16 problems

Let $M=\mathbb T^d$ be the standard flat torus, let $e_\lambda$ be a toral eigenfunction with frequency radius $\lambda$, and let $\Sigma\subset\mathbb T^d$ be a real-analytic hype…

Let $Y$ be a compact hyperbolic surface, let $\phi_i$ be a Laplace eigenfunction on $Y$ with eigenvalue $\mu_i$, and let $\ell$ be a geodesic on $Y$. Write…

Let $\Sigma\subset\mathbb T^2$ be a real-analytic curve and let $e_\lambda$ be a toral eigenfunction. Real-analytic curve restriction conjecture. The uniform bound should hold: … U…

Let $T_{d,q}(\lambda,L;C)$ denote the maximal cardinality of an admissible set of lattice points on $\lambda S^{d-1}$ with the parameters $d,q,\lambda,L,C$. Transverse-configuratio…

Let $A_{k,d,\lambda}$ denote the maximal number of lattice points in the intersection of $k$ unit-width, $\nu$-transverse bands on the sphere $\lambda S^{d-1}$, with…

Let $e_\lambda$ be a toral eigenfunction on the standard flat torus $\mathbb T^d$, and let $\Sigma\subset\mathbb T^d$ be a smooth submanifold of codimension $m$. Huang–Zhang conjec…

Bounded multiplicity conjecture. The desired bound is

Let $(M,g)$ be a compact manifold all of whose sectional curvatures are negative. For $q\in(2,q_c)$, where $q_c=\tfrac{2(n+1)}{n-1}$, let $\chi_{[\lambda,\lambda+\delta(\lambda)]}$…

Let $(M,g)$ be a compact manifold all of whose sectional curvatures are negative. For $q_c=\tfrac{2(n+1)}{n-1}$, let $\chi_{[\lambda,\lambda+\delta(\lambda)]}$ denote the spectral…

Let $T$ be the dilated averaging operator considered in Theorem 1, with dilation parameter $\omega\in(0,2)$, and let $(1/p,1/q)$ denote the reciprocal exponent pair for its…

Restriction conjecture. The restriction estimate holds if and only if

Let $d$ be the spatial dimension, let $\Sigma$ be the elliptic hyperboloid equipped with its Lorentz-invariant measure $\mu$, and let $\mathcal{E}$ denote the associated extension…

Let $d\ge3$, let $\Sigma\subset\mathbb{T}^d$ be a smooth submanifold of dimension $k$ with $1\le k\le d-2$, and let $e_\lambda$ be an eigenfunction of $\mathbb{T}^d$. Higher-dimens…

Let $M$ be a Riemannian manifold, let $H\subset M$ be an interior hypersurface, let $u$ be an eigenfunction, and let $h$ be the associated semiclassical parameter. Write $\Delta_H$…

Let $S_1,\ldots,S_{n+1}\subset\mathbb{R}^{n+1}$ be hypersurfaces, and let $d\sigma_i$ denote the surface measure on $S_i$. For functions $f_i$ on $S_i$, consider the multilinear re…

Subpolynomial restriction norm conjecture. For any fixed $\varepsilon>0$ and every $\epsilon>0$,