32 problems

Let $K\subset\mathbb E^n$ be a smooth manifold of codimension at least $2$ with a regular embedding, and let $\mathcal W(\epsilon)$ denote the volume of the trapping locus…

Scattering conjecture for critical NLS. There exist functions

Scattering conjecture for critical gKdV. There exist functions

Uniform boundedness conjecture. There is a universal constant $B_{V_0,R_0}$ such that

Let $\gamma_0$ denote the portion of $\gamma$ lying in the strip $[-d/2,d/2) \times \mathbb{R}$ and define … These are the quasi-periodic Hilbert transform and Laplace double-layer…

Let $\mathcal{H}$ and $\mathcal{I}$ denote the horizon and null infinