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Terence Tao

November 25, 2023

Terence Chi-Shen Tao FAA FRS (Chinese: 陶哲軒; born 17 July 1975) is an Australian mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. 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]

Terence Tao

FAA FRS
Tao in 2021
Born (1975-07-17) 17 July 1975 (age 48)
Adelaide, South Australia, Australia
Citizenship
  • Australia
  • United States[3]
Alma mater
Known forPartial Differential Equations Analytic Number Theory Random matrices Compressed Sensing Combinatorics Dynamical Systems
SpouseLaura Tao
Children2
AwardsFields Medal (2006)
List
Scientific career
FieldsHarmonic analysis
InstitutionsUniversity of California, Los Angeles
ThesisThree Regularity Results in Harmonic Analysis[3] (1996)
Doctoral advisorElias M. Stein
Doctoral studentsMonica Vișan
Website
  • www.math.ucla.edu/~tao/
  • terrytao.wordpress.com
mathstodon.xyz/@tao

Tao was born to ethnic 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 and has been referred to as the "Mozart of mathematics."[6][7][8][9][10]

Life and career edit

Family edit

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 living in Australia. Both formerly represented the states at the International Mathematical Olympiad.[15] Furthermore, Trevor 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: Riley[c] and daughter Madeleine.[18][19]

Childhood edit

A child prodigy,[20] 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.[21][6] 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.[22]

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.[23]

Career edit

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.[24] 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.[24] 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.[24]

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:[25][26]

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:

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.[30] 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.[31]

Recognition edit

British mathematician and Fields medalist Timothy Gowers remarked on Tao's breadth of knowledge:[32]

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[33] 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].[34]

Tao has won numerous mathematician honours and awards over the years.[35] 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,[36] and the American Mathematical Society.[37] 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.[34][38] 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.[39] 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.[40] 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.[41] Tao was a finalist to become Australian of the Year in 2007.[42]

As of 2022, Tao has published over three hundred articles, along with sixteen books.[43] He has an Erdős number of 2.[44] He is a highly cited researcher.[45][46]

Research contributions edit

Dispersive partial differential equations edit

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]

 
Tao at the age of 10 with mathematician Paul Erdős in 1985

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][47] 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.[48] 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][49]

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.[50] 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][27] However, such results remain (as of 2022) conjectural.

Harmonic analysis edit

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.[51] 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.[52][53][54][55][56][57] Similar problems were analyzed 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.[58][59]

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.[60][61][62] 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.[63][64][65] 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][66] 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][67]

Compressed sensing and statistics edit

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.[68]

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.[69] 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 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.[70] 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.[71] Similar problems and results have also been considered by a number of other authors.[72][73][74][75][76]

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.[77] Trevor Hastie, Robert Tibshirani, and Jerome H. Friedman conclude that it is "somewhat unsatisfactory" in a number of cases.[78] Nonetheless, it remains of significant interest in the statistical literature.

Random matrices edit

In the 1950s, Eugene Wigner initiated the study of random matrices and their eigenvalues.[79][80] 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 average 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.[81][82]

Analytic number theory and arithmetic combinatorics edit

 
Tao (second from left) with UCLA undergraduate students in 2021

In 2004, Tao, together with Jean Bourgain and Nets Katz, studied the additive and multiplicative structure of subsets of finite fields of prime order.[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.[83][84]

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 edit

"his work in Lp harmonic analysis and on related questions in geometric measure theory and partial differential equations."
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."
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
"their exceptional achievements in the area of analytic and combinatorial number theory"
their expository article "Honeycombs and Sums of Hermitian Matrices" (Notices of the AMS. 48 (2001), 175–186.)
"his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory"
"his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis"
"his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics". His Lars Onsager lecture was entitled "Structure and randomness in the prime numbers" at NTNU, Norway.[91]
"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:[97]
"Hilbert's Fifth Problem and Related Topics" ISBN 978-1-4704-1564-8

Major publications edit

Textbooks

Research articles. Tao is the author of over 300 articles. The following, among the most cited, are surveyed above.

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.
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.
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.
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.
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])
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.
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.
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. doi:10.4310/MRL.2002.v9.n5.a9. MR 1906069. Zbl 1152.35491.
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.
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.
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.
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.
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.
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.
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.
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.
T04b.
Tao, Terence (2004). "Fuglede's conjecture is false in 5 and higher dimensions". Mathematical Research Letters. 11 (2–3): 251–258. doi:10.4310/MRL.2004.v11.n2.a8. MR 2067470. Zbl 1092.42014.
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.
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. doi:10.1016/j.jfa.2005.08.004. MR 2204680. Zbl 1090.35162.
BCT06.
Bennett, Jonathan; Carbery, Anthony; Tao, Terence (2006). "On the multilinear restriction and Kakeya conjectures". Acta Mathematica. 196 (2): 261–302. doi:10.1007/s11511-006-0006-4. MR 2275834. Zbl 1203.42019.
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.
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.
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.
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. doi:10.1214/009053606000001523. MR 2382644. Zbl 1139.62019.
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.
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.
GT08.
Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. Second Series. 167 (2): 481–547. doi:10.4007/annals.2008.167.481. MR 2415379. Zbl 1191.11025.
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. doi:10.4171/JEMS/180. MR 2557134. Zbl 1187.35237.
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.
GT10.
Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. Second Series. 167 (2): 481–547. doi:10.4007/annals.2008.167.481. MR 2415379. Zbl 1191.11025.
TV10.
Tao, Terence; Vu, Van (2010). With an appendix by Manjunath Krishnapur. "Random matrices: universality of ESDs and the circular law". Annals of Probability. 38 (5): 2023–2065. doi:10.1214/10-AOP534. MR 2722794. Zbl 1203.15025.
TV11.
Tao, Terence; Vu, Van (2011). "Random matrices: universality of local eigenvalue statistics". Acta Mathematica. 206 (1): 127–204. doi:10.1007/s11511-011-0061-3. MR 2784665. Zbl 1217.15043.
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. doi:10.4007/annals.2012.176.2.11. MR 2950773. Zbl 1282.11007.
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. doi:10.1090/jams/838. MR 3486169. Zbl 1342.35227.

Notes edit

  1. ^ Chinese: 陶象國; pinyin: Táo Xiàngguó
  2. ^ Chinese: 梁蕙蘭; Jyutping: Loeng⁴ Wai⁶-laan⁴
  3. ^ Being non-binary, Riley's pronouns are they/them.

