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Terence Tao AC FRS

Professor of Mathematics; James and Carol Collins Chair, University of California, Los Angeles

tao@math.ucla.edu · https://www.math.ucla.edu/~tao/

Biography

Terence Tao was born in Adelaide, Australia in 1975. He has been a professor of mathematics at UCLA since 1999, having completed his PhD under Elias Stein at Princeton in 1996. Tao's areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Salem Prize in 2000, the Fields Medal in 2006, the MacArthur Fellowship in 2007, the Crafoord Prize in 2012, and the Breakthrough Prize in Mathematics in 2015. Terence Tao also holds the James and Carol Collins Chair in mathematics at UCLA, and is a Fellow of the Royal Society, the Australian Academy of Sciences, the National Academy of Sciences, and the American Academy of Arts and Sciences. From 2020 to 2024, he served on the President's Council of Advisors on Science and Technology.

Research interests

PrimaryOscillatory integrals and Kakeya maximal functions; Nonlinear dispersive equations; Multilinear operators (e.g. the bilinear Hilbert transform); Arithmetic combinatorics, and connections with number theory and ergodic theory; Compressed sensing
SecondaryRepresentation and symplectic theory of U(n); honeycombs; Geometric PDE (e.g. Yang-Mills); General relativity; General harmonic analysis; Inverse scattering, nonlinear Fourier transforms, and integrable systems for rough data

Education

1996Ph.D., Princeton University (advisor: Elias Stein)
Thesis: Three regularity results in harmonic analysis
1992M.Sc., Flinders University (advisor: Garth Gaudry)
Thesis: Convolution operators generated by right-monogenic and harmonic kernels
1991B.Sc. (Hons), Flinders University (advisor: Garth Gaudry)

Appointments

1992–1994Assistant Researcher, Flinders Medical Centre
1993–1994Assistant Researcher, Princeton University
1996–1998Hedrick Assistant Professor, UCLA
1997–1997Member (Fall), MSRI
1999–1999Acting Assistant Professor, UCLA
1999–1999Visiting Fellow, UNSW
2000–2000Assistant Professor, UCLA
2000–2000Visiting Professor, UNSW
2000–2009Professor, UCLA
2009–Distinguished Professor, UCLA
2001–2003CMI Long-term Prize Fellow, Clay Mathematics Institute
2001–2003Honorary Professor, Australian National University

Awards and honors

1986Bronze Medal — International Mathematical Olympiad
1987Silver Medal — International Mathematical Olympiad
1988Gold Medal — International Mathematical Olympiad
1992University Medal — Flinders University
1992–1995Fulbright Postgraduate Scholar — Australian-American Fulbright Commission
1995Postgraduate Fellowship — Sloan Foundation
1997–2000Summer grant DMS-9706964 — National Science Foundation
1999–2001Research Fellowship — Sloan Foundation
1999–2006Foundation Fellowship — David and Lucile Packard Foundation
2000Salem Prize — Salem Prize
2002Bôcher Memorial Prize — American Mathematical Society
2003Clay Research Award — Clay Mathematics Institute
2004Levi L. Conant Award (with Allen Knutson) — American Mathematical Society
2005Robert Sorgenfrey Distinguished Teaching Award — UCLA
2005Australian Mathematical Society Medal — Australian Mathematical Society
2005ISAAC Award — ISAAC
2006Plenary Speaker — International Congress of Mathematicians
2006Fields Medal — International Mathematical Union
2006Corresponding Member — Australian Academy of Science
2006SASTRA Ramanujan Prize — SASTRA
2007–2011MacArthur Fellowship — MacArthur Foundation
2007Ostrowski Prize — Ostrowski Foundation
2007SA finalist, Australian of the Year — National Australia Day Council
2007–James and Carol Collins Chair — UCLA
2007–2011Research Award DMS-0649473 — National Science Foundation
2007–Fellow — The Royal Society
2008–2010Alan T. Waterman Award — National Science Foundation
2008Achievement Award — CESASC
2008–Foreign Affiliate (regular member since 2022) — National Academy of Sciences
2008Information Theory Society Paper Award (with Emmanuel Candès) — Information Theory Society
2008Convocation Award — Flinders University Alumni Association
2008Onsager Medal — NTNU
2009Member — American Academy of Arts and Sciences
2009Pólya Prize (with Emmanuel Candès) — SIAM
2010King Faisal International Prize (Mathematics) — King Faisal Foundation
2010Nemmers Prize — Northwestern University
2012Crafoord Prize (Mathematics) — Royal Swedish Academy of Sciences
2012Member — American Philosophical Society
2012Fellow — American Mathematical Society
2012–2022Simons Investigator — Simons Foundation
2013Overseas Member — Spanish Royal Academy of Sciences
2013NSF grant DMS-1301620 — National Science Foundation
2013–2018NSF grant DMS-1266164 — National Science Foundation
2013Joseph Lieberman Award — Center for Excellence in Education
2013Distinguished Alumni Award — Center for Talented Youth
2014Royal Medal — The Royal Society
2014Foreign Member — Royal Swedish Academy of Sciences
2015Breakthrough Prize in Mathematics — Breakthrough Prize
2015PROSE Award, Mathematics — PROSE Awards
2018–2023NSF grant DMS-1764034 — National Science Foundation
2019Riemann Prize — RISM
2020Princess of Asturias Award for Technical and Scientific Research — Princess of Asturias Foundation
2020János Bolyai International Mathematical Prize — Hungarian Academy of Sciences
2020IEEE Jack S. Kilby Signal Processing Medal (2021) — IEEE
2021Foreign Member — Istituto Lombardo Accademia di Scienze e Lettere
2022Honorary Doctorate of Science — Harvey Mudd College
2022Global Australian of the Year — Advance
2023Grande Médaille — French Academy of Sciences
2023Alexanderson Award — American Institute of Mathematics
2024NSF grant DMS-2347850 — National Science Foundation
2025Gala Award — Embassy of Australia (US)
2025Honorary Member — Real Sociedad Matemática Española
2025Frontiers of Science Award (declined) — International Congress of Basic Science
2025Science and Education Pioneer Award — UCLA
2026Moonshot seed grant (co-PI) — Laude Institute
2026–2029expMath grant (co-PI) — DARPA
2026Madison Medal — Princeton University
2026Companion of the Order of Australia — Australia
2026Golden Plate Award — American Academy of Achievement

Books

2026Six Math Essentials — Quanta Books
2015Expansion in finite simple groups of Lie type — American Mathematical Society
2014Hilbert’s fifth problem and related topics — American Mathematical Society
2013Compactness and contradiction — American Mathematical Society
2012Higher order Fourier analysis — American Mathematical Society
2012Topics in random matrix theory — American Mathematical Society
2011An introduction to measure theory — American Mathematical Society
2010An epsilon of room: pages from year three of a mathematical blog — American Mathematical Society
2009Poincaré’s legacies: pages from year two of a mathematical blog — American Mathematical Society
2008Structure and Randomness: pages from year one of a mathematical blog — American Mathematical Society
2006Additive Combinatorics (with Van Vu) — Cambridge University Press
2006Analysis I — Hindustan Book Agency
2006Analysis II — Hindustan Book Agency
2006Nonlinear dispersive equations: local and global analysis — American Mathematical Society
2006Solving Mathematical Problems (Second edition) — Oxford University Press
2000Three Regularity Results in Harmonic Analysis — World Scientific Publishing Co.
1992Solving Mathematical Problems: A personal perspective — Deakin University Press
Climbing the cosmic distance ladder (with Tanya Klowden) — Yale University Press
Spending symmetry

Publications

377 papers; the full, searchable list is on the papers page.

