Dispersive partial differential equations From 2001 to 2010, Tao was part of a collaboration with
James Colliander, Markus Keel,
Gigliola Staffilani, and Hideo Takaoka. They found a number of novel results, many to do with the
well-posedness of
weak solutions, for
Schrödinger equations,
KdV equations, and KdV-type equations. 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. 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. 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. 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.
Rowan Killip, Tao, and Vișan later made notable progress on the two-dimensional problem in radial symmetry. An article by Tao in 2001 considered the
wave maps equation with two-dimensional domain and spherical range. He built upon earlier innovations of
Daniel Tataru, who considered wave maps valued in
Minkowski space. 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. In 2016, Tao constructed a variant of the
Navier–Stokes equations which possess solutions exhibiting irregular behavior in finite time. 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. 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. 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. 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 spaces. This unified and extended earlier notable results of
Ronald Coifman,
Carlos Kenig,
Michael Lacey,
Yves Meyer,
Elias Stein, and Thiele, among others. Similar problems were analysed by Tao in 2001 in the context of Bourgain spaces, rather than the usual spaces. 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. 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. 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 spaces. Such multilinear problems originated in the 1990s, including in notable work of
Jean Bourgain,
Sergiu Klainerman, and
Matei Machedon. 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. In 2003, Tao adapted ideas developed by
Thomas Wolff for bilinear restriction to conical sets into the setting of restriction to quadratic hypersurfaces. 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.
Compressed sensing and statistics In collaboration with
Emmanuel Candes and Justin Romberg, Tao has made notable contributions to the field of
compressed sensing. In mathematical terms, most of their results identify settings in which a convex optimisation problem correctly computes the solution of an optimisation problem which seems to lack a computationally tractable structure. These problems are of the nature of finding the solution of an underdetermined linear system with the minimal possible number of nonzero entries, referred to as "sparsity". Around the same time,
David Donoho considered similar problems from the alternative perspective of high-dimensional geometry. Motivated by striking numerical experiments, Candes, Romberg, and Tao first studied the case where the matrix is given by the discrete Fourier transform. 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. 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. These ideas and results were later improved by Candes. Candes and Tao also considered relaxations of the sparsity condition, such as power-law decay of coefficients. 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. 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. 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.
Trevor Hastie,
Robert Tibshirani, and
Jerome H. Friedman conclude that it is "somewhat unsatisfactory" in a number of cases. 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. 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. Improved results were later found by Recht. Similar problems and results have also been considered by a number of other authors.
Random matrices In the 1950s,
Eugene Wigner initiated the study of
random matrices and their eigenvalues. 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 is large and the entries of a matrix are selected randomly according to any fixed
probability distribution of
expectation 0 and
standard deviation 1, then the eigenvalues of will tend to be uniformly scattered across the disk of radius around the origin; this can be made precise using the language of
measure theory. 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. Similar results were obtained around the same time by László Erdös,
Horng-Tzer Yau, and Jun Yin.
Analytic number theory and arithmetic combinatorics In 2004, Tao, together with
Jean Bourgain and
Nets Katz, studied the additive and multiplicative structure of subsets of
finite fields of prime order. 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. 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. 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 matrix and a matrix whose components are all integers, Green and Tao give conditions on when there exist infinitely many matrices such that all components of are prime numbers. 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
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