See also edit

References edit

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External links edit

 
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November 25, 2023
terence, terence, shen, chinese, 陶哲軒, born, july, 1975, australian, mathematician, professor, mathematics, university, california, angeles, ucla, where, holds, james, carol, collins, chair, research, includes, topics, harmonic, analysis, partial, differential,. Terence Chi Shen Tao FAA FRS Chinese 陶哲軒 born 17 July 1975 is an Australian mathematician He is a professor of mathematics at the University of California Los Angeles UCLA where he holds the James and Carol Collins chair 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 Terence TaoFAA FRSTao in 2021Born 1975 07 17 17 July 1975 age 48 Adelaide South Australia AustraliaCitizenshipAustraliaUnited States 3 Alma materFlinders University BS MSc Princeton University PhD Known forPartial Differential Equations Analytic Number Theory Random matrices Compressed Sensing Combinatorics Dynamical SystemsSpouseLaura TaoChildren2AwardsFields Medal 2006 List Salem Prize 2000 Bocher Memorial Prize 2002 Clay Research Award 2003 Australian Mathematical Society Medal 2005 Ostrowski Prize 2005 Levi L Conant Prize 2005 MacArthur Award 2006 SASTRA Ramanujan Prize 2006 Sloan Fellowship 2006 Fellow of the Royal Society 2007 Alan T Waterman Award 2008 Onsager Medal 2008 King Faisal International Prize 2010 1 Nemmers Prize in Mathematics 2010 Polya Prize 2010 2 Crafoord Prize 2012 Simons Investigator 2012 Breakthrough Prize in Mathematics 2014 Royal Medal 2014 PROSE Award 2015 Riemann Prize 2019 Princess of Asturias Award 2020 Bolyai Prize 2020 IEEE Jack S Kilby Signal Processing Medal 2021 Global Australian of the Year Award 2022 Grande Medaille 2023 Scientific careerFieldsHarmonic analysisInstitutionsUniversity of California Los AngelesThesisThree Regularity Results in Harmonic Analysis 3 1996 Doctoral advisorElias M SteinDoctoral studentsMonica VișanWebsitewww wbr math wbr ucla wbr edu wbr tao wbr terrytao wbr wordpress wbr commathstodon wbr xyz wbr taoTerence TaoTraditional Chinese陶哲軒Simplified Chinese陶哲轩TranscriptionsStandard MandarinHanyu PinyinTao ZhexuanWuSuzhouneseDau Tseh shieYue CantoneseYale RomanizationTouh Jit hinJyutpingTou4 Zit3 hin1IPA tʰou tsiːt hiːn Tao was born to ethnic 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 and has been referred to as the Mozart of mathematics 6 7 8 9 10 Contents 1 Life and career 1 1 Family 1 2 Childhood 1 3 Career 1 4 Recognition 2 Research contributions 2 1 Dispersive partial differential equations 2 2 Harmonic analysis 2 3 Compressed sensing and statistics 2 4 Random matrices 2 5 Analytic number theory and arithmetic combinatorics 3 Notable awards 4 Major publications 5 Notes 6 See also 7 References 8 External linksLife and career editFamily edit 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 living in Australia Both formerly represented the states at the International Mathematical Olympiad 15 Furthermore Trevor 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 Riley c and daughter Madeleine 18 19 Childhood edit A child prodigy 20 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 21 6 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 22 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 23 Career edit 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 24 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 24 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 24 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 25 26 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 27 his 2015 resolution of the Erdos discrepancy problem which used entropy estimates within analytic number theory 28 his 2019 progress on the Collatz conjecture in which he proved the probabilistic claim that almost all Collatz orbits attain almost bounded values 29 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 30 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 31 Recognition edit British mathematician and Fields medalist Timothy Gowers remarked on Tao s breadth of knowledge 32 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 33 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 34 Tao has won numerous mathematician honours and awards over the years 35 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 36 and the American Mathematical Society 37 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 34 38 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 39 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 40 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 41 Tao was a finalist to become Australian of the Year in 2007 42 As of 2022 Tao has published over three hundred articles along with sixteen books 43 He has an Erdos number of 2 44 He is a highly cited researcher 45 46 Research contributions editDispersive partial differential equations editFrom 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 Schrodinger equations KdV equations and KdV type equations C 03 nbsp Tao at the age of 10 with mathematician Paul Erdos in 1985Michael Christ Colliander and Tao developed methods of