2026Gilbreath's conjecture: a Cramér random model and a deterministic analysis. with Zachary Chase, Zach Hunter. Preprint, to be submitted
2026Polynomial towers and inverse Gowers theory for bounded-exponent groups. with Asgar Jamneshan, Or Shalom. Submitted
2026Products of consecutive integers with unusual anatomy. Submitted, Acta Arith. (special issue in honor of Heath-Brown)
2026Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond. with Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang. To appear
2026Local Bernstein theory, and lower bounds for Lebesgue constants. Submitted, Constructive Analysis
2026Mathematical methods and human thought in the age of AI. with Tanya Klowden. Submitted, Blackwell Companion to Philosophy of Mathematics
2025New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach. with Timothy Trudgian, Andrew Yang. To appear, Math. Comp.
2025Decomposing a factorial into large factors. with Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew V. Sutherland, Markus Uhr, Kevin Ventullo. To appear, Math. Comp.
2025Some variants of the periodic tiling conjecture. Submitted, Comm. Amer. Math. Soc.
2025On the number of exceptional intervals to the prime number theorem in short intervals. with Ayla Gafni. To appear, Essential Number Theory
2025Rough numbers between consecutive primes. with Ayla Gafni. Submitted, Proc Lond. Math. Soc.
2025Growth rates of sequences governed by the squarefree properties of its translates. with Wouter van Doorn. To appear, Acta Arith.
2025Quantitative correlations and some problems on prime factors of consecutive integers. with Joni Teräväinen. Submitted, GAFA
2025New Nikodym set constructions over finite fields. Submitted, Pacific J. Math.
2025Sum-difference exponents for boundedly many slopes, and rational complexity. Submitted, Discrete Analysis
2025The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree. To appear, Journal d'Analyse Mathematique
2025Mathematical exploration and discovery at scale. with Bogdan Georgiev, Javier Gómez-Serrano, Adam Zsolt Wagner. Preprint
2025The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale. with Matthew Bolan, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, Andrés Goens, Aaron Hill, Harald Husum, Hernán Ibarra Mejia, Zoltan Kocsis, Bruno Le Floch, Amir Livne Bar-on, Lorenzo Luccioli, Douglas McNeil, Alex Meiburg, Pietro Monticone, Pace P. Nielsen, Emmanuel Osalotioman Osazuwa, Giovanni Paolini, Marco Petracci, Bernhard Reinke, David Renshaw, Marcus Rossel, Cody Roux, Jérémy Scanvic, Shreyas Srinivas, Anand Rao Tadipatri, Vlad Tsyrklevich, Fernando Vaquerizo-Villar, Daniel Weber, Fan Zheng. Submitted
2024Marton's conjecture for abelian groups of finite torsion. with Timothy Gowers, Ben Green, Freddie Manners. Submitted
2024On product representations of squares. Submitted, Acta Math. Hungarica
2024Dense sets of natural numbers with unusually large least common multiples. To appear, Integers
2024Planar point sets with forbidden 4-point patterns and few distinct distances. To appear, Combin. Disc. Geom.
2024Higher uniformity of arithmetic functions in short intervals II. Almost all intervals. with Kaisa Matomäki, Maksym Radziwiłł, Xuansheng Shao, Joni Teräväinen. Invent. math. (2026)
2024On several irrationality problems for Ahmes series. with Vjekoslav Kovač. Submitted, Acta Math. Hungarica
2024On the distribution of eigenvalues of GUE and its minors at fixed index. Submitted, Random Matrix Theory and Applications
2023Infinite partial sumsets in the primes. with Tamar Ziegler. J. d'Analyse Jerusalem Volume 151, pages 375–389, (2023) 10.1007/s11854-023-0323-y
2023A Host--Kra Fω2-system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the U6(Fn2) norm. with Asgar Jamneshan, Or Shalom. Submitted, Math. Annalen
2023The structure of totally disconnected Host--Kra--Ziegler factors, and the inverse theorem for the Uk Gowers uniformity norms on finite abelian groups of bounded torsion. with Asgar Jamneshan, Or Shalom. To appear, Annales de la Faculté des Sciences de Toulouse
2023An upper bound on the mean value of the Hooley delta function. with Dimitris Koukoulopoulos. To appear, Proc LMS
2023Sumsets and entropy revisited. with Ben Green, Freddie Manners. Random Structures and Algorithms (2023). doi:10.1002/rsa.21252
2023The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture. To appear, Comm. AMS
2023A lower bound on the mean value of the Hooley delta function. with Kevin Ford, Dimitris Koukoulopoulos. Proc LMS 127, December 2023, 1865–1885 http://dx.doi.org/10.1112/plms.12572
2023Monotone non-decreasing sequences of the Euler totient function. La Matematica (2024). doi:10.1007/s44007-024-00115-z
2023Undecidability of translational monotilings. with Rachel Greenfeld. To appear, J. Eur. Math. Soc.
2023On a conjecture of Marton. with Timothy Gowers, Ben Green, Freddie Manners. Submitted, Annals Math.
2023Adjoint Brascamp-Lieb inequalities. with Jon Bennett. Submitted, Proc. Lond. Math. Soc.
2023A Maclaurin type inequality. To appear, Proc. Amer. Math. Soc.
2023Sums of GUE matrices and concentration of hives from correlation decay of eigengaps. with Hariharan Narayanan, Scott Sheffield. Probab. Theory Relat. Fields 190 (2024), 1121–1165.
2022Measurable tilings by abelian group actions. with Jan Grebík, Rachel Greenfeld, Václav Rozhoň. IMRN, rnad048, https://doi.org/10.1093/imrn/rnad048
2022Higher uniformity of arithmetic functions in short intervals I. All intervals. with Kaisa Matomäki, Xuansheng Shao, Joni Teräväinen. Forum of Mathematics, Pi. 2023;11:e29. doi:10.1017/fmp.2023.28
2022A counterexample to the periodic tiling conjecture. with Rachel Greenfeld. submitted, Annals Math.
2022Perfectly packing a square by squares of nearly harmonic sidelength. Comp. Disc. Geom.(2023). https://doi.org/10.1007/s00454-023-00523-y
2021Singmaster's conjecture in the interior of Pascal's triangle. with Kaisa Matomäki, Maksym Radziwiłł, Xuansheng Shao, Joni Teräväinen. The Quarterly Journal of Mathematics, haac006, https://doi.org/10.1093/qmath/haac006
2021Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions. with Joni Teräväinen. J. Eur. Math. Soc. 27 (2025), no. 4, 1321–1384.
2021Undecidable translational tilings with only two tiles, or one nonabelian tile. with Rachel Greenfeld. to appear, Disc. Comp. Geom.
2021The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero. with Joni Teräväinen. J. Lond. Math. Soc. https://doi.org/10.1112/jlms.12663
2021The structure of arbitrary Conze-Lesigne systems. with Asgar Jamneshan, Or Shalom. submitted, Comm. AMS
2021The inverse theorem for the U3 Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches. with Asgar Jamneshan. Discrete Analysis https://doi.org/10.19086/da.84268
2021The effective potential of an $M$-matrix. with Marcel Filoche, Svetlana Mayboroda. J. Math. Phys. 62, 041902 (2021)
2021Elias M. Stein (2013-2018). with Lillian Pierce et al.. Not. Amer. Math. Soc. 68 (2021), 546–563
2021Optimal sine and sawtooth inequalities. with Louis Esser, Burt Totaro, Chengxi Wang. J. Fourier Anal. Appl. (2022) 28:14
2020Szemeredi's proof of Szemeredi's theorem. Acta Math. Hungar. 161 (2020), no. 2, 443–487.
2020Higher uniformity of bounded multiplicative functions in short intervals on average. with Kaisa Matomäki, Maksym Radziwiłł, Joni Teräväinen, Tamar Ziegler. Annals Math Pages 739–857 from Volume 197 (2023), Issue 2
2020Zarankiewicz's problem for semilinear hypergraphs. with Abdul Basit, Artem Chernikov, Sergei Starchenko, Chiu-Minh Tran. Forum of Mathematics, Sigma , Volume 9 , 2021 , e59
2020An uncountable Mackey-Zimmer theorem. with Asgar Jamneshan. Studia Math. 266 (2022), no. 3, 241–289.
2020The structure of translational tilings in Z^d. with Rachel Greenfeld. Discrete Analysis 2021:16, 28 pp.
2020Additive energy of regular measures in one and higher dimensions, and the fractional uncertainty principle. with Laura Cladek. Ars Inven. Anal. 2021, Paper No. 1, 38 pp.
2020Pointwise ergodic theorems for non-conventional bilinear polynomial averages. with Ben Krause, Mariusz Mirek. Annals Math. Pages 997–1109 from Volume 195 (2022), Issue 3
2020The Ionescu-Wainger multiplier theorem and the adeles. Mathematika 67 (2021), no. 3, 647–677.
2020Homogenization of iterated singular integrals with applications to random quasiconformal maps. with Kari Astala, Steffen Rohde, Eero Saksman. Revista Iberoamericana VOL. 38, NO. 7PP. 2285–2336
2020Fractional free convolution powers. With an appendix by David Jekel. with Dimitri Shlyakhtenko. To appear, Indiana U. Math. J.
2020Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration. with Asgar Jamneshan. Fundamenta Mathematicae 261 (2023), 1–98
2020Sendov's conjecture for sufficiently high degree polynomials. Acta Mathematica, Vol. 229, No. 2 (2022), pp. 347–392
2019Least singular value, circular law, and Lindeberg exchange. Random matrices, 461–498,IAS/Park City Math. Ser., 26, Amer. Math. Soc., Providence, RI, 2019.
2019Value patterns of multiplicative functions and related sequences. with Joni Teräväinen. Forum Math Sigma Volume 7 (2019), e33
2019Effective approximation of heat flow evolution of the Riemann xi function, and a new upper bound for the de Bruijn-Newman constant. with D.H.J. Polymath. Research in the Mathematical Sciences (2019) 6: 31. https://doi.org/10.1007/s40687-019-0193-1
2019Large prime gaps and probabilistic models. with William Banks, Kevin Ford. Inventiones Volume 233, pages 1471–1518, (2023)
2019Almost all orbits of the Collatz map attain almost bounded values. Forum Math. Pi 10 (2022), Paper No. e12, 56 pp.
2019An uncountable Moore-Schmidt theorem. with Asgar Jamneshan. Ergodic Theory and Dynamical Systems Volume 43 , Issue 7 , July 2023 , pp. 2376 - 2403 https://doi.org/10.1017/etds.2022.36
2019Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint. Mathematika 66 (2020), 517–576
