Terence Tao

Template:Short description Template:Use dmy dates Template:Use Australian English Template:Infobox scientist Template:Infobox Chinese

Terence Chi-Shen Tao Template:Post-nominals (Template:Zh; born 17 July 1975) is an Australian-American mathematician, Fields medalist, and professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref>

Tao was born to Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014, and is a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers, and is widely regarded as one of the greatest living mathematicians.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="singmind">Template:Cite news</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref><ref name="Terence Tao: the Mozart of maths" />

Life and careerEdit

FamilyEdit

Tao's parents are first generation immigrants from Hong Kong to Australia.<ref name="ReferenceA">Wen Wei Po, Page A4, 24 August 2006.</ref> Tao's father, Billy Tao,Template:Efn was a Chinese paediatrician who was born in Shanghai and earned his medical degree (MBBS) from the University of Hong Kong in 1969.<ref>Dr Billy Tao, Healthshare.</ref> Tao's mother, Grace Leong,Template:Efn was born in Hong Kong; she received a first-class honours degree in mathematics and physics at the University of Hong Kong.<ref name="Terence Tao: the Mozart of maths">Terence Tao: the Mozart of maths, 7 March 2015, Stephanie Wood, The Sydney Morning Herald.</ref> She was a secondary school teacher of mathematics and physics in Hong Kong.<ref>Oriental Daily, Page A29, 24 August 2006.</ref> Billy and Grace met as students at the University of Hong Kong.<ref>Terence Chi-Shen Tao, MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland.</ref> They then emigrated from Hong Kong to Australia in 1972.<ref name="ReferenceA"/><ref name="Terence Tao: the Mozart of maths"/>

Tao also has two brothers, Trevor and Nigel, who are currently living in Australia. Both formerly represented Australia at the International Mathematical Olympiad.<ref name="NigelIMO">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="TrevorIMO">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Furthermore, Trevor Tao has been representing Australia internationally in chess and holds the title of Chess International Master.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Tao speaks Cantonese but cannot write Chinese. Tao is married to Laura Tao, an electrical engineer at NASA's Jet Propulsion Laboratory.<ref name="Terence Tao: the Mozart of maths" /><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> They live in Los Angeles, California, and have two children.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

ChildhoodEdit

A child prodigy,<ref>Template:Citation.</ref> Terence Tao skipped 5 grades.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He is one of only three children in the history of the Johns Hopkins Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just eight years old; Tao scored a 760.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Julian Stanley, Director of the Study of Mathematically Precocious Youth, stated that Tao had the greatest mathematical reasoning ability he had found in years of intensive searching.<ref name="singmind" /><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref>

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

CareerEdit

At age 14, Tao attended the Research Science Institute, a summer program for secondary students. In 1991, he received his bachelor's and master's degrees at the age of 16 from Flinders University under the direction of Garth Gaudry.<ref name="theage.com.au">It's prime time as numbers man Tao tops his Field Stephen Cauchi, 23 August 2006. Retrieved 31 August 2006.</ref> In 1992, he won a postgraduate Fulbright Scholarship to undertake research in mathematics at Princeton University in the United States. From 1992 to 1996, Tao was a graduate student at Princeton University under the direction of Elias Stein, receiving his PhD at the age of 21.<ref name="theage.com.au" /> In 1996, he joined the faculty of the University of California, Los Angeles. In 1999, when he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution.<ref name="theage.com.au" />

He is known for his collaborative mindset; by 2006, Tao had worked with over 30 others in his discoveries,<ref name="singmind" /> reaching 68 co-authors by October 2015.

Tao has had a particularly extensive collaboration with British mathematician Ben J. Green; together they proved the Green–Tao theorem, which is well known among both amateur and professional mathematicians. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:<ref>Template:Cite news Template:Cbignore</ref><ref>Template:Cite news Template:Cbignore</ref>

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.{{#if:|{{#if:|}}
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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<ref name="quantanavier">{{#invoke:citation/CS1|citation

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his 2015 resolution of the Erdős discrepancy problem, which used entropy estimates within analytic number theory<ref>{{#invoke:citation/CS1|citation

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his 2019 progress on the Collatz conjecture, in which he proved the probabilistic claim that almost all Collatz orbits attain almost bounded values.<ref>Template:Cite arXiv</ref>

Tao has also resolved or made progress on a number of conjectures. In 2012, Green and Tao announced proofs of the conjectured "orchard-planting problem," which asks for the maximum number of lines through exactly 3 points in a set of n points in the plane, not all on a line. In 2018, with Brad Rodgers, Tao showed that the de Bruijn–Newman constant, the nonpositivity of which is equivalent to the Riemann hypothesis, is nonnegative.<ref>Template:Cite journal.</ref> In 2020, Tao proved Sendov's conjecture, concerning the locations of the roots and critical points of a complex polynomial, in the special case of polynomials with sufficiently high degree.<ref>Template:Cite journal</ref>

RecognitionEdit

File:Terence Tao.jpg

Tao at ICM 2006.