Carlos Kenig Gustavo Ponce and Luis Vega to establish ill posedness of certain Schrodinger and KdV equations for Sobolev data of sufficiently low exponents CCT03 47 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 Schrodinger 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 Schrodinger 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 48 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 49 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 50 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 27 However such results remain as of 2022 conjectural Harmonic analysis edit 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 51 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 52 53 54 55 56 57 Similar problems were analyzed 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 58 59 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 60 61 62 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 63 64 65 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 66 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 67 Compressed sensing and statistics edit 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 68 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 69 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 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 70 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 71 Similar problems and results have also been considered by a number of other authors 72 73 74 75 76 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 77 Trevor Hastie Robert Tibshirani and Jerome H Friedman conclude that it is somewhat unsatisfactory in a number of cases 78 Nonetheless it remains of significant interest in the statistical literature Random matrices edit In the 1950s Eugene Wigner initiated the study of random matrices and their eigenvalues 79 80 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 average 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 Laszlo Erdos Horng Tzer Yau and Jun Yin 81 82 Analytic number theory and arithmetic combinatorics edit nbsp Tao second from left with UCLA undergraduate students in 2021In 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 83 84 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 Szemeredi 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 Szemeredi 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 Szemeredi 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 edit1999 Packard Fellowship 2000 Salem Prize for 85 his work in Lp harmonic analysis and on related questions in geometric measure theory and partial differential equations dd 2002 Bocher Memorial Prize for Global regularity of wave maps I Small critical Sobolev norm in high dimensions Internat Math Res Notices 2001 no 6 299 328 Global regularity of wave maps II Small energy in two dimensions Comm Math Phys 2244 2001 no 2 443 544 dd 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 dd 2005 Australian Mathematical Society Medal 2005 Ostrowski Prize with Ben Green for their exceptional achievements in the area of analytic and combinatorial number theory dd 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 dd 2006 Fields Medal for his contributions to partial differential equations combinatorics harmonic analysis and additive number theory dd 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 dd 2008 Onsager Medal 90 for his combination of mathematical depth width and volume in a manner unprecedented in contemporary mathematics His Lars Onsager lecture was entitled Structure and randomness in the prime numbers at NTNU Norway 91 dd 2009 Inducted into the American Academy of Arts and Sciences 92 2010 King Faisal International Prize 2010 Nemmers Prize in Mathematics 93 2010 Polya Prize with Emmanuel Candes 2012 Crafoord Prize 94 95 2012 Simons Investigator 96 2014 Breakthrough Prize in Mathematics For numerous breakthrough contributions to harmonic analysis combinatorics partial differential equations and analytic number theory dd 2014 Royal Medal 2015 PROSE award in the category of Mathematics for 97 Hilbert s Fifth Problem and Related Topics ISBN 978 1 4704 1564 8 dd 2019 Riemann Prize 98 2020 Princess of Asturias Award for Technical and Scientific Research 99 with Emmanuel Candes for their work on compressed sensing 2020 Bolyai Prize 100 2021 IEEE Jack S Kilby Signal Processing Medal 101 2021 USIA Award 2022 Education amp Research award finalist 2022 Global Australian of the Year Advance Global Australians Advance org 102 103 2022 Research com Mathematics in United States Leader Award 2023 Grande Medaille 2023 Research com Mathematics in United States Leader AwardMajor publications editTextbooks 2006 Solving mathematical problems A personal perspective Second edition of 1992 original ed Oxford 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 Cambridge University Press doi 10 1017 CBO9780511755149 ISBN 978 0 521 85386 6 MR 2289012 Zbl 1127 11002 104 105 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 Poincare 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 Poincare 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 PDF 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 