2019A Close Call: How a Near Failure Propelled Me to Succeed. Living Proof: Stories of Resilience Along the Mathematical Journey, American Math. Society, Math. Assoc. America, 2019. A. Henderson, E. Lawrence, M. Pons, D. Taylor eds., 96–99.
2019Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra.. with Peter Denton, Stephen Parke, Xining Zhang. Bull. Amer. Math. Soc. (N.S.) 59 (2022), no. 1, 31–58.
2019On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows. Pure Appl. Funct. Anal. 5 (2020), no. 6, 1425–1443.
2019Searching for singularities in the Navier–Stokes equations. Nature Reviews Physics 1,418–419(2019)
2019Quantitative bounds for critically bounded solutions to the Navier-Stokes equations. Nine mathematical challenges—an elucidation, 149–193, Proc. Sympos. Pure Math., 104, Amer. Math. Soc., Providence, RI, [2021], ©2021.
2018The de Bruijn-Newman constant is nonnegative. with Brad Rodgers. Forum of Math. Pi, Volume 8 (2020), e6
2018Long gaps in sieved sets. with Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance. J. Eur. Math. Soc. (JEMS) 23 (2021), no. 2, 667–700.
2018The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures. with Joni Teräväinen. Algebra & Number Theory 13-9 (2019), 2103–2150. DOI 10.2140/ant.2019.13.2103
2018Fourier uniformity of bounded multiplicative functions in short intervals on average. with Kaisa Matomäki, Maksym Radziwiłł. Inventiones, 220 (2020), no. 1, 1–58
2018Commutators close to the identity. J. Op. Thy. 82:2(2019), 369–382
2018Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion. Rev. Mat. Iberoam. 37 (2021), no. 1, 1–44.
2017New bounds for Szemeredi's Theorem, III: A polylog bound for r_4(N). with Ben Green. Mathematika Volume 63, Issue 3 2017 , pp. 944–1040
2017Some remarks on the lonely runner conjecture. Contrib. Discrete Math. 13 (2018), no. 2, 1–31.
2017A bound on partitioning clusters. with Daniel Kane. Electron. J. Combin. 24 (2017), no. 2, Paper 2.31, 13 pp.
2017Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. with Kaisa Matomäki, Maksym Radziwiłł. Proc. Lond. Math. Soc. (3) 118 (2019), no. 2, 284–350.
2017The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures. with Joni Teräväinen. Duke Math. J. 168 (2019), no. 11, 1977–2027
2017Odd order cases of the logarithmically averaged Chowla conjecture. with Joni Teräväinen. J. Théor. Nombres Bordeaux 30 (2018), no. 3, 997–1015.
2017An inverse theorem for an inequality of Kneser. Proc. Steklov Inst. Math. 303 (2018), no. 1, 193–219
2017Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges. with Kaisa Matomäki, Maksym Radziwiłł. Math. Ann. 374 (2019), no. 1–2, 793–840.
2017On the sign patterns of entrywise positivity preservers in fixed dimension. with Apoorva Khare. Amer. J. Math. 143 (2021), 1863–1929.Extended abstract: FPSAC 2018, Volume 80B of Seminaire Lotharingien de Combinatoire (2018)
2017On the universality of potential well dynamics. Dynamics of PDE 14 (2017), 219–238.
2017On the universality of the incompressible Euler equation on compact manifolds. Disc. Cont. Dynam. Sys. 38 (2018), 1553–1565
2016Sum-avoiding subsets in groups. with Van Vu. Discrete Analysis 2016:15, 31 pp.Survey: J. Comb. 8n3, 2017, 541–552
2016Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors. with Tamar Ziegler. Discrete Anal. 2016, Paper No. 13, 60 pp.
2016Polynomial patterns in the primes. with Tamar Ziegler. Forum Math. Pi 6 (2018), e1, 60 pp.
2016Equivalence of the logarithmically averaged Chowla and Sarnak conjectures. Number theory—Diophantine problems, uniform distribution and applications, 391–421, Springer, Cham, 2017.
2016An integration approach to the Toeplitz square peg problem. Forum Math. Sigma 5(2017), e30
2016Finite time blowup for a supercritical defocusing nonlinear wave system. Anal. PDE 9 (2016), no. 8, 1999–2030.
2016Finite time blowup for a high dimensional nonlinear wave systems with bounded smooth nonlinearity. Comm. Partial Differential Equations 41 (2016), no. 8, 1204–1229.
2016Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation. Ann. PDE 2 (2016), no. 2, Art. 9, 79 pp.
2016Finite time blowup for a supercritical defocusing nonlinear Schrödinger system. Analysis & PDE 11-2 (2018), 383–438. DOI 10.2140/apde.2018.11.383
2015An averaged form of Chowla’s conjecture. with Kaisa Matomäki, Maksym Radziwiłł. Algebra & Number Theory 9-9 (2015), 2167–2196. DOI 10.2140/ant.2015.9.2167
2015Random matrices: tail bounds for gaps between eigenvalues. with Van Vu, Hoi Nguyen. Probab. Theory Related Fields 167 (2017), no. 3–4, 777–816.
2015Inverse theorems for sets and measures of polynomial growth. Q. J. Math. 68 (2017), no. 1, 13–57
2015Sign patterns for the Liouville and Mobius functions. with Kaisa Matomäki, Maksym Radziwiłł. Forum Math. Sigma 4 (2016), e14, 44 pp.
2015The logarithmically averaged Chowla and Elliott conjectures for two-point correlations. Forum Math. Pi 4 (2016), e8, 36 pp.
2015The Erdos discrepancy problem. Discrete Analysis 2016:1, 26 pp.
2015Chains of large gaps between primes. with Kevin Ford, James Maynard. Submitted
2015Failure of the $L^1$ pointwise and maximal ergodic theorems for the free group. Forum Math. Sigma 3 (2015), e27, 19 pp.
2015Cancellation in the multilinear Hilbert transform. Collectanea Mathematica 67 (2016), 1–16
2014New equidistribution results of Zhang type. with D.H.J. Polymath. Algebra & Number Theory 8-9 (2014), 2067–2199
2014Variants of the Selberg sieve, and bounded intervals containing many primes. with D.H.J. Polymath. Research in the Mathematical Sciences 1:12 (2014)
2014Large gaps between consecutive prime numbers. with Kevin Ford, Ben Green, Sergei Konyagin. Annals Math. 183 (2016), 935–974.
2014Narrow progressions in the primes. with Tamar Ziegler. Analytic number theory, 357–379, Springer, Cham, 2015.
2014The “bounded gaps between primes” Polymath project - a retrospective. with D.H.J. Polymath. Newsletter of the European Mathematical Society, Dec 2014. Issue 94, 13–23
2014The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture. Algebra & Number Theory 9-4 (2015), 1005–1034. DOI 10.2140/ant.2015.9.1005
2014Random matrices have simple spectrum. with Van Vu. Combinatorica 37 (2017), no. 3, 539–553
2014Long gaps in the primes. with Kevin Ford, Ben Green, Sergei Konyagin, James Maynard. J. Amer. Math. Soc. 31 (2018), no. 1, 65–105
2014Finite time blowup for an averaged three-dimensional Navier-Stokes equation. J. Amer. Math. Soc. 29 (2016), no. 3, 601–674.
2013Expansion in simple groups of Lie type. with Emmanuel Breuillard, Ben Green, Bob Guralnick. J. Europ. Math. Soc. 17 (2015), 1367–1434
2013Multiple recurrence and convergence results associated to F_p^omega-actions. with Vitaly Bergelson, Tamar Ziegler. J. Anal. Math. 127 (2015), 329–378.
2013A multi-dimensional Szemerédi theorem for the primes via a correspondence principle. with Tamar Ziegler. Israel J. Math., Feb 2015
2013Local universality of zeroes of random polynomials. with Van Vu. International Mathematics Research Notices 2014;doi: 10.1093/imrn/rnu084
2013Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory. EMS surveys in the mathematical sciences 1 (2014), 1–46
2013Concentration compactness for critical wave maps, by Joachim Krieger and Wilhelm Schlag. Bull. Amer. Math. Soc. 50 (2013), 655–662
2012Random matrices: Sharp concentration of eigenvalues. with Van Vu. Random matrices: Theory and Applications 2 (2013), 1350007
2012Every odd number greater than 1 is the sum of at most five primes. Mathematics of Computation 83 (2014), 997–1038
2012Random matrices: The Universality phenomenon for Wigner ensembles. with Van Vu. In: Modern Aspects of Random Matrix Theory, Proceedings of Symposia in Applied Mathematics 72 (2013), V. Vu Editor, 121–172.
2012The asymptotic distribution of a single eigenvalue gap of a Wigner matrix. Probability Theory and Related Fields 157 (2013), 81–106
2012E pluribus unum: from complexity, universality. Daedalus 141 (3) (Summer 2012)
2012New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited. with Ben Green. Submitted, Proc. Lond. Math. Soc.
2012Random matrices: Universality of local spectral statistics of non-Hermitian matrices. with Van Vu. Annals of Prob. 43 (2015), 782–874
2012On sets defining few ordinary lines. with Ben Green. Disc. Comp. Geom. 50 (2013), 409–468
2012Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets. Contrib. Discrete Math. 10 (2015), no. 1, 22–98
2012Multiple recurrence in quasirandom groups. with Vitaly Bergelson. Geom. Func. Anal. 24 (2014), 1–48
2012Mixing for progressions in nonabelian groups. Forum of Mathematics, Sigma 1 (2013), e2
2012Nonlinear Fourier Analysis. with Christoph Thiele. To be submitted
2011Random matrices: Localization of the eigenvalues and the necessity of four moments. with Van Vu. Acta Mathematica Vietnamica 36 (2011), 431–449
2011The inverse conjecture for the Gowers norm over finite fields in low characteristic. with Tamar Ziegler. Annals of Combinatorics 16 (2012), 121–188
2011A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups. with Emmanuel Breuillard, Ben Green. Submitted,
2011The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. with Van Vu. Electronic Journal of Probability 16 (2011), 2104–2121
2011Random matrices: Universal properties of Eigenvectors. with Van Vu. Random matrices: Theory and Applications 1 (2012), 1150001
2011An incidence theorem in higher dimensions. with Jozsef Solymosi. Disc. Comp. Geom. 48 (2012), 255–280
2011Noncommutative sets of small doubling. European Journal of Combinatorics 34 (2013), 1459–1465
2011Counting the number of solutions to the Erdös-Straus equation on unit fractions. with Christian Elsholtz. J. Aust. Math. Soc. 94 (2013), 50–105
2011The structure of approximate groups. with Emmanuel Breuillard, Ben Green. Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221
2011A central limit theorem for the determinant of a Wigner matrix. with Van Vu. Adv. Math. 231 (2012), 74–101
