8 problems

For $j=1,\ldots,n$, let $\mathcal{T}_j$ be a family of $\delta$-tubes in $\mathbb{R}^n$. Call the families transversal when, for each $j$, every tube in $\mathcal{T}_j$ points in a…

Let $\{U_j:j\in[d]\}$ be a collection of $C^2$ hypersurfaces in $\mathbb{R}^d$, with the normal to $U_j$ at every point within $\frac{1}{100}$ of the $x_j$-axis. Let $\mathcal E_j$…

The $k$-restriction conjecture for the cone. One expects

Let $1\leq k\leq n$, and let $\mathbf{T}^{\lambda}=(T_1^{\lambda},\dots,T_k^{\lambda})$ be a $\nu$-transverse $k$-tuple of H"ormander-type operators with positive-definite phase fu…

Endpoint multilinear restriction conjecture. The estimate should remain valid without the factor $\delta^{-\epsilon}$. This is equivalent to the endpoint $k$-linear restriction est…

Let $S_1,\ldots,S_n$ be compact $C^2$ hypersurfaces in $\mathbb{R}^n$ with surface measures $d\sigma_1,\ldots,d\sigma_n$, and suppose they are transverse, meaning their unit normal…

Let $S_i\subset\mathbb{R}^{n+1}$ be hypersurfaces, and let $\mathcal R^*(2\times\cdots\times 2\rightarrow p)$ denote the $k$-linear restriction estimate associated with the surface…

Multilinear restriction conjecture. Under these assumptions, there is a constant $C$, depending on finitely many derivatives of the $\Sigma_i$, the sets $U_i$, $\nu$, and $n$, such…