Scientists / Open questions

Terence Tao

Harmonic analysis; partial differential equations; number theory

2006 Fields Medal

17 questions

Invariant sub-tiling structure of measurable tilings of tori in dimensions $d \geq 3$
Is the excess $M(x)-\pi(x)$ of the longest $\varphi$ -nondecreasing sequence over the primes bounded?
Mass concentration and blow-up dynamics above the ground-state threshold for the focusing mass-critical nonlinear Schrödinger equation
Extending the effective-potential theory of M-matrices to random ensembles: resonances, sharp decay constants, weaker decoupling, and Poisson spectral statistics
Uniform estimates for modulation-symmetric multi-linear multipliers: the higher-rank degeneration and the exponent range
Endpoint estimates for one-dimensional Fourier multipliers of bounded 2-variation: do $V_2$ multipliers map $H^1$ and $L\log^{1/2}L$ into $L^{1,\infty}$ ?
General singular symbols for the biest: $L^p$ boundedness beyond product structure and its necessity
Strongly dense free subgroups of semisimple algebraic groups: existence inside an arbitrary dense subgroup and dominance of double word maps (Problems 1 and 2 of Breuillard–Green–Guralnick–Tao)
Endpoint uniform estimates ( $p_i=\infty$ and BMO) for the Muscalu-Tao-Thiele paraproduct family
Completing the relative-efficiency theory of kernel versus local log-polynomial density estimators: full-support asymptotics, sharp worst-case constants, and structural classification
The Fourier-transform $M^{q,*}$ Carleson maximal estimate for exponents $1<q<2$
Boundary cases of the Tao–Wright $L^p$ -improving characterization for averaging operators along curves
Endpoint theory of rough homogeneous singular integrals beyond the L log L threshold: weak (1,1) bounds for odd and Hardy-space kernels, and L^2 boundedness on homogeneous groups
Carleson–Hunt endpoint maximal estimates for the continuous AKNS-ZS/Dirac scattering transform with square-integrable potential
Endpoint estimates for the biest operator, the boundedness regime of reparametrized simplex multipliers, and maximal higher-order ordering operators
L^p bounds for multi-linear operators with singular multipliers beyond the rank threshold k<n/2, and the dependence of the bounds on the geometry of the singular subspace
Extremal structure of the optimal sine and sawtooth inequalities: full characterization of optimal coefficient vectors and classification of extremal measures

Back to open problems

Invariant sub-tiling structure of measurable tilings of tori in dimensions d≥3d \geq 3d≥3

Is the excess M(x)−π(x)M(x)-\pi(x)M(x)−π(x) of the longest φ\varphiφ-nondecreasing sequence over the primes bounded?

Mass concentration and blow-up dynamics above the ground-state threshold for the focusing mass-critical nonlinear Schrödinger equation

Extending the effective-potential theory of M-matrices to random ensembles: resonances, sharp decay constants, weaker decoupling, and Poisson spectral statistics

Uniform estimates for modulation-symmetric multi-linear multipliers: the higher-rank degeneration and the exponent range

Endpoint estimates for one-dimensional Fourier multipliers of bounded 2-variation: do V2V_2V2​ multipliers map H1H^1H1 and Llog⁡1/2LL\log^{1/2}LLlog1/2L into L1,∞L^{1,\infty}L1,∞?

General singular symbols for the biest: LpL^pLp boundedness beyond product structure and its necessity

Strongly dense free subgroups of semisimple algebraic groups: existence inside an arbitrary dense subgroup and dominance of double word maps (Problems 1 and 2 of Breuillard–Green–Guralnick–Tao)

Endpoint uniform estimates (pi=∞p_i=\inftypi​=∞ and BMO) for the Muscalu-Tao-Thiele paraproduct family

Completing the relative-efficiency theory of kernel versus local log-polynomial density estimators: full-support asymptotics, sharp worst-case constants, and structural classification

The Fourier-transform Mq,∗M^{q,*}Mq,∗ Carleson maximal estimate for exponents 1<q<21<q<21<q<2

Boundary cases of the Tao–Wright LpL^pLp-improving characterization for averaging operators along curves

Endpoint theory of rough homogeneous singular integrals beyond the L log L threshold: weak (1,1) bounds for odd and Hardy-space kernels, and L^2 boundedness on homogeneous groups

Carleson–Hunt endpoint maximal estimates for the continuous AKNS-ZS/Dirac scattering transform with square-integrable potential

Endpoint estimates for the biest operator, the boundedness regime of reparametrized simplex multipliers, and maximal higher-order ordering operators

L^p bounds for multi-linear operators with singular multipliers beyond the rank threshold k<n/2, and the dependence of the bounds on the geometry of the singular subspace

Extremal structure of the optimal sine and sawtooth inequalities: full characterization of optimal coefficient vectors and classification of extremal measures

Invariant sub-tiling structure of measurable tilings of tori in dimensions $d \geq 3$

Is the excess $M(x)-\pi(x)$ of the longest $\varphi$ -nondecreasing sequence over the primes bounded?

Endpoint estimates for one-dimensional Fourier multipliers of bounded 2-variation: do $V_2$ multipliers map $H^1$ and $L\log^{1/2}L$ into $L^{1,\infty}$ ?

General singular symbols for the biest: $L^p$ boundedness beyond product structure and its necessity

Endpoint uniform estimates ( $p_i=\infty$ and BMO) for the Muscalu-Tao-Thiele paraproduct family

The Fourier-transform $M^{q,*}$ Carleson maximal estimate for exponents $1<q<2$

Boundary cases of the Tao–Wright $L^p$ -improving characterization for averaging operators along curves