64-9. MR 3309891. Zbl 1300.26002.
- — (2014) [2006]. Analysis. II. Texts and Readings in Mathematics. Vol. 38 (3rd ed.). New Delhi: Hindustan Book Agency. ISBN 978-93-80250-65-6. MR 3310023. Zbl 1300.26003.
- — (2014). Hilbert's fifth problem and related topics. Graduate Studies in Mathematics. Vol. 153. Providence, RI: American Mathematical Society. doi:10.1090/gsm/153. ISBN 978-1-4704-1564-8. MR 3237440. Zbl 1298.22001.
- — (2015). Expansion in finite simple groups of Lie type. Graduate Studies in Mathematics. Vol. 164. Providence, RI: American Mathematical Society. doi:10.1090/gsm/164. ISBN 978-1-4704-2196-0. MR 3309986. S2CID 118288443. Zbl 1336.20015.[111]
- — (2022). Analysis I. Texts and Readings in Mathematics. Vol. 37 (4th ed.). Springer Singapore. ISBN 978-981-19-7261-4.
- — (2022). Analysis II. Texts and Readings in Mathematics. Vol. 38 (4th ed.). Springer Singapore. ISBN 978-981-19-7284-3.
Research articles
KT98. | Keel, Markus; Tao, Terence (1998). "Endpoint Strichartz estimates". American Journal of Mathematics. 120 (5): 955–980. CiteSeerX 10.1.1.599.1892. doi:10.1353/ajm.1998.0039. JSTOR 25098630. MR 1646048. S2CID 13012479. Zbl 0922.35028.
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TVV98. | Tao, Terence; Vargas, Ana; Vega, Luis (1998). "A bilinear approach to the restriction and Kakeya conjectures". Journal of the American Mathematical Society. 11 (4): 967–1000. doi:10.1090/S0894-0347-98-00278-1. MR 1625056. Zbl 0924.42008.
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KT99. | Knutson, Allen; Tao, Terence (1999). "The honeycomb model of tensor products. I. Proof of the saturation conjecture". Journal of the American Mathematical Society. 12 (4): 1055–1090. doi:10.1090/S0894-0347-99-00299-4. MR 1671451. Zbl 0944.05097.
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C+01. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2001). "Global well-posedness for Schrödinger equations with derivative". SIAM Journal on Mathematical Analysis. 33 (3): 649–669. arXiv:math/0101263. doi:10.1137/S0036141001384387. MR 1871414. Zbl 1002.35113.
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T01a. | Tao, Terence (2001). "Global regularity of wave maps. II. Small energy in two dimensions". Communications in Mathematical Physics. 224 (2): 443–544. arXiv:math/0011173. Bibcode:2001CMaPh.224..443T. doi:10.1007/PL00005588. MR 1869874. S2CID 119634411. Zbl 1020.35046. (Erratum: [1])
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T01b. | Tao, Terence (2001). "Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations". American Journal of Mathematics. 123 (5): 839–908. arXiv:math/0005001. doi:10.1353/ajm.2001.0035. JSTOR 25099087. MR 1854113. S2CID 984131. Zbl 0998.42005.
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C+02a. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2002). "A refined global well-posedness result for Schrödinger equations with derivative". SIAM Journal on Mathematical Analysis. 34 (1): 64–86. arXiv:math/0110026. doi:10.1137/S0036141001394541. MR 1950826. S2CID 9007785. Zbl 1034.35120.
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C+02b. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2002). "Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation". Mathematical Research Letters. 9 (5–6): 659–682. arXiv:math/0203218. doi:10.4310/MRL.2002.v9.n5.a9. MR 1906069. Zbl 1152.35491.
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MTT02. | Muscalu, Camil; Tao, Terence; Thiele, Christoph (2002). "Multi-linear operators given by singular multipliers". Journal of the American Mathematical Society. 15 (2): 469–496. doi:10.1090/S0894-0347-01-00379-4. MR 1887641. Zbl 0994.42015.