2011Random matrices: The Four Moment Theorem for Wigner matrices. with Van Vu. Random matrix theory, interacting particle systems, and integrable systems, 509–528, Math. Sci. Res. Inst. Publ., 65, Cambridge Univ. Press, New York, 2014.
2011A nilpotent Freiman dimension lemma. with Emmanuel Breuillard, Ben Green. European Journal of Combinatorics 34 (2013), 1287–1292
2011Effective limiting absorption principles, and applications. with Igor Rodnianski. Comm. Math. Phys. 333 (2015), 1–95
2011Localisation and compactness properties of the Navier-Stokes global regularity problem. Analysis & PDE 6 (2013), 25–107
2010Linear approximate groups. with Emmanuel Breuillard, Ben Green. Electronic research announcements 17 (2010), 57–67
2010An arithmetic regularity lemma, an associated counting lemma, and applications. with Ben Green. An Irregular Mind: Szemeredi is 70, Bolyai Society Mathematical Studies, 261–334
2010Yet another proof of Szemeredi's theorem. with Ben Green. An Irregular Mind: Szemeredi is 70, Bolyai Society Mathematical Studies, 335–342
2010The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi. with Van Vu. Combinatorica, 32 (2012), 363–372
2010Suzuki groups as expanders. with Emmanuel Breuillard, Ben Green. Groups, Geometry, and Dynamics 5 (2011), no. 2, 281–-299.
2010Approximate subgroups of linear groups. with Emmanuel Breuillard, Ben Green. GAFA 21 (2011), 774–819
2010Strongly dense free subgroups of semisimple algebraic groups. with Emmanuel Breuillard, Ben Green, Bob Guralnick. Israel J. Math. 192 (2012), 347–379
2010An inverse theorem for the Gowers U^s+1[N] norm. with Ben Green, Tamar Ziegler. Annals of Math. 176 (2012), no. 2, 1231–1372.
2010Deterministic methods to find primes. with Ernie Croot, Harald Helfgott. Mathematics of Computation 81 (2012), 1233–1246
2010Large values of the Gowers-Host-Kra seminorms. with Tanja Eisner. J. d’Analyse Mathematique 117 (2012), 133–186
2010Outliers in the spectrum of iid matrices with bounded rank permutations. Probability theory and related fields155 (2013), 231–263
2010Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem. with Assaf Naor. Israel J. Math. 192 (2012), 489–504
2010Asymptotic decay for a one-dimensional nonlinear wave equation. with Hans Lindblad. Analysis & PDE 5 (2012), 411–422.
2009An inverse theorem for the uniformity seminorms associated with the action of F^omega. with Vitaly Bergelson, Tamar Ziegler. Geom. Funct. Anal. 19 (2010), no. 6, 1539–1596.
2009A sharp inverse Littlewood-Offord theorem. with Van Vu. Random Structures and Algorithms37 (2010), 525–539
2009Random matrices: the distribution of smallest singular values. with Van Vu. GAFA, 20 (2010), 260–297
2009Random matrices: universality of local eigenvalue statistics. with Van Vu. Acta Math 206 (2011), 127–204
2009An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm. with Ben Green. Math. Proc. Camb. Phil. Soc. 149 (2010), 1–19
2009Freiman’s theorem for solvable groups. Contributions to Discrete Mathematics 5 (2010), no. 2, 137–184,
2009Sumset and inverse sumset theorems for Shannon entropy. Combinatorics, Probability, and Computing 19 (2010), 603–639
2009Bulk universality for Wigner hermitian matrices with subexponential decay. with Laszlo Erdos, Benjamin Schlein, Van Vu, Horng-Tzer Yau, Jose Ramírez. Math. Res. Lett. 17 (2010), 793–794
2009Random matrices: universality of local eigenvalue statistics up to the edge. with Van Vu. Communications in Mathematical Physics, 298 (2010), 549–572
2009A remark on partial sums involving the Mobius function. Bull. Aust. Math. Soc. 81 (2010), 343–349
2009A new proof of the density Hales-Jewett theorem. with D.H.J. Polymath. Annals of Math. 175 (2012), 1283–1327.
2009A finitary version of Gromov's polynomial growth theorem. with Yehuda Shalom. GAFA 20 (2010), no. 6, 1502–1547.
2009An inverse theorem for the Gowers U^4 norm. with Ben Green, Tamar Ziegler. Glasgow Mathematical Journal 53(2011), no. 1, 1–50
2009Random covariance matrices: Universality of local statistics of eigenvalues. with Van Vu. Annals of Probability 40 (2012), 1283–1315.
2009Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems. with Tim Austin, Tanja Eisner. Pacific Journal of Mathematics 250-1 (2011), 1–60. DOI 10.2140/pjm.2011.250.1
2009Random Martingales and localization of maximal inequalities. with Assaf Naor. J. Funct. Anal. 259 (2010), 731–779
2009The Kakeya set and maximal conjectures for algebraic varieties over finite fields. with Jordan Ellenberg, Richard Oberlin. Mathematika 56 (2010) 1–25
2009A variation norm Carleson theorem. with Richard Oberlin, Andreas Seeger, Christoph Thiele, Jim Wright. J. Europ. Math. Soc. 14 (2012), 421–464.
2009The high exponent limit p to infty for the one-dimensional nonlinear wave equation. Anal. PDE 2 (2009), no. 2, 235–259.
2009An inverse theorem for the bilinear L^2 Strichartz estimate for the wave equation. Unpublished preprint
2009Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions(“Project Heatwave”, part 4 of 5.). Unpublished preprint
2009Global regularity of wave maps VII. Control of delocalised or dispersed solutions(“Project Heatwave”, part 5 of 5.). Unpublished preprint
2009Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation. Analysis & PDE 2 (2009), 361–366
2009Operator splitting for the KdV equation. with Helge Holden, Kenneth Karlsen, Nils Risebro. Math. Comp. 80 (2011) 821–846.
2009Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm. with Mark Keel, Tristan Roy. Discrete Cont. Dynam. Systems 30 (2011), 573–621
2009The power of matrix completion: near-optimal convex relaxation. with Emmanuel Candes. To appear, IEEE Inf. Theory
2008The Möbius function is asymptotically orthogonal to nilsequences. with Ben Green. Annals of Math., 175 (2012), 541–566
2008On the testability and repair of hereditary hypergraph properties. with Tim Austin. Random Structures and Algorithms 36 (2010), 373–463
2008A remark on primality testing and decimal expansions. J. Aust. Math. Soc. 91 (2011), 405–413
2008On the permanent of random Bernoulli matrices. with Van Vu. Adv. Math. 220 (2009), 657—669.
2008Smooth analysis of the condition number and the least singular value. with Van Vu. Mathematics of Computation, 79 (2010), 2333–2352
2008The sum-product phenomenon in arbitrary rings. Contrib. Discrete Math. 4 (2009), no. 2, 59–82.
2008Random matrices: Universality of ESDs and the circular law. with Van Vu, Manjunath Krishnapur. Annals of Probability 38 (2010), no. 5, 2023–2065.
2008From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices. with Van Vu. Bull. Amer. Math. Soc. 46 (2009), 377–396
2008The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. with Tamar Ziegler. Analysis & PDE 3 (2010), 1–20
2008Classification of almost quarter-pinched manifolds. with Peter Petersen. Proc. Amer. Math. Soc. 137 (2009), 2437—2440.
2008Why are solitons stable?. Bull. Amer. Math. Soc. 46 (2009), 1–33.
2008A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential. Dynamics of PDE 5 (2008), 101—116.
2008Global regularity of wave maps III. Large energy from R^1+2 to hyperbolic spaces.("Project Heatwave", part 1 of 5.). Unpublished preprint
2008Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class("Project Heatwave", part 2 of 5.). Unpublished preprint
2008Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equation. Anal. PDE 2 (2009), no. 1, 61–81.
2008Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Inventiones Math.181 (2010), 39–113
2008Global regularity of wave maps V. Large data local well-posedness in the energy class("Project Heatwave", part 3 of 5.). Unpublished preprint
2008Reflections on compressed sensing. with Emmanuel Candes. IEEE Information Theory Society Newsletter, Dec 2008 58 (4), 20–23
2007John-type theorems for generalized arithmetic progressions and iterated sumsets. with Van Vu. Adv. in Math. 219 (2008), 428—449.
2007A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields. with Ben Green. J. Aust. Math. Soc. 86 (2009), 61–74.
2007The condition number of a randomly perturbed matrix. with Van Vu. Proceedings of the thirty-ninth annual ACM symposium on Theory of computing (STOC) 2007, 248–255
2007Freiman's theorem in finite fields via extremal set theory. with Ben Green. Combin. Probab. Comput. 18 (2009), no. 3, 335–355
2007Norm convergence of multiple ergodic averages for commuting transformations. Ergodic Theory and Dynamical Systems 28 (2008), 657–688
2007Structure and randomness in combinatorics. Proceedings of the 48th annual symposium on Foundations of Computer Science (FOCS) 2007, 3–18
2007Random Matrices: The circular Law. with Van Vu. Communications in Contemporary Mathematics, 10 (2008), 261–307
2007The quantitative behaviour of polynomial orbits on nilmanifolds. with Ben Green. Annals of Math. Volume 175 (2012), Issue 2, 465–540.
2007The distribution of polynomials over finite fields, with applications to the Gowers norms. with Ben Green. Contrib. Discrete Math. 4 (2009), no. 2, 1–36.
2007A quantitative version of the Besicovitch projection theorem via multiscale analysis. Proc. Lond. Math. Soc. 98 (2009), 559–584.
2007What is good mathematics?. Mathematical Perspectives, Bull. Amer. Math. Soc. 44 (2007), 623–634.
2007Multi-linear multipliers associated to simplexes of arbitrary length. with Camil Muscalu, Christoph Thiele. Submitted, Analysis & PDE
2007The Walsh model for $M_2^*$ Carleson. with Ciprian Demeter, Michael Lacey, Christoph Thiele. Revista Mat. Iber. 24 (2008), 721–744
2007Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R^2. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Disc. Cont. Dynam. Systems A 21 (2008), 665–686
2007The cubic nonlinear Schrödinger equation in two dimensions with radial data. with Rowan Killip, Monica Visan. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203–1258.
2007A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation. Dynamics of PDE 4 (2007), 293–302.