106 2010 An epsilon of room II Pages from year three of a mathematical blog PDF 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 PDF 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 107 2012 Topics in random matrix theory PDF 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 PDF 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 PDF Providence RI American Mathematical Society doi 10 1090 mbk 081 ISBN 978 0 8218 9492 7 MR 3026767 Zbl 1276 00007 2014 Analysis I Texts and Readings in Mathematics Vol 37 Third edition of 2006 original ed New Delhi Hindustan Book Agency ISBN 978 93 80250 64 9 MR 3309891 Zbl 1300 26002 2014 Analysis II Texts and Readings in Mathematics Vol 38 Third edition of 2006 original 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 108 Research articles Tao is the author of over 300 articles The following among the most cited are surveyed above 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 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 KT99 Knutson Allen Tao Terence 1999 The honeycomb model of G L n C displaystyle GL n mathbb C nbsp 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 C 01 Colliander J Keel M Staffilani G Takaoka H Tao T 2001 Global well posedness for Schrodinger equations with derivative SIAM Journal on Mathematical Analysis 33 3 649 669 arXiv math 0101263 doi 10 1137 S0036141001384387 MR 1871414 Zbl 1002 35113 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 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 C 02a Colliander J Keel M Staffilani G Takaoka H Tao T 2002 A refined global well posedness result for Schrodinger 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 C 02b Colliander J Keel M Staffilani G Takaoka H Tao T 2002 Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation Mathematical Research Letters 9 5 6 659 682 doi 10 4310 MRL 2002 v9 n5 a9 MR 1906069 Zbl 1152 35491 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 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 C 03 Colliander J Keel M Staffilani G Takaoka H Tao T 2003 Sharp global well posedness for KdV and modified KdV on R displaystyle mathbb R nbsp and T displaystyle mathbb T nbsp Journal of the American Mathematical Society 16 3 705 749 doi 10 1090 S0894 0347 03 00421 1 MR 1969209 Zbl 1025 35025 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 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 C 04 Colliander J Keel M Staffilani G Takaoka H Tao T 2004 Global existence and scattering for rough solutions of a nonlinear Schrodinger 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 KTW04 Knutson Allen Tao Terence Woodward Christopher 2004 The honeycomb model of G L n C displaystyle GL n mathbb C nbsp 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 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 T04b Tao Terence 2004 Fuglede s conjecture is false in 5 and higher dimensions Mathematical Research Letters 11 2 3 251 258 doi 10 4310 MRL 2004 v11 n2 a8 MR 2067470 Zbl 1092 42014 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 BT06 Bejenaru Ioan Tao Terence 2006 Sharp well posedness and ill posedness results for a quadratic non linear Schrodinger equation Journal of Functional Analysis 233 1 228 259 doi 10 1016 j jfa 2005 08 004 MR 2204680 Zbl 1090 35162 BCT06 Bennett Jonathan Carbery Anthony Tao Terence 2006 On the multilinear restriction and Kakeya conjectures Acta Mathematica 196 2 261 302 doi 10 1007 s11511 006 0006 4 MR 2275834 Zbl 1203 42019 CRT06a Candes 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 CRT06b Candes 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 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 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 doi 10 1214 009053606000001523 MR 2382644 Zbl 1139 62019 TVZ07 Tao Terence Visan Monica Zhang Xiaoyi 2007 The nonlinear Schrodinger 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 C 08 Colliander J Keel M Staffilani G Takaoka H Tao T 2008 Global well posedness and scattering for the energy critical nonlinear Schrodinger equation in ℝ3 Annals of Mathematics Second Series 167 3 767 865 doi 10 4007 annals 2008 167 767 MR 2415387 Zbl 1178 35345 GT08 Green Ben Tao Terence 2008 The primes contain arbitrarily long arithmetic progressions Annals of Mathematics Second Series 167 2 481 547 doi 10 4007 annals 2008 167 481 MR 2415379 Zbl 1191 11025 KTV09 Killip Rowan Tao Terence Visan Monica 2009 The cubic nonlinear Schrodinger equation in two dimensions with radial data Journal of the European Mathematical Society 11 6 1203 1258 doi 10 4171 JEMS 180 MR 2557134 Zbl 1187 35237 CT10 Candes 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 GT10 Green Ben Tao Terence 2008 The primes contain arbitrarily long arithmetic progressions Annals of Mathematics Second Series 167 2 481 547 doi 10 4007 annals 2008 167 481 MR 2415379 Zbl 1191 11025 TV10 Tao Terence Vu Van 2010 With an appendix by Manjunath Krishnapur Random matrices universality of ESDs and the circular law Annals of Probability 38 5 2023 2065 doi 10 1214 10 AOP534 MR 2722794 Zbl 1203 15025 TV11 Tao Terence Vu Van 2011 Random matrices universality of local