7 problems

Let $g_0$ be a Liouville metric on $\mathbb{T}^d$, let $r$ be the regularity exponent appearing in the $C^r$ norm, and let $g$ be another Riemannian metric. Say that $g$ is Laplace…

Let $u$ be an eigenfunction of the Laplacian $-\Delta$ on $\mathbb{T}^3$ with eigenvalue $\lambda^2$. Bourgain's eigenfunction conjecture. One should have … The loss…

Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, let $\lambda>2$ and $\delta<1$, and define … Let … so that $P_{\lambda,\delta}f=\Phi_{\lambda,\delta}*f$. Kernel conjecture. If…

Spectral projector conjecture. If $p\geq2$ and $\kappa>0$, then, whenever $\delta>\lambda^{-1+\kappa}$,

Product decomposition conjecture. Given $\mu\in\mathrm{Dens}({\mathbb T}^n)$, there exist $\eta\in\mathrm{Diff}_{\mathrm{d}x}({\mathbb T}^n)$ and functions…

Let $\mathbb T^2$ be the two-dimensional torus, let $\varphi$ be an eigenfunction with eigenvalue $E=\lambda^2$, and let $C\subset\mathcal N_\varphi$ be a regular arc. For a convex…

Even-dimensional parameter-count conjecture. There exist non-extensible combinatorial torus cube packings with $n+2$ cubes and