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CCT03. | Christ, Michael; Colliander, James; Tao, Terrence (2003). "Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations". American Journal of Mathematics. 125 (6): 1235–1293. arXiv:math/0203044. doi:10.1353/ajm.2003.0040. MR 2018661. S2CID 11001499. Zbl 1048.35101.
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C+03. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2003). "Sharp global well-posedness for KdV and modified KdV on and ". Journal of the American Mathematical Society. 16 (3): 705–749. doi:10.1090/S0894-0347-03-00421-1. MR 1969209. Zbl 1025.35025.
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T03. | Tao, T. (2003). "A sharp bilinear restrictions estimate for paraboloids". Geometric and Functional Analysis. 13 (6): 1359–1384. arXiv:math/0210084. doi:10.1007/s00039-003-0449-0. MR 2033842. S2CID 15873489. Zbl 1068.42011.
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BKT04. | Bourgain, J.; Katz, N.; Tao, T. (2004). "A sum-product estimate in finite fields, and applications". Geometric and Functional Analysis. 14 (1): 27–57. arXiv:math/0301343. doi:10.1007/s00039-004-0451-1. MR 2053599. S2CID 14097626. Zbl 1145.11306.
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C+04. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2004). "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3". Communications on Pure and Applied Mathematics. 57 (8): 987–1014. arXiv:math/0301260. doi:10.1002/cpa.20029. MR 2053757. S2CID 16423475. Zbl 1060.35131.
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KTW04. | Knutson, Allen; Tao, Terence; Woodward, Christopher (2004). "The honeycomb model of tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone". Journal of the American Mathematical Society. 17 (1): 19–48. doi:10.1090/S0894-0347-03-00441-7. MR 2015329. Zbl 1043.05111.
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T04a. | Tao, Terence (2004). "Global well-posedness of the Benjamin–Ono equation in H1(ℝ)". Journal of Hyperbolic Differential Equations. 1 (1): 27–49. arXiv:math/0307289. doi:10.1142/S0219891604000032. MR 2052470. Zbl 1055.35104.
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T04b. | Tao, Terence (2004). "Fuglede's conjecture is false in 5 and higher dimensions". Mathematical Research Letters. 11 (2–3): 251–258. arXiv:math/0306134. doi:10.4310/MRL.2004.v11.n2.a8. MR 2067470. Zbl 1092.42014.
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CT05. | Candes, Emmanuel J.; Tao, Terence (2005). "Decoding by linear programming". IEEE Transactions on Information Theory. 51 (12): 4203–4215. arXiv:math/0502327. doi:10.1109/TIT.2005.858979. MR 2243152. S2CID 12605120. Zbl 1264.94121.
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BT06. | Bejenaru, Ioan; Tao, Terence (2006). "Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation". Journal of Functional Analysis. 233 (1): 228–259. arXiv:math/0508210. doi:10.1016/j.jfa.2005.08.004. MR 2204680. Zbl 1090.35162.
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BCT06. | Bennett, Jonathan; Carbery, Anthony; Tao, Terence (2006). "On the multilinear restriction and Kakeya conjectures". Acta Mathematica. 196 (2): 261–302. arXiv:math/0509262. doi:10.1007/s11511-006-0006-4. MR 2275834. Zbl 1203.42019.
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CRT06a. | Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence (2006). "Stable signal recovery from incomplete and inaccurate measurements". Communications on Pure and Applied Mathematics. 59 (8): 1207–1223. arXiv:math/0503066. doi:10.1002/cpa.20124. MR 2230846. S2CID 119159284. Zbl 1098.94009.
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CRT06b. | Candès, Emmanuel J.; Romberg, Justin; Tao, Terence (2006). "Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information". IEEE Transactions on Information Theory. 52 (2): 489–509. arXiv:math/0409186. doi:10.1109/TIT.2005.862083. MR 2236170. S2CID 7033413. Zbl 1231.94017.
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CT06. | Candes, Emmanuel J.; Tao, Terence (2006). "Near-optimal signal recovery from random projections: universal encoding strategies?". IEEE Transactions on Information Theory. 52 (12): 5406–5425. arXiv:math/0410542. doi:10.1109/TIT.2006.885507. MR 2300700. S2CID 1431305. Zbl 1309.94033.
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CT07. | Candes, Emmanuel; Tao, Terence (2007). "The Dantzig selector: statistical estimation when p is much larger than n". Annals of Statistics. 35 (6): 2313–2351. arXiv:math/0506081. doi:10.1214/009053606000001523. MR 2382644. Zbl 1139.62019.
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TVZ07. | Tao, Terence; Visan, Monica; Zhang, Xiaoyi (2007). "The nonlinear Schrödinger equation with combined power-type nonlinearities". Communications in Partial Differential Equations. 32 (7–9): 1281–1343. arXiv:math/0511070. doi:10.1080/03605300701588805. MR 2354495. S2CID 15109526. Zbl 1187.35245.
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C+08. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2008). "Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3". Annals of Mathematics. Second Series. 167 (3): 767–865. doi:10.4007/annals.2008.167.767. MR 2415387. Zbl 1178.35345.
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GT08. | Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. Second Series. 167 (2): 481–547. arXiv:math/0404188. doi:10.4007/annals.2008.167.481. MR 2415379. Zbl 1191.11025.
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KTV09. | Killip, Rowan; Tao, Terence; Visan, Monica (2009). "The cubic nonlinear Schrödinger equation in two dimensions with radial data". Journal of the European Mathematical Society. 11 (6): 1203–1258. arXiv:0707.3188. doi:10.4171/JEMS/180. MR 2557134. Zbl 1187.35237.
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CT10. | Candès, Emmanuel J.; Tao, Terence (2010). "The power of convex relaxation: near-optimal matrix completion". IEEE Transactions on Information Theory. 56 (5): 2053–2080. arXiv:0903.1476. doi:10.1109/TIT.2010.2044061. MR 2723472. S2CID 1255437. Zbl 1366.15021.
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GT10. | Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. Second Series. 167 (2): 481–547. arXiv:math/0404188. doi:10.4007/annals.2008.167.481. MR 2415379. Zbl 1191.11025.
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TV10. | Tao, Terence; Vu, Van (2010). "Random matrices: universality of ESDs and the circular law". Annals of Probability. 38 (5). With an appendix by Manjunath Krishnapur: 2023–2065. arXiv:0807.4898. doi:10.1214/10-AOP534. MR 2722794. Zbl 1203.15025.
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TV11. | Tao, Terence; Vu, Van (2011). "Random matrices: universality of local eigenvalue statistics". Acta Mathematica. 206 (1): 127–204. arXiv:0908.1982. doi:10.1007/s11511-011-0061-3. MR 2784665. Zbl 1217.15043.
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GTZ12. | Green, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers Us+1[N]-norm". Annals of Mathematics. Second Series. 176 (2): 1231–1372. arXiv:1006.0205. doi:10.4007/annals.2012.176.2.11. MR 2950773. Zbl 1282.11007.
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T16. | Tao, Terence (2016). "Finite time blowup for an averaged three-dimensional Navier–Stokes equation". Journal of the American Mathematical Society. 29 (3): 601–674. arXiv:1402.0290. doi:10.1090/jams/838. MR 3486169. Zbl 1342.35227.
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Notes
See also
References
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- ^ a b "Vitae and Bibliography for Terence Tao". 12 October 2009. Retrieved 21 January 2010.
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- ^ Tataru, Daniel. On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37–77.
- ^ 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.
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External links
- Terence Tao's home page
- Tao's research blog
- Tao's MathOverflow page
- O'Connor, John J.; Robertson, Edmund F., "Terence Tao", MacTutor History of Mathematics Archive, University of St Andrews
- Terence Tao at the Mathematics Genealogy Project
- Terence Tao's entry in the Numericana Hall of Fame
- Terence Tao's results at International Mathematical Olympiad
- 1975 births
- 21st-century American male writers
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