eigenvalue statistics Acta Mathematica 206 1 127 204 doi 10 1007 s11511 011 0061 3 MR 2784665 Zbl 1217 15043 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 doi 10 4007 annals 2012 176 2 11 MR 2950773 Zbl 1282 11007 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 doi 10 1090 jams 838 MR 3486169 Zbl 1342 35227 Notes edit Chinese 陶象國 pinyin Tao Xiangguo Chinese 梁蕙蘭 Jyutping Loeng Wai laan Being non binary Riley s pronouns are they them See also editErdos discrepancy problem Inscribed square problem Goldbach s weak conjecture Cramer conjectureReferences edit King Faisal Foundation retrieved 11 January 2010 SIAM George Polya Prize Archived from the original on 23 October 2021 Retrieved 5 September 2015 a b Vitae and Bibliography for Terence Tao 12 October 2009 Retrieved 21 January 2010 Mathematician Proves Huge Result on Dangerous Problem 11 December 2019 Archived from the original on 23 October 2021 Search arXiv e print repository a b c d Cook Gareth 24 July 2015 The Singular Mind of Terry Tao Published 2015 The New York Times ISSN 0362 4331 Retrieved 15 February 2021 Primed for Success 2 October 2007 PRESIDENT S COUNCIL OF ADVISORS ON SCIENCE AND TECHNOLOGY Terence Tao PhD 2021 Terence Tao Mozart of Math is first UCLA math prof to win Fields Medal 8 August 2006 a b c d Terence Tao the Mozart of maths 7 March 2015 Stephanie Wood The Sydney Morning Herald a b Wen Wei Po Page A4 24 August 2006 Dr Billy Tao Healthshare Oriental Daily Page A29 24 August 2006 Terence Chi Shen Tao MacTutor History of Mathematics archive School of Mathematics and Statistics University of St Andrews Scotland Nigel makes Waves Google s bid to overthrow email Asher Moses Sydney Morning Herald 2 October 2009 Tao Trevor History Travel Arts Science People Places Smithsonian Archived from the original on 10 September 2012 Retrieved 5 September 2015 Wood Stephanie 4 March 2015 Terence Tao the Mozart of maths The Sydney Morning Herald Retrieved 13 February 2023 Winners of Our Fourth Annual Podcast Contest The New York Times 1 July 2021 Retrieved 26 March 2023 Clements M A Ken 1984 Terence Tao Educational Studies in Mathematics 15 3 213 238 doi 10 1007 BF00312075 JSTOR 3482178 S2CID 189827772 Radical acceleration in Australia Terence Tao Radical Acceleration in Australia Terence Tao www davidsongifted org Archived from the original on 23 October 2021 International Mathematical Olympiad a b c It s prime time as numbers man Tao tops his Field Stephen Cauchi 23 August 2006 Retrieved 31 August 2006 Kenneth Chang 13 March 2007 Journeys to the Distant Fields of Prime The New York Times Archived from the original on 23 October 2021 Corrections For the Record The New York Times 13 March 2007 Archived from the original on 23 October 2021 a b Quanta Magazine 24 February 2014 Terence Tao s Answer to the Erdos Discrepancy Problem Quanta Magazine October 2015 Archived from the original on 26 February 2019 Tao Terence 2019 Almost all orbits of the Collatz map attain almost bounded values arXiv 1909 03562 math PR Rodgers Brad Tao Terence 6 April 2020 The De Bruijn Newman constant is non negative Forum of Mathematics Pi 8 e6 arXiv 1801 05914 doi 10 1017 fmp 2020 6 Tao Terence 2020 Sendov s conjecture for sufficiently high degree polynomials arXiv 2012 04125 math CV Mathematical Reviews MR2523047 Review by Timothy Gowers of Terence Tao s Poincare s legacies part I http mathscinet NewScientist com Prestigious Fields Medals for mathematics awarded 22 August 2006 a b 2006 Fields Medals awarded PDF Notices of the American Mathematical Society 53 9 1037 1044 October 2006 Archived from the original PDF on 2 November 2006 Vitae UCLA Retrieved 5 September 2015 APS Member History search amphilsoc org Retrieved 19 March 2021 List of Fellows of the American Mathematical Society retrieved 25 August 2013 Reclusive Russian turns down math world s highest honour Canadian Broadcasting Corporation CBC 22 August 2006 Archived from the original on 23 October 2021 Retrieved 26 August 2006 Media information UCLA Archived from the original on 23 October 2021 Retrieved 5 September 2015 President Biden Announces Members of President s Council of Advisors on Science and Technology White House 22 September 2021 Archived from the original on 23 October 2021 Terence Tao il matematico con il QI piu alto al mondo Non so cantare e ho fallito un paio di esami Huffington Post Italy 21 September 2021 Archived from the original on 23 October 2021 National Australia Day Committee Terence Tao Australian of the Year Retrieved 3 February 2023 Terence C Tao MathSciNet American Mathematical Society Retrieved 24 November 2022 Who am I UCLA Archived from the original on 23 October 2021 Retrieved 5 September 2015 Terence Tao s Publons profile publons com Archived from the original on 23 October 2021 Retrieved 6 February 2021 Highly Cited Researchers publons com Archived from the original on 23 October 2021 Retrieved 6 February 2021 Kenig Carlos E Ponce Gustavo Vega Luis On the ill posedness of some canonical dispersive equations Duke Math J 106 2001 no 3 617 633 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 Lemarie Rieusset Pierre Gilles 2016 The Navier Stokes