Tao has won numerous mathematician honours and awards over the years.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> He is a Fellow of the Royal Society, the Australian Academy of Science (Corresponding Member), the National Academy of Sciences (Foreign member), the American Academy of Arts and Sciences, the American Philosophical Society,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and the American Mathematical Society.<ref>List of Fellows of the American Mathematical Society, retrieved 25 August 2013.</ref> In 2006 he received the Fields Medal; he was the first Australian, the first UCLA faculty member, and one of the youngest mathematicians to receive the award.<ref name="2006 Fields Medals awarded">Template:Cite journal</ref><ref name="cbc.ca">Template:Cite news Template:Cbignore</ref> He was also awarded the MacArthur Fellowship. He has been featured in The New York Times, CNN, USA Today, Popular Science, and many other media outlets.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In 2021, Tao was awarded the Riemann Prize Week as recipient of the inaugural Riemann Prize 2019 by the Riemann International School of Mathematics at the University of Insubria.<ref>Template:Cite news</ref> Tao was a finalist to become Australian of the Year in 2007.<ref>National Australia Day Committee, Terence Tao – Australian of the Year. Retrieved 3 February 2023.</ref>

As of 2022, Tao had published over three hundred articles, along with sixteen books.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> He has an Erdős number of 2.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> He is a highly cited researcher.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

An article by New Scientist<ref>NewScientist.com, Prestigious Fields Medals for mathematics awarded, 22 August 2006.</ref> writes of his ability:

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

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.{{#if:|{{#if:|}}
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Research contributionsEdit

Dispersive partial differential equationsEdit

From 2001 to 2010, Tao was part of a collaboration with James Colliander, Markus Keel, Gigliola Staffilani, and Hideo Takaoka. They found a number of novel results, many to do with the well-posedness of weak solutions, for Schrödinger equations, KdV equations, and KdV-type equations.Template:Ran

File:Paul Erdos with Terence Tao.jpg

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.Template:Ran<ref>Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106 (2001), no. 3, 617–633.</ref> In many cases these results were sharp enough to perfectly complement well-posedness results for sufficiently large exponents as due to Bourgain, Colliander−Keel−Staffilani−Takaoka−Tao, and others. Further such notable results for Schrödinger equations were found by Tao in collaboration with Ioan Bejenaru.Template:Ran

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.Template:Ran 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.Template:Ran Rowan Killip, Tao, and Vișan later made notable progress on the two-dimensional problem in radial symmetry.Template:Ran

An article by Tao in 2001 considered the wave maps equation with two-dimensional domain and spherical range.Template:Ran He built upon earlier innovations of Daniel Tataru, who considered wave maps valued in Minkowski space.<ref>Tataru, Daniel. On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37–77.</ref> Tao proved the global well-posedness of solutions with sufficiently small initial data. The fundamental difficulty is that Tao considers smallness relative to the critical Sobolev norm, which typically requires sophisticated techniques. Tao later adapted some of his work on wave maps to the setting of the Benjamin–Ono equation; Alexandru Ionescu and Kenig later obtained improved results with Tao's methods.Template:Ran<ref>Ionescu, Alexandru D.; Kenig, Carlos E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Amer. Math. Soc. 20 (2007), no. 3, 753–798.</ref>

In 2016, Tao constructed a variant of the Navier–Stokes equations which possess solutions exhibiting irregular behavior in finite time.Template:Ran 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.<ref>Template:Cite book</ref> Tao speculated that the Navier–Stokes equations might be able to simulate a Turing complete system, and that as a consequence it might be possible to (negatively) resolve the existence and smoothness problem using a modification of his results.<ref name="singmind" /><ref name="quantanavier" /> However, such results remain (as of 2024) conjectural.

Harmonic analysisEdit

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 Template:Math<ref>Fuglede, Bent. Commuting self-adjoint partial differential operators and a group theoretic problem. J. Functional Analysis 16 (1974), 101–121.</ref> Tao resolved the conjecture in the negative for dimensions larger than 5, based upon the construction of an elementary counterexample to an analogous problem in the setting of finite groups.Template:Ran

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

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

Compressed sensing and statisticsEdit

In collaboration with Emmanuel Candes and Justin Romberg, Tao has made notable contributions to the field of compressed sensing. In mathematical terms, most of their results identify settings in which a convex optimisation problem correctly computes the solution of an optimisation problem which seems to lack a computationally tractable structure. These problems are of the nature of finding the solution of an underdetermined linear system with the minimal possible number of nonzero entries, referred to as "sparsity". Around the same time, David Donoho considered similar problems from the alternative perspective of high-dimensional geometry.<ref>Donoho, David L. Compressed sensing. IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306.</ref>

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

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.Template:Ran 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.<ref>Bickel, Peter J.; Ritov, Ya'acov; Tsybakov, Alexandre B. Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 (2009), no. 4, 1705–1732.</ref> Trevor Hastie, Robert Tibshirani, and Jerome H. Friedman conclude that it is "somewhat unsatisfactory" in a number of cases.<ref>Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome The elements of statistical learning. Data mining, inference, and prediction. Second edition. Springer Series in Statistics. Springer, New York, 2009. xxii+745 pp. Template:ISBN</ref> Nonetheless, it remains of significant interest in the statistical literature.

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

Random matricesEdit

In the 1950s, Eugene Wigner initiated the study of random matrices and their eigenvalues.<ref>Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics (2) 62 (1955), 548–564.</ref><ref>Wigner, Eugene P. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958), 325–327.</ref> Wigner studied the case of hermitian 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 Template:Mvar is large and the entries of a Template:Math matrix Template:Mvar are selected randomly according to any fixed probability distribution of expectation 0 and standard deviation 1, then the eigenvalues of Template:Mvar will tend to be uniformly scattered across the disk of radius Template:Math around the origin; this can be made precise using the language of measure theory.Template:Ran 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.Template:Ran Similar results were obtained around the same time by László Erdös, Horng-Tzer Yau, and Jun Yin.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Analytic number theory and arithmetic combinatoricsEdit

File:Professor Tao with his Devoted Students.jpg

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.Template:Ran 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.<ref>Bourgain, J. More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (2005), no. 1, 1–32.</ref><ref>Bourgain, J.; Glibichuk, A.A.; Konyagin, S.V. Estimates for the number of sums and products and for exponential sums in fields of prime order. J. London Math. Soc. (2) 73 (2006), no. 2, 380–398.</ref>

Tao and Ben Green 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.Template:Ran The source of Green and Tao's arithmetic progressions is Endre Szemerédi's 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 Template:Math matrix Template:Mvar and a Template:Math matrix Template:Mvar whose components are all integers, Green and Tao give conditions on when there exist infinitely many Template:Math matrices Template:Mvar such that all components of Template:Math are prime numbers.Template:Ran 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.Template:Ran

Notable awardsEdit

Terence Tao has won numerous awards for his work. He won the Fields Medal, the highest award of mathematics, in 2006. Template:Refbegin

1999 – Packard Fellowship
2000 – Salem Prize for:<ref>Mathematics People. Notices of the AMS</ref>

"his work in Template:Math harmonic analysis and on related questions in geometric measure theory and partial differential equations."

2002 – Bôcher Memorial Prize for:<ref>Template:Cite journal</ref>

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:<ref>{{#invoke:citation/CS1|citation

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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<ref name="Alladi SASTRA">Template:Cite journal</ref>
2006 – Sloan Fellowship
2007 – Fellow of the Royal Society<ref>Fellows and Foreign Members of the Royal Society, retrieved 9 June 2010.</ref>
2008 – Alan T. Waterman Award for:<ref>National Science Foundation, Alan T. Waterman Award. Retrieved 18 April 2008.</ref>

"his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis"

2008 – Onsager Medal<ref>{{#invoke:citation/CS1|citation

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"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.<ref>Template:YouTube</ref>

2009 – Inducted into the American Academy of Arts and Sciences<ref>{{#invoke:citation/CS1|citation

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His 2009 induction ceremony is here.</ref>

2010 – King Faisal International Prize<ref>Template:Cite journal</ref>
2010 – Nemmers Prize in Mathematics<ref>{{#invoke:citation/CS1|citation

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2010 – Polya Prize (with Emmanuel Candès)
2012 – Crafoord Prize<ref>{{#invoke:citation/CS1|citation

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2012 – Simons Investigator<ref>{{#invoke:citation/CS1|citation

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2014 – Breakthrough Prize in Mathematics

"For numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations and analytic number theory."

2014 – Royal Medal
2015 – PROSE award in the category of "Mathematics" for:<ref>{{#invoke:citation/CS1|citation

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"Hilbert's Fifth Problem and Related Topics" Template:ISBN

2019 – Riemann Prize<ref>{{#invoke:citation/CS1|citation

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2019 – The Carnegie Corporation of New York honored Tao with 2019 Great Immigrant Award.<ref>{{#invoke:citation/CS1|citation

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2020 – Princess of Asturias Award for Technical and Scientific Research,<ref>{{#invoke:citation/CS1|citation

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2020 – Bolyai Prize<ref>{{#invoke:citation/CS1|citation

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2021 – IEEE Jack S. Kilby Signal Processing Medal<ref name="IEEE Awards">{{#invoke:citation/CS1|citation

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2022 – Global Australian of the Year (Advance Global Australians; Advance.org)<ref name="2022-09-08_A">World’s greatest mathematician named 2022 Global Australian of the Year, Advance.org, media release 2022-09-08, accessed 2022-09-14</ref><ref name="2022-09-07_FR">Why this maths genius refuses to work for a hedge fund, Tess Bennett, Australian Financial Review, 2022-09-07, accessed 2022-09-14</ref>
2022 – Grande Médaille<ref>{{#invoke:citation/CS1|citation

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