22 problems

Let $11\le p\le 12$. Assume $a\ge b\ge 0$ and $a+b=\frac{p}{2}-3$. Let … and let $\Phi(t)=(t,t^2,t^3,t^4)$ be the moment curve in $\mathbb{R}^4$. Here $n\sim N$ denotes summation o…

Let $n,d\ge 1$. For each $n$, let $U(n)$ denote uniform decoupling for the polynomial class $\mathcal P_{n,d}$, let $Z(n)$ denote uniform degenerate decoupling for…

Hyperbolic-paraboloid decoupling conjecture. For $p>3$, for every $\varepsilon>0$ there exists $C_\varepsilon>0$ such that for all $R\geq1$,

Let $S$ be a compact piece of a smooth surface in $\mathbb{R}^n$, and let $N_\delta(M)$ denote its $\delta$-neighborhood. A family of parallelograms has width at least $\delta$ if…

Let $\phi:\mathbb{R}^{n-1}\to\mathbb{R}$ be smooth, and let $\mathcal{M}_{\phi}=\{(\xi,\phi(\xi)):\xi\in[0,1]^{n-1}\}$. A set $S\subset[0,1]^{n-1}$ is $(\phi,\delta)$-flat when it…

Large-set planar conjecture. If $|Y|\geq R^{1/2}$, then is true with

Let $R$ be a large parameter, let $B_R$ be a ball or cube of radius or side length $R$ in ${\mathbb R}^3$, and let $E_SF$ be the extension operator for the two-dimensional parabolo…

Let $R$ be a large parameter, let $B_R$ be a ball or cube of radius or side length $R$ in ${\mathbb R}^3$, and let $E_SF$ denote the Fourier extension operator associated to the tw…

Let $d\geq 3$ be an integer, let $\Gamma=\{(\xi,\xi^d)\in\mathbb R^2:1\leq|\xi|\leq2\}$, and let $P_j=(\xi_j,\xi_j^d)$ for $j=1,\ldots,N$ be equally distributed in $\Gamma$. For ea…

Let … where $a_m$ is supported in … For $\lambda>0$, define the superlevel set $U_\lambda(\nu)=\{(x,t):|\nu(x,t)|\geq\lambda\}$. Periodic restriction conjecture. If $|a_m|\leq 1$ f…

Let $N\geq 1$, $0\leq \sigma\leq 2$, and $s\geq 1$, and write $e(t)=e^{2\pi i t}$. Then the small cap decoupling conjecture. There is a constant $C_\varepsilon$ such that … This co…

Bourgain–Demeter–Kemp decoupling conjecture. There is a partition $\mathcal P_\delta=\mathcal P_\delta(\phi)$ of $[-1,1]^2$ into $\delta$-flat subsets $S$ such that, for every $f$…

Let $s\in[1,2)$ and $2\le p\le 2+2s$. Write $D(N,N^s,p)$ for the decoupling constant associated with the parabola at spatial scale $N^s$ and exponent $p$. Uniform decoupling conjec…

Local moment conjecture. For every $j\geq 0$,

Let $N\geq 1$, $T>0$, and let $B_T$ be any ball of radius $T$. For coefficients $\{b_n\}_{n=N+1}^{2N}\subset\mathbb{C}$, write … for the associated Dirichlet polynomial, and let…

Square-root cancellation in $L^p$. For $11\le p\le12$ and $\alpha\ge\beta\ge0$ satisfying $\alpha+\beta=p/2-3$, one should have

Let $\phi_3,\phi_4$ be real analytic functions on $(0,3)$ satisfying the derivative bounds and nondegeneracy conditions … … … For a finite interval $I\subset {\mathbb Z}$, define ……

Let ${\mathcal C}^2=\{(\xi_1,\xi_2,\sqrt{\xi_1^2+\xi_2^2}):\frac14\leq\xi_1^2+\xi_2^2\leq4\}$, and let ${\mathcal N}_\delta({\mathcal C}^2)$ denote its $\delta$-neighborhood. Suppo…

Let $S$subset${\mathbb R}^3$ be the graph of a nonconstant real analytic function $g:[-1,1]^2\to{\mathbb R}$. For $0<\delta\leq 1$, let ${\mathcal N}_\delta(S)$ be its $\delta$-nei…

The Main Conjecture. For all integers $s,k\geq 1$,

Let $2\leq k\leq n+1$. For a function $F$ on $\mathbb{R}^{n+1}$ with Fourier support in $N_{1/D^2}\Sigma$, let $F_{D,V}$ denote its contribution associated with a $k$-dimensional s…

Let $P^{n-1}=\{(\xi_1,\ldots,\xi_{n-1},\xi_1^2+\cdots+\xi_{n-1}^2):|\xi_i|\le 1/2\}$, let $S$ be either $P^{n-1}$ or $S^{n-1}$, and let ${\mathcal N}_\delta$ be the $\delta$-neighb…