17 problems

Let $M=\Gamma\backslash\mathbb H^2$ be a convex co-compact hyperbolic surface, and let $\delta$ denote the dimension of the limit set of $\Gamma$. Jakobson--Naud finiteness conject…

Non-entire continuation conjecture. The function $\eta_D$ cannot be prolonged as an entire function.

Let $\Sigma_{D,1}$ denote the set of pathological points corresponding to the spectral behavior of $L_0$ at infinity, and let $\operatorname{Res} L_{\varepsilon}$ be the set of sca…

Let $\eta_D$ be the Dirichlet dynamical zeta function associated with the oriented periodic rays of a dispersing billiard flow, initially defined by its convergent Dirichlet series…

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…

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 $P(h)$ have a compact hyperbolic trapped set $K_E$ at energy $E$, and suppose the underlying manifold $M$ has effective meromorphic continuation of $(P(h)-z)^{-1}$. Resonance-f…

Suppose that $P(\lambda)$ is the operator referred to in the source, and that its coefficients are real analytic near $x_1=0$. Let $U$ be a sufficiently small neighbourhood of…

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…

The modular surface is the quotient of the hyperbolic plane whose Laplacian resonances are identified with the zeros of the Riemann zeta function. The decay rate is represented by…

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…

Symmetry conjecture. There exists $n_\star\in\mathcal A_{\mathrm{sym}}$ such that

Bragg-width conjecture. The interval widths satisfy

Let $l_{\star}(\epsilon)$ denote the mode index of the scattering resonances nearest the real axis for a compressible fluid, where $\epsilon$ is the compressibility parameter and…

Let ${\mathcal P}$ be the set of primitive periodic rays. For each $\gamma\in{\mathcal P}$, define … and let $r_\gamma=0$ if $m_\gamma$ is even and $r_\gamma=1$ if $m_\gamma$ is od…