7 problems

Hypoelliptic damping conjecture. The second part of the theorem for the elliptically perturbed operator $P_\epsilon$ holds with $\widetilde P_\epsilon$ in place of $P_\epsilon$. Th…

Let $N(\Omega):=N_{P(h)}(\Omega)$ be the resonance counting function, and let $\mathcal P_E(s)$ be the topological pressure. Resonance concentration conjecture. For every…

Let $N(\Omega):=N_{P(h)}(\Omega)$ be the resonance counting function, and let $\mathcal P_E(s)$ denote the topological pressure from the source. Pressure threshold conjecture. For…

Let $N(\Omega):=N_{P(h)}(\Omega)$ be the resonance counting function, and let $\delta_E$ be the trapped-set dimension parameter defined in the source. Fractal Weyl law conjecture.…

For Kerr--de Sitter parameters $a$, $\Lambda$, and $M$ satisfying … and lying in the shaded region specified by the source, quasinormal modes are complex frequencies $\lambda$ for…

Let $V\in L^\infty_{\operatorname{comp}}(\mathbb R^n)$ be real-valued and non-zero, with $n$ odd. Let $N_V(r)$ denote the number of resonances in $\{|\lambda|\le r\}$. Optimal reso…

Let $n_\star(x)$ be a locally optimal one-dimensional refractive index on the interval $[0,L]$, taking the values $n_+$ and $n_-=1$ on its material intervals. The point $x=L/2$ is…