2006New bounds for Szemeredi's Theorem, II: A new bound for r_4(N). with Ben Green. Analytic number theory: essays in honour of Klaus Roth, W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, R. C. Vaughan, eds, Cambridge University Press, 2009. 180–204.
2006Quadratic uniformity of the Möbius function. with Ben Green. Annales de l’Institut Fourier 58 (2009), 1863—1935.
2006Linear equations in primes. with Ben Green. Annals of Math. 171 (2010), 1753–1850
2006Product set estimates in noncommutative groups. Combinatorica 28 (2008), 547–594
2006A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma. J. d’Analyse Mathematique 103 (2007), 1–45.
2006The ergodic and combinatorial approaches to Szemerédi's theorem. Centre de Recerches Mathématiques, CRM Proceedings and Lecture Notes Vol. 43 (2007), 145–193
2006The primes contain arbitrarily long polynomial progressions. with Tamar Ziegler. Acta Math. 201 (2008), 213—305.Errata: Acta Math. 210, (2013), Page 403–404
2006Perelman’s proof of the Poincare conjecture – a nonlinear PDE perspective. Unpublished
2006Breaking duality in the return times theorem. with Ciprian Demeter, Michael Lacey, Christoph Thiele. Duke Math. J. 143 (2008), 281–355
2006Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dynamics of PDE 3 (2006), 93–110
2006Scattering for the quartic generalised Korteweg-de Vries equation. J. Diff. Eq. 232 (2007), 623—651
2006Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data. J. Hyperbolic Diff. Eq. 4 (2007), 259–266
2006Two remarks on the generalised Korteweg-de Vries equation. Discrete Cont. Dynam. Systems 18 (2007), 1–14
2006A pseudoconformal compactification of the nonlinear Schrodinger equation and applications. New York J. Math. 15 (2009), 265–282.
2006Global behaviour of nonlinear dispersive and wave equations. Current Developments in Mathematics 2006, International Press. 255–340.
2006Minimal-mass blowup solutions of the mass-critical NLS. with Monica Visan, Xiaoyi Zhang. Forum Mathematicum 20 (2008), 881–919
2006Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions. with Monica Visan, Xiaoyi Zhang. Duke Math J. 140 (2007), 165–202
2006A counterexample to an endpoint bilinear Strichartz inequality. Electron. J. Diff. Eq. 2006 (2006) 151, 1—6.
2006A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations. Dynamics of PDE 4 (2007), 1–53
2006A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. with Michael Christ, Jim Colliander. J. Funct. Anal 254 (2007), 368–395
2006Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. with Emmanuel Candes, Justin Romberg. IEEE Inf. Theory 52 (2006) Vol 2., 489–509
2005New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries. with Ben Green. Proc. Lond. Math. Soc. 98 (2009), 365–392
2005On the singularity probability of random Bernoulli matrices. with Van Vu. J. Amer. Math. Soc. 20 (2007), 603–628
2005The Gaussian primes contain arbitrarily shaped constellations. J. d’Analyse Mathematique 99 (2006), 109–176
2005An inverse theorem for the Gowers $U^3(G)$ norm. with Ben Green. Proc. Edin. Math. Soc. 51 (2008), 73–153
2005A variant of the hypergraph removal lemma. J. Combin. Thy. A 113 (2006), 1257–1280
2005Szemeredi's regularity lemma revisited. Contrib. Discrete Math. 1 (2006), 8–28
2005Random symmetric matrices are almost surely non-singular. with Van Vu, Kevin Costello. Duke Math. J. 135 (2006), 395–413
2005Obstructions to uniformity, and arithmetic patterns in the primes. Quarterly J. Pure Appl. Math. 2 (2006), 199–217 [Special issue in honour of John H. Coates, Vol. 1 of 2]
2005Compressions, convex geometry, and the Freiman-Bilu theorem. with Ben Green. Quarterly J. Math. 57 (2006), 495–504
2005Inverse Littlewood-Offord theorems and the condition number of random discrete matrices. with Van Vu. Annals of Math. 169 (2009), 595–632
2005The dichotomy between structure and randomness, arithmetic progressions, and the primes. 2006 ICM proceedings, Vol. I., 581–608
2005The Brascamp--Lieb Inequalities: Finiteness, Structure and Extremals. with Tony Carbery, Michael Christ, Jon Bennett. GAFA 17 (2008), 1343–1415
2005Finiteness bounds for Holder-Brascamp-Lieb multilinear inequalities. with Tony Carbery, Michael Christ, Jon Bennett. Math. Res. Letters, 17 (2010), 647–666
2005On the Multilinear Restriction and Kakeya conjectures. with Tony Carbery, Jon Bennett. Acta Math. 196 (2006) 261—302
2005Maximal multilinear operators. with Ciprian Demeter, Christoph Thiele. Trans. Amer. Math. Soc., 360 (2008), 4989–5042
2005Stability of energy-critical nonlinear Schrödinger equations in high dimensions. with Monica Visan. Electron. J. Diff. Eq. Vol. 2005 (2005), No. 118, 1–28.
2005Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. with Ioan Bejenaru. J. Funct. Anal. Vol. 233 (2006), 228–259
2005Velocity averaging, kinetic formulations, and regularizing effects in quasilinear PDE.. with Eitan Tadmor. CPAM 61 (2007), 1–34
2005The nonlinear Schrödinger equation with combined power-type nonlinearities. with Monica Visan, Xiaoyi Zhang. Comm. PDE 32 (2007), 1281–1343.
2005Decoding by Linear Programming. with Emmanuel Candes. IEEE Inf. Theory 51 (2005), 4203–4215
2005Stable Signal Recovery from Incomplete and Inaccurate Measurements. with Emmanuel Candes, Justin Romberg. Comm. Pure Appl. Math. 59 (2006), 1207–1223
2005Error Correction via Linear Programming. with Emmanuel Candes, Mark Rudelson, Roman Vershynin. Proc. 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS05), IEEE, 2005. pp. 295–308
2005The Dantzig selector: statistical estimation when $p$ is much larger than $n$. with Emmanuel Candes. Annals of Statistics 35 (2007), 2313–2351
2004The primes contain arbitrarily long arithmetic progressions. with Ben Green. Annals of Math. 167 (2008), 481–547
2004Restriction theory of the Selberg Sieve, with applications. with Ben Green. Journal de Théorie des Nombres de Bordeaux 18 (2006), 137—172
2004A quantitative ergodic theory proof of Szemerédi's theorem. Electron. J. Combin. 13 (2006). 1 No. 99, 1–49.
2004On random pm 1 matrices: Singularity and Determinant. with Van Vu. Random Structures and Algorithms 28 (2006), 1—23.[An extended abstracted is also in: STOC’05: Proceedings of the 37thannual ACM symposium on the theory of computing, 431—440, New York 2005.]
2004Arithmetic progressions and the primes. Collectanea Mathematica (2006), Vol. Extra., 37–88.[Proceedings, 7th International Conference on Harmonic Analysis and Partial Differential Equations]
2004The Bi-Carleson operator. with Camil Muscalu, Christoph Thiele. GAFA 16 (2006), 230—277
2004Multi-parameter paraproducts. with Camil Muscalu, Jill Pipher, Christoph Thiele. Revista Mat. Iber. 22 (2006), 963–976
2004Symplectic nonsqueezing of the KdV flow. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Acta Math. 195 (2005), 197–252
2004Global well-posedness and scattering in the energy space for the critical nonlinear Schrodinger equation in R^3("Project Gopher"). with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Annals of Math. 167 (2007), 767–865[A survey article is in Contemp. Math. 439, "Recent Developments in Nonlinear Partial Differential Equations: The second symposium on Analysis and PDEs June 7–10 2004, Purdue University, West Lafayette Indiana", D. Danielli, Ed., pp. 69–80. American Mathematial Society, Providence RI 2007]
2004Long-time decay estimates for Schrodinger equations on manifolds. with Igor Rodnianski. Mathematical aspects of nonlinear dispersive equations, 223–253, Ann. of Math. Stud. 163, Princeton University Press, Princeton NJ 2007
2004Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data. New York Journal of Mathematics 11 (2005), 57–80
2004Sharp Strichartz estimates on non-trapping asymptotically conic manifolds. with Andrew Hassell, Jared Wunsch. Amer. J. Math. 128 (2006), 963—1024.
2004Geometric renormalization of large energy wave maps. Journees “Equations aux derives partielles”, Forges les Eaux, 7–11 June 2004, XI 1–32
2004Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?. with Emmanuel Candes. IEEE Inf. Theory 52 (2006), 5406–5425
2003A sum-product estimate for finite fields, and applications. with Jean Bourgain, Nets Katz. GAFA 14 (2004), 27–57
2003Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Letters 11 (2004), 251–258
2003An uncertainty principle for cyclic groups of prime order. Math. Res. Letters 12 (2005), 121–127
2003A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. with Allen Knutson, Chris Woodward. Electron. J. Combin. 11 (2004), Research Paper 61, 18 pp. (electronic);
2003Some recent progress on the Restriction conjecture. Fourier analysis and convexity, 217–243, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston,MA, 2004.
2003Recent progress on the Restriction conjecture. to appear, Park City proceedings
2003Bi-parameter paraproducts. with Camil Muscalu, Jill Pipher, Christoph Thiele. Acta Math. 193 (2004), 269–296
2003On multilinear oscillatory integrals, nonsingular and singular. with Michael Christ, Xiaochun Li, Christoph Thiele. Duke Math. J. 130 (2005), 321—351.
2003Global existence and scattering for rough solutions of a nonlinear Schrodinger equation in R^3. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. CPAM 57 (2004), 987–1014
2003Global regularity for the Maxwell-Klein-Gordon equation in high dimensions. with Igor Rodnianski. Comm. Math. Phys. 251 (2004), 377–426
2003Ill-posedness for nonlinear Schrodinger and wave equations. with Michael Christ, Jim Colliander. Unpublished preprint
2003A Strichartz inequality for the Schrodinger equation on non-trapping asymptotically conic manifolds. with Andrew Hassell, Jared Wunsch. Comm. PDE 30 (2004), 157–205
2003Global well-posedness of the Benjamin-Ono equation in H^1(R). J. Hyperbolic Diff. Eq. 1 (2004) 27–49
2003Instability of the periodic nonlinear Schrodinger equation. with Michael Christ, Jim Colliander. Unpublished preprint
2003On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation. Dynamics of PDE 1 (2004), 1–48
2002Endpoint mapping properties of spherical maximal operators. with Andreas Seeger, Jim Wright. J. Institut Math. Jussieu 2 (2003) 1, 109–144