problem in the 21st century Boca Raton FL CRC Press doi 10 1201 b19556 ISBN 978 1 4665 6621 7 MR 3469428 S2CID 126089972 Zbl 1342 76029 Fuglede Bent Commuting self adjoint partial differential operators and a group theoretic problem J Functional Analysis 16 1974 101 121 Coifman R R Meyer Yves On commutators of singular integrals and bilinear singular integrals Trans Amer Math Soc 212 1975 315 331 Coifman R Meyer Y Commutateurs d integrales singulieres et operateurs 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the Fourier transform Bull Amer Math Soc 81 1975 477 478 Strichartz Robert S Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations Duke Math J 44 1977 no 3 705 714 Bourgain J Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I Schrodinger equations Geom Funct Anal 3 1993 no 2 107 156 Bourgain J Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II The KdV equation Geom Funct Anal 3 1993 no 3 209 262 Klainerman S Machedon M Space time estimates for null forms and the local existence theorem Comm Pure Appl Math 46 1993 no 9 1221 1268 Wolff Thomas A sharp bilinear cone restriction estimate Ann of Math 2 153 2001 no 3 661 698 Bourgain Jean Guth Larry Bounds on oscillatory integral operators based on multilinear estimates Geom Funct Anal 21 2011 no 6 1239 1295 Donoho David L Compressed sensing IEEE Trans Inform Theory 52 2006 no 4 1289 1306 Candes Emmanuel J The restricted isometry property and its implications for compressed sensing C R Math Acad Sci Paris 346 2008 no 9 10 589 592 Candes Emmanuel J Recht Benjamin Exact matrix completion via convex optimization Found Comput Math 9 2009 no 6 717 772 Recht Benjamin A simpler approach to matrix completion J Mach Learn Res 12 2011 3413 3430 Keshavan Raghunandan H Montanari Andrea Oh Sewoong Matrix completion from a few entries IEEE Trans Inform Theory 56 2010 no 6 2980 2998 Recht Benjamin Fazel Maryam Parrilo Pablo A Guaranteed minimum rank solutions of linear matrix equations via nuclear norm minimization SIAM Rev 52 2010 no 3 471 501 Candes Emmanuel J Plan Yaniv Tight oracle inequalities for low rank matrix recovery from a minimal number of noisy random measurements IEEE Trans Inform Theory 57 2011 no 4 2342 2359 Koltchinskii Vladimir Lounici Karim Tsybakov Alexandre B Nuclear norm penalization and optimal rates for noisy low rank matrix completion Ann Statist 39 2011 no 5 2302 2329 Gross David Recovering low rank matrices from few coefficients in any basis IEEE Trans Inform Theory 57 2011 no 3 1548 1566 Bickel Peter J Ritov Ya acov Tsybakov Alexandre B Simultaneous analysis of lasso and Dantzig selector Ann Statist 37 2009 no 4 1705 1732 Hastie Trevor Tibshirani Robert Friedman Jerome The elements of statistical learning Data mining inference and prediction Second edition Springer Series in Statistics Springer New York 2009 xxii 745 pp ISBN 978 0 387 84857 0 Wigner Eugene P Characteristic vectors of bordered matrices with infinite dimensions Annals of Mathematics 2 62 1955 548 564 Wigner Eugene P On the distribution of the roots of certain symmetric matrices Ann of Math 2 67 1958 325 327 Erdos Laszlo Yau Horng Tzer Yin Jun 2012 Rigidity of eigenvalues of generalized Wigner matrices Advances in Mathematics 229 3 1435 1515 arXiv 1007 4652 doi 10 1016 j aim 2011 12 010 Erdos Laszlo Yau Horng Tzer Yin Jun 2012 Bulk universality for generalized Wigner matrices Probability Theory and Related Fields 154 1 2 341 407 doi 10 1007 s00440 011 0390 3 S2CID 253977494 Bourgain J More on the sum product phenomenon in prime fields and its applications Int J Number Theory 1 2005 no 1 1 32 Bourgain J Glibichuk A A Konyagin S V Estimates for the number of sums and products and for exponential sums in fields of prime order J London Math Soc 2 73 2006 no 2 380 398 Mathematics People Notices of the AMS Clay Research Awards Alladi Krishnaswami 9 December 2019 Ramanujan s legacy the work of the SASTRA prize winners Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences The Royal Society 378 2163 20180438 doi 10 1098 rsta 2018 0438 ISSN 1364 503X PMID 31813370 S2CID 198231874 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 NTNU s Onsager Lecture by Terence Tao on 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 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 Candes 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 Green Ben 2009 Review Additive combinatoricsby Terence C Tao and Van H Vu PDF Bull Amer Math Soc N S 46 3 489 497 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 Lubotzky Alexander 25 January 2018 Review of Expansion in finite simple groups of Lie type by Terence Tao Bull Amer Math Soc N S 1 doi 10 1090 bull 1610 review published electronically a href Template Cite journal html title Template Cite journal cite journal a CS1 maint postscript link External links edit nbsp Wikiquote has quotations related to Terence Tao nbsp Wikimedia Commons has media related to Terence Tao 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 nbsp Retrieved from https en wikipedia org w index php title Terence Tao amp oldid 1186041093, wikipedia, wiki, book, books, library,

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