Terence Tao
Terence Tao FAA FRS | |
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Born | Adelaide, South Australia, Australia | 17 July 1975
Citizenship |
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Alma mater |
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Known for | Dynamical Systems |
Spouse | Laura Tao |
Children | 2 |
Awards | Fields Medal (2006) List
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Scientific career | |
Fields | Harmonic analysis |
Institutions | University of California, Los Angeles |
Thesis | Three Regularity Results in Harmonic Analysis[3] (1996) |
Doctoral advisor | Elias M. Stein |
Doctoral students | Monica Vișan |
Website | mathstodon |
Terence Tao | |
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Suzhounese | Dau Tseh-shie |
Yue: Cantonese | |
Yale Romanization | Tòuh Jit-hīn |
Jyutping | Tou4 Zit3-hin1 |
IPA | [tʰou˩ tsiːt̚˧.hiːn˥] |
Terence Chi-Shen Tao
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. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers.[5] He is widely regarded as one of the greatest living mathematicians.[6][7][8][9][10]
Life and career
Family
Tao's parents are first-generation
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
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
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 Conjectureby 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.
As of 2022, Tao has published over three hundred articles, along with sixteen books.
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
In 2016, Tao constructed a variant of the
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
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
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,
In 2011, Tao and Vu established a "four
Analytic number theory and arithmetic combinatorics
In 2004, Tao, together with
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
- "Hilbert's Fifth Problem and Related Topics"
- 2019 – Riemann Prize[99]
- 2020 – Princess of Asturias Award for Technical and Scientific Research,[100] with Emmanuel Candès, for their work on compressed sensing
- 2020 – Bolyai Prize[101]
- 2021 – IEEE Jack S. Kilby Signal Processing Medal[102]
- 2022 – Global Australian of the Year (Advance Global Australians; Advance.org)[103][104]
- 2023 - Grande Médaille
Major publications
Textbooks
- — (2006). Solving mathematical problems. A personal perspective (Second edition of 1992 original ed.). Oxford: Zbl 1098.00006.
- — (2006). Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics. Vol. 106. Providence, RI: Zbl 1106.35001.
- —;
- — (2008). Structure and randomness. Pages from year one of a mathematical blog. Providence, RI: Zbl 1245.00024.
- — (2009). Poincaré's legacies, pages from year two of a mathematical blog. Part I. Providence, RI: Zbl 1171.00003.
- — (2009). Poincaré's legacies, pages from year two of a mathematical blog. Part II. Providence, RI: Zbl 1175.00010.
- — (2010). An epsilon of room, I: real analysis. Pages from year three of a mathematical blog (PDF). Zbl 1216.46002.[107]
- — (2010). An epsilon of room, II. Pages from year three of a mathematical blog (PDF). Providence, RI: Zbl 1218.00001.
- — (2011). An introduction to measure theory (PDF). Zbl 1231.28001.[108]
- — (2012). Topics in random matrix theory (PDF). Zbl 1256.15020.
- — (2012). Higher order Fourier analysis (PDF). Zbl 1277.11010.
- — (2013). Compactness and contradiction (PDF). Providence, RI: Zbl 1276.00007.
- — (2014). Analysis. I. Texts and Readings in Mathematics. Vol. 37 (Third edition of 2006 original ed.). New Delhi: Hindustan Book Agency. Zbl 1300.26002.
- — (2014). Analysis. II. Texts and Readings in Mathematics. Vol. 38 (Third edition of 2006 original ed.). New Delhi: Hindustan Book Agency. Zbl 1300.26003.
- — (2014). Hilbert's fifth problem and related topics. Zbl 1298.22001.
- — (2015). Expansion in finite simple groups of Lie type. Zbl 1336.20015.[109]
Research articles. Tao is the author of over 300 articles. The following, among the most cited, are surveyed below.
KT98. | Keel, Markus; Tao, Terence (1998). "Endpoint Strichartz estimates".
Zbl 0922.35028 . |
TVV98. | Tao, Terence; Vargas, Ana;
Zbl 0924.42008 . |
KT99. | Knutson, Allen; Tao, Terence (1999). "The honeycomb model of tensor products. I. Proof of the saturation conjecture".
Zbl 0944.05097 . |
C+01. | Zbl 1002.35113 . |
T01a. | Tao, Terence (2001). "Global regularity of wave maps. II. Small energy in two dimensions".
Zbl 1020.35046. (Erratum: [1] ) |
T01b. | Tao, Terence (2001). "Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations".
Zbl 0998.42005 . |
C+02a. | Zbl 1034.35120 . |
C+02b. | Zbl 1152.35491 . |
MTT02. | Muscalu, Camil; Tao, Terence;
Zbl 0994.42015 . |
CCT03. | Zbl 1048.35101 . |
C+03. | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. (2003). "Sharp global well-posedness for KdV and modified KdV on and ".
Zbl 1025.35025 . |
T03. | Tao, T. (2003). "A sharp bilinear restrictions estimate for paraboloids".
Zbl 1068.42011 . |
BKT04. | Zbl 1145.11306 . |
C+04. | Zbl 1060.35131 . |
KTW04. | Knutson, Allen; Tao, Terence; Woodward, Christopher (2004). "The honeycomb model of tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone".
Zbl 1043.05111 . |
T04a. | Tao, Terence (2004). "Global well-posedness of the Benjamin–Ono equation in H1(ℝ)".
Zbl 1055.35104 . |
T04b. | Tao, Terence (2004). "Fuglede's conjecture is false in 5 and higher dimensions". Mathematical Research Letters. 11 (2–3): 251–258.
Zbl 1092.42014 . |
CT05. | Zbl 1264.94121 . |
BT06. | Bejenaru, Ioan; Tao, Terence (2006). "Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation".
Zbl 1090.35162 . |
BCT06. | Zbl 1203.42019 . |
CRT06a. | Zbl 1098.94009 . |
CRT06b. | Zbl 1231.94017 . |
CT06. | Zbl 1309.94033 . |
CT07. | Zbl 1139.62019 . |
TVZ07. | Tao, Terence;
Zbl 1187.35245 . |
C+08. | Zbl 1178.35345 . |
GT08. | Zbl 1191.11025 . |
KTV09. | Killip, Rowan; Tao, Terence;
Zbl 1187.35237 . |
CT10. | Zbl 1366.15021 . |
GT10. | Zbl 1191.11025 . |
TV10. | Tao, Terence;
Zbl 1203.15025 . |
TV11. | Tao, Terence;
Zbl 1217.15043 . |
GTZ12. | Zbl 1282.11007 . |
T16. | Tao, Terence (2016). "Finite time blowup for an averaged three-dimensional Navier–Stokes equation".
Zbl 1342.35227 . |
Notes
See also
- Erdős discrepancy problem
- Inscribed square problem
- Goldbach's weak conjecture
- Cramer conjecture
References
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- ^ Fellows and Foreign Members of the Royal Society, retrieved 9 June 2010.
- ^ National Science Foundation, Alan T. Waterman Award. Retrieved 18 April 2008.
- ^ "The Lars Onsager Lecture and Professorship – IMF". Archived from the original on 3 February 2009. Retrieved 13 January 2009.
- YouTube
- ^ "Alphabetical Index of Active AAAS Members" (PDF). amacad.org. Archived from the original (PDF) on 5 October 2013. Retrieved 21 November 2013.
His 2009 induction ceremony is here. - ^ "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the American Mathematical Society. 57 (5): 642–643. May 2010.
- ^ "Major Math and Science Awards Announced: Northwestern University News". Archived from the original on 16 April 2010. Retrieved 5 September 2015.
- ^ "The Crafoord Prize in Mathematics 2012 and The Crafoord Prize in Astronomy 2012". Royal Swedish Academy of Sciences. 19 January 2012. Archived from the original on 23 October 2021. Retrieved 13 November 2014.
- ^ "4 Scholars Win Crafoord Prizes in Astronomy and Math – The Ticker – Blogs – The Chronicle of Higher Education". 19 January 2012. Archived from the original on 23 October 2021. Retrieved 5 September 2015.
- ^ "Simons Investigators Awardees". Simons Foundation. Archived from the original on 23 October 2021. Retrieved 9 September 2017.
- ^ PROSE 2015 winners
- ^ "Riemann Prize laureate 2019: Terence Tao". Archived from the original on 20 December 2019. Retrieved 23 November 2019.
- ^ "Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès, Princess of Asturias Award for Technical and Scientific Research 2020". Princess of Asturias Foundation. Archived from the original on 26 June 2020. Retrieved 23 June 2020.
- ^ "Vitae and Bibliography for Terence Tao". UCLA. Retrieved 13 November 2020.
- ^ "IEEE Awards". IEEE Awards. 27 June 2022. Retrieved 10 September 2022.
- ^ World’s greatest mathematician named 2022 Global Australian of the Year, Advance.org, media release 2022-09-08, accessed 2022-09-14
- ^ Why this maths genius refuses to work for a hedge fund, Tess Bennett, Australian Financial Review, 2022-09-07, accessed 2022-09-14
- doi:10.1090/s0273-0979-09-01231-2. Archived from the original(PDF) on 11 March 2012.
- ^ Vestal, Donald L. (6 June 2007). "Review of Additive Combinatorics by Terence Tao and Van H. Vu". MAA Reviews, Mathematical Association of America.
- ^ Stenger, Allen (4 March 2011). "Review of A Epsilon of Room, I: Real Analysis: Pages from year three of a mathematical blog by Terence Tao". MAA Reviews, Mathematical Association of America.
- ^ Poplicher, Mihaela (14 April 2012). "Review of An Introduction to Measure Theory by Terence Tao". MAA Reviews, Mathematical Association of America.
- doi:10.1090/bull/1610; review published electronically)
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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