2002Pointwise convergence of lacunary spherical means. with Andreas Seeger, Jim Wright. Mt. Holyoke Proceedings (2001), Contemporary Mathematics (2003), 341–352
2002Singular maximal functions and Radon transforms near L^1. with Andreas Seeger, Jim Wright. Amer. J. Math. 126 (2004), 607–647.
2002Weak-type (1,1) bounds for Fourier integral operators. J. Aust. Math. Soc. 75 (2003), 1–21.
2002On the Maximal Bochner-Riesz conjecture in the plane, for p<2. Trans. Amer. Math. Soc. 354 (2002), 1947–1959
2002Kakeya and restriction phenomena for finite fields. with Gerd Mockenhaupt. Duke Math. J. 121 (2004), 35–74
2002Bochner-Riesz summability for analytic functions on the m-complex unit sphere and for cylindrically symmetric functions on R^n-1 times R. with Adam Sikora. Comm. Anal. Geom. 12 (2004), 43–57
2002A new bound for finite field Besicovitch sets in four dimensions. Pacific J. Math 222 (2005), 337–363
2002A sharp bilinear restriction estimate for paraboloids. GAFA 13 (2003), 1359–1384
2002Relative efficiencies of kernel and local likelihood density estimators. with Peter Hall. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 (2002), 537–547
2002A discrete model for the bi-carleson operator. with Camil Muscalu, Christoph Thiele. Geom. Func. Anal. 12 (2002), 1324–1364
2002A Carleson-type theorem for a Cantor group model of the Scattering Transform. with Camil Muscalu, Christoph Thiele. Nonlinearity 19 (2003), 219–246
2002L^p bounds for a maximal dyadic sum operator. with Loukas Grafakos, Erin Terwilleger. Math. Z. 246 (2004), 321–337
2002Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Math. Res. Letters 9 (2002), 659–682.
2002Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Discrete Cont. Dynam. Systems 9 (2003), 31–54
2002Polynomial growth and orbital instability bounds for $L^2$-subcritical NLS below the energy norm. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Comm. Pure Appl. Anal. 2 (2003), 33–50
2002A singularity removal theorem for Yang-Mills fields in higher dimensions. with Gang Tian. J. Amer. Math. Soc. 17 (2004), 557–593.
2002Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocussing equations. with Michael Christ, Jim Colliander. Amer. J. Math. 125 (2003), 1235–1293
2002Upper and lower bounds for Dirichlet eigenfunctions. with Andrew Hassell. Math. Res. Letters 9 (2002), 289–305
2002Local and global well-posedness for nonlinear dispersive equations. Proc. Centre Math. Appl. Austral. Nat. Univ. 40 (2002), 19–48
2002Existence globale et diffusion pour l'équation de Schrödinger nonlinéaire répulsive cubique sur R^3 en dessous l'espace d'énergie. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), Exp. No. X, 14 pp., Univ. Nantes, Nantes,2002
2001L^p improving estimates for averages along curves. with Jim Wright. J. Amer. Math. Soc. 16 (2003), 605–638
2001The Fuglede spectral conjecture holds for convex bodies in the plane. with Alex Iosevich, Nets Katz. Math. Res. Letters 10 (2003), 559–570
2001Carleson measures, trees, extrapolation, and T(b) theorems. with Pascal Auscher, Steve Hofman, Camil Muscalu, Christoph Thiele. Publications Matematiques Barcelona 46 (2002), 257–325
2001The honeycomb model of GL_n(C) tensor products II. Facets of the Littlewood-Richardson cone. with Allen Knutson, Chris Woodward. J. Amer. Math. Soc. 17 (2004), 19–48.
2001Puzzles and (equivariant) cohomology of Grassmannians. with Allen Knutson. Duke Math. J. 119 (2003), 221–260
2001Some connections between the Falconer and Furstenburg conjectures. with Nets Katz. New York J. Math. 7 (2001), 148–187
2001New bounds for Kakeya problems. with Nets Katz. Journal d'Analyse de Jerusalem, 87 (2002), 231–263
2001Uniform estimates on paraproducts. with Camil Muscalu, Christoph Thiele. Journal d'Analyse de Jerusalem 87 (2002), 369–384
2001Uniform estimates for multi-linear operators with modulation symmetry. with Camil Muscalu, Christoph Thiele. Journal d'Analyse de Jerusalem 88 (2002), 255–309
2001L^p estimates for the "biest" I. The Walsh model.. with Camil Muscalu, Christoph Thiele. Math. Annalen 329 (2004), 401–426
2001L^p estimates for the "biest" II. The Fourier model.. with Camil Muscalu, Christoph Thiele. Math. Annalen 329 (2004), 427–461
2001A counterexample to a multilinear endpoint question of Christ and Kiselev. with Camil Muscalu, Christoph Thiele. Math. Res. Letters 10 (2003), 237–246
2001Multilinear interpolation between adjoint operators. with Loukas Grafakos. J. Funct. Anal. 199 (2003), 379–385
2001Global well-posedness result for KdV in Sobolev spaces of negative index. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. EJDE 2001 (2001) No 26, 1–7
2001Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. J. Amer. Math. Soc. 16 (2003), 705–749.
2001Multi-linear estimates for periodic KdV equations, and applications. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. J. Funct. Anal. 211 (2004), 173–218
2001Global well-posedness for the Schrodinger equations with derivative. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Siam J. Math. 33 (2001), 649–669
2001A refined global well-posedness for the Schrodinger equations with derivative. with Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka. Siam J. Math. 34 (2002), 64–86.
2001A physical approach to wave equation bilinear estimates. with Sergiu Klainerman, Igor Rodnianski. J. Anal. Math. 87 (2002), 299—336
2000Almost everywhere convergence of general wavelet shrinkage estimators. with Brani Vidakovic. ACHA 9 (2000), 72–82
2000Endpoint multiplier theorems of Marcinkiewicz type. with Jim Wright. Revista Mat. Iber. 17 (2001), 521–558
2000Some light on Littlewood-Paley theory. with Michael Cowling. Math. Annalen 321 (2001) 4, 885–888
2000Honeycombs and sums of Hermitian matrices. with Allen Knutson. Notices Amer. Math. Soc. 48 (2001) No. 2, 175–186
2000A bilinear approach to cone multipliers I. Restriction estimates. with Ana Vargas. GAFA 10 (2000), 185–215
2000A bilinear approach to cone multipliers II. Applications. with Ana Vargas. GAFA10 (2000), 216–258
2000An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension. with Izabella Laba. GAFA 11 (2001), 773–806
2000From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE.. Notices Amer. Math. Soc. 48 (2001) No 3, 294–303
2000Recent progress on the Kakeya conjecture. with Nets Katz. Publicacions Matematiques, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, U. Barcelona 2002, 161–180
2000Local well-posedness for the Yang-Mills equation below the energy norm. JDE 189 (2003), 366–382
2000Multilinear weighted convolution of L^2 functions, and applications to non-linear dispersive equations. Amer. J. Math. 123 (2001), 839–908
2000Global regularity of wave maps I. Small critical Sobolev norm in high dimension. IMRN 7 (2001), 299–328
2000Global regularity of wave maps II. Small energy in two dimensions. Comm. Math. Phys. 224 (2001), 443–544
1999The weak-type (1,1) of L log L homogeneous convolution operators. Indiana U. Math. J. 48 (1999), 1547–1584
1999Convex bodies with a point of curvature do not admit exponential bases. with Alex Iosevich, Nets Katz. Amer. J. Math. 123 (2001), 115–120
1999Sharp Lorentz space estimates for rough operators. with Andreas Seeger. Math. Annalen 320 (2001) 2, 381–415
1999A converse extrapolation theorem for translation-invariant operators. J. Funct. Anal. 180 (2001), 1–10
1999The Bochner-Riesz conjecture implies the restriction conjecture. Duke Math. J. 96 (1999), 363–376
1999An improved bound on the Minkowski dimension of Besicovitch sets in R^3. with Nets Katz, Izabella Laba. Annals of Math. 152 (2000), 383–446
1999Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238 (2001), 215–268
1999Bounds on arithmetic projections, and applications to the Kakeya conjecture. with Nets Katz. Math. Res. Letters 6 (1999), 625–630
1999An x-ray transform estimate in R^n. with Izabella Laba. Revista Mat. Iber. 17 (2001), 375–408
1999Multilinear operators given by singular multipliers. with Camil Muscalu, Christoph Thiele. J. Amer. Math. Soc. 15 (2002), 469–496
1999Small data blowup for semilinear Klein-Gordon equations. with Mark Keel. Amer. J. Math. 121 (1999), 629–669
1998The honeycomb model of GL_n(C) tensor products I. Proof of the saturation conjecture. with Allen Knutson. J. Amer. Math. Soc.12 (1999), 1055–1090
1998The weak-type endpoint Bochner-Riesz conjecture and related topics. Indiana U. Math. J. 47 (1998), 1097–1124
1998A bilinear approach to the restriction and Kakeya conjectures. with Ana Vargas, Luis Vega. J. Amer. Math. Soc. 11 (1998), 967–1000
1998Endpoint Strichartz Estimates. with Mark Keel. Amer. J. Math., 120 (1998), 955–980
1998Local and global well-posedness of wave maps in R^1+1 for rough data. with Mark Keel. IMRN 21 (1998), 1117–1156
1998Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm. PDE 25 (2000), 1471–1485
1998Ill-posedness for one-dimensional wave maps at the critical regularity. Amer. J. Math., 122 (2000), 451–463
1997Low regularity semi-linear wave equations. Comm. PDE 24 (1999), 599—630
1996On the almost-everywhere convergence of wavelet summation methods. ACHA 3 (1996), 384–387
1996Harmonic convolution operators on Lipschitz graphs. Adv. Appl. Clifford Alg. 6 (1996), 207–218
1996Weak-type endpoint bounds for Riesz means. Proc. Amer. Math. Soc. 124 (1996), 2797–2805
1996On the Structure of Projective Group Representations in Quaternionic Hilbert Space. with Andrew Millard. J. Math. Phys. 37 (1996), 5848–5857
Szemeredi's theorem. with Ben Green. Scholarpedia, p. 15573
Analysis and applications: the work of Elias Stein. with Charles Fefferman, Alex Ionescu, Stephen Wainger. Bull. Amer. Math. Soc. https://doi.org/10.1090/bull/1691
Machine assisted proof. Submitted, Notices Amer. Math. Soc.
Why global regularity for Navier-Stokes is hard (translated, Chinese). "Mathematical Advances in Translation", vol. 33, No.3. p.212-221.

Teaching

Spring 2026Math 246C: Complex analysis
Fall 2025Math 245A: Measure theory
Winter 2025Math 254A: Analytic number theory
Fall 2024Math 246A: Complex analysis
Winter 2024Math 246B: Complex analysis
Fall 2023Math 246A: Complex analysis
Spring 2023Math 246C: Complex analysis
Fall 2022Math 254A: Analytic prime number theory
Spring 2022Math 246C: Complex analysis
Winter 2022Math 246B: Complex analysis
Fall 2021Math 246A: Complex analysis
Winter 2021Math 246B: Complex analysis
Fall 2020Math 246A: Complex analysis
Spring 2020Math 247B: Graduate Fourier analysis
Fall 2019Math 254A: Analytic number theory
Winter 2019Math 255B: Incompressible Euler equations
Fall 2018Math 254A: Incompressible fluids
Spring 2018Math 246C: Complex analysis
Spring 2017Math 245C: Real analysis
Fall 2016Math 246A: Complex analysis
Spring 2016Math 245C: Real analysis
Fall 2015Math 275A: Probability
Winter 2015Math 254A: Analytic prime number theory
Spring 2014Math 245C: Real analysis
Winter 2014Math 245B: Real analysis
Spring 2013Math 245C: Real analysis
Winter 2013Math 245B: Real analysis
Winter 2012Math 254B: Expansion in groups of Lie type
Fall 2011Math 254A: Hilbert's fifth problem
Spring 2011Math 245C: Real analysis
Fall 2010Math 245A: Real analysis
Spring 2010Math 254B: Higher order Fourier analysis
Winter 2010Math 254A: Random matrices
Spring 2009Math 245C: Graduate real analysis
Winter 2009Math 245B: Graduate real analysis
Spring 2008Math 285G: Poincaré conjecture
Winter 2008Math 254A: Ergodic theory
Winter 2007Math 247B: Graduate Fourier analysis
Fall 2006Math 247A: Graduate Fourier analysis
Spring 2006Math 251B: Introduction to PDE
Winter 2006Math 251A: Introduction to PDE
Spring 2005Math 245C: Real analysis
Winter 2005Math 33A: Linear algebra
Winter 2005Math 245B: Real analysis
Fall 2004Math 245A: Real analysis
Winter 2004Math 133: Fourier analysis
Winter 2004Math 245B: Real analysis
Semester 2 2003Math 3228: Complex variables (Honors) (ANU)
Spring 2003Math 131BH: Honors real analysis
Winter 2003Math 131AH: Honors real analysis
Winter 2003Math 254A: Additive combinatorics
Fall 2002Math 115A: Linear algebra
Winter 2001Math 254A: Time-frequency analysis
Semester 2 2000Math 2620: Higher complex analysis (UNSW)
Spring 2000Math 199: Problem solving
Spring 2000Math 121: Introduction to topology
Winter 2000Math 132: Complex analysis
Semester 2 1999Math 2620: Higher complex analysis (UNSW)
Spring 1999Math 254B: Restriction/Kakeya
Winter 1999Math 254A: Oscillatory integrals
Fall 1998Math 132: Complex analysis
Spring 1998Math 32A: Calculus of several variables
Spring 1998Math 132: Complex analysis
Winter 1998Math 32A: Calculus of several variables
Spring 1997Math 132: Complex analysis
Spring 1997Math 121: Introduction to topology
Winter 1997Math 31B: Calculus and analytic geometry (Honors)
Fall 1996Math 132: Complex analysis

Students

2000Deborah Cromer (Honours undergraduate, UCLA / UNSW)
Uniform distribution of sequences mod 1
2000–2001Thomas Lam (Honours undergraduate, UNSW)
Cycles of even length in graphs
2002–2004Julia Garibaldi (PhD, co-advised with John Garnett, UCLA)
Erdős distance problem for convex metrics
2002–2003Roman Sasyk (PhD, co-advised with Sorin Popa, UCLA)
Operator algebra methods in ergodic theory
2003–2006Monica Visan (PhD, UCLA)
Critical nonlinear Schrödinger equations in higher dimensions
2003–2006Jon Bueti (PhD, UCLA)
Finite field x-ray and k-plane estimates
2005Umut Isik (Undergraduate, UCLA)
Maximal non-commuting subsets of groups
2006–2008Shuanglin Shao (PhD, UCLA)
Restriction estimates for paraboloids and cones in the cylindrically symmetric case
2006–2008Soonsik Kwon (PhD, UCLA)
Low-regularity problems of fifth-order KdV and modified KdV equations
2006–2008Tristan Roy (PhD, UCLA)
Global well-posedness of the defocusing cubic wave equation in dimension 3
2006–2010Tim Austin (PhD, UCLA)
Multiple recurrence and the structure of probability-preserving systems
2007–2010Le Thai Hoang (PhD, UCLA)
Topics in additive combinatorics in finite fields
2008–2011Paul Smith (PhD, UCLA)
Subthreshold geometric renormalization and energy-critical Schrödinger maps
2008–2011Zaher Hani (PhD, UCLA)
Global and dynamical aspects of nonlinear Schrödinger equations on compact manifolds
2008–2011Kenny Maples (PhD, UCLA)
Arithmetic properties of random matrices
2009–2013Josh Zahl (PhD, UCLA)
Some algebraic and geometric variants of restriction and Kakeya problems
2011–2013Brad Rodgers (PhD, UCLA)
The statistics of the Riemann zeta function and related topics
2012–2015Pietro Carolino (PhD, UCLA)
The structure of locally compact groups
2012–2015Jacques Benatar (PhD, UCLA)
The existence of small gaps in subsets of the integers
2013–2015Anand Rajagopalan (PhD, UCLA)
Outlier eigenvalue fluctuations of perturbed IID matrices
2013–2015Ben Krause (PhD, UCLA)
Some results in pointwise ergodic theory
2014–2016Jordy Greenblatt (PhD, UCLA)
Dimensional asymptotics for norms of maximal averaging operators on Cartesian powers of finite graphs
2013–2016Nick Cook (PhD, UCLA)
Spectral properties of non-Hermitian random matrices
2015–2017Yilong Yang (PhD, UCLA)
Shapes of finite groups through covering properties and Cayley graphs
2015–2017Stephen Ge (PhD, UCLA)
The eigenvalue spacing of IID random matrices and related least singular value results
2017–2019Zane Li (PhD, UCLA)
Decoupling for the parabola and connections to efficient congruencing
2018–2021Bjoern Bringmann (PhD, UCLA)
Probabilistic perspectives on dispersive partial differential equations
2018–2021Khang Huynh (PhD, UCLA)
Intrinsic harmonic analysis on manifolds with boundary, and Onsager's conjecture
2019–2021Redmond McNamara (PhD, UCLA)
Dynamical methods for the Sarnak and Chowla conjectures
2019–2021Alex Dobner (PhD, UCLA)
Zeros of Riemann zeta-type functions
2020–2023Stan Palasek (PhD, UCLA)
Some quantitative regularity theorems for the Navier–Stokes equations
2021–2023Jaume de Dios (PhD, UCLA)
Uniform estimates for operators involving polynomial curves; decoupling estimates for fractal and product sets
2022–2025James Leng (PhD, UCLA)
Szemerédi's theorem, primes, and nilsequences
2022–2025Ben Johnsrude (PhD, UCLA)
Topics in the theory of Fourier decoupling
2022–2026Zi Li Lim (PhD, UCLA)
2023–2026Rushil Raghavan (PhD, UCLA)
2024–Arian Nadjimzadah (PhD, UCLA)
2024–Siddharth Mulherkar (PhD, UCLA)
2025–Ben Goldman (PhD, co-advised with Dominique Maldague, UCLA)

Professional service and editorial

2025–Co-founder — Foundation for Science and AI Research (SAIR)
2025–Scientific Advisory Board — Mathlib Initiative
2025–Director of Special Projects — IPAM
2025–Scientific Advisory Board — AI for Math
2025–Senior Advisor — Renaissance Philanthropy
2024–Advisory Committee, AIMO Prize — XTX Markets
2024–2025Academic Senate AI workgroup — UCLA
2023–Mathematics for Humanity scientific committee — ICMS
2023–2023Stein Memorial Prize Committee — AMS
2023–2023Planning committee, AI to assist Mathematical Reasoning — National Academy of Sciences
2021–2021Infosys Mathematics Prize Jury — Infosys Science Foundation
2021–2024President's Council of Advisors on Science and Technology — Office of Science and Technology Policy
2021–Scientific Board — Riemann International School of Mathematics
2021–2023Organizing committee, Machine Assisted Proof (chair) — IPAM
2020–2023Chair, Math Circle Steering Committee — UCLA
2019–2025Doob Prize Selection Committee — AMS
2019–2023Board of Trustees, Vice-Chair — MSRI / SLMath
2019–2021Editorial Boards Committee — AMS
2019–2025Structure Committee (Chair, 2019–2024) — International Mathematical Union
2019–2023Advisory Board — SMRI
2018–2023Sloan Fellowship Committee — Sloan Foundation
2018–2023Committee on National Awards and Public Representation — AMS
2017–2018Conference organiser — Clay Mathematics Institute
2016–2018Fields Medal Committee — International Mathematical Union
2016–2022Advisory Board — Bhaumik Institute, UCLA
2015–2025Editor — Discrete Analysis
2015–2021Newton Fellowships committee (Physical Sciences) — The Royal Society
2015–2022Scientific Committee — MATRIX
2015–2017Analytic Number Theory program organiser (chair) — MSRI
2014–2014Joint Lecture Committee — AMS / MAA
2014–2014von Neumann symposium selection committee — AMS
2013–Epi-Math committee — Episciences
2013–Evaluation / visiting committees — MSRI
2012–Editor — Forum of Mathematics (Cambridge University Press)
2012–2013Scientific Reviewing Award committee — National Academy of Sciences
2012–2013Subcommittee chair, decadal plan — National Committee for the Mathematical Sciences
2011–2013Abel Prize Committee — Norwegian Academy of Sciences
2009–Scientific Advisory / Research Committee — AMSI
2008–2010ICM 2010 Program Committee — International Mathematical Union
2007–2025Editor — Analysis & PDE
2007–2020Scientific Advisory Board — IPAM
2005–2011Editor — J. Amer. Math. Soc.
2004–2006Sectional Committee, Analysis — International Mathematical Union
2003–2008Advisory Board — International Mathematics Research Surveys
2003–Associate Editor — Dynamics of Partial Differential Equations
2002–2012Global moderator / math.CA moderator — arXiv.org
2002–2004Associate Editor — J. Amer. Math. Soc.
2002–Associate Editor — American Journal of Mathematics
2002–2022Scientific Advisory Board — Australian Mathematical Sciences Institute
2000–2001math.AP moderator — arXiv.org
1998–2005Reviewer — Zentralblatt Math.

Selected lectures

2002Zygmund–Calderón Lectures, University of Chicago
Low-regularity behavior of the Korteweg–de Vries equations
2002Edinburgh Lectures, Edinburgh
The Kakeya problem
2003Wolff Memorial Lectures, Caltech
Long-time behavior of semi-linear dispersive equations
2003Park City Lectures, Park City
Recent progress on the restriction conjecture
2004Phillips Lectures, Michigan State University
The nonlinear Fourier transform
2004Hardy Lectures, United Kingdom
2005CBMS Lectures, New Mexico State University
Nonlinear dispersive and wave equations
2006Chern Lectures, UC Berkeley
Nonlinear geometric PDE
2006Erdős Lectures, University of Memphis
Arithmetic progressions and the primes
2006Aisenstadt Lectures, Montreal
Szemerédi's theorem and related topics
2007Science Faculty Research Colloquium, UCLA
Structure and randomness in the prime numbers
2007Simons Lectures, MIT
The dichotomy between structure and randomness
2007Milliman Lectures, University of Washington
Recent developments in arithmetic combinatorics
2009Mahler Lectures, Australia
2010Einstein Public Lecture, AMS / UCLA
The cosmic distance ladder
2022MasterClass, online
Mathematical thinking

Patents

US Patent 7,890,942 B2, issued Feb 15, 2011A computer-implemented method for correcting transmission errors using linear programming (Emmanuel Candès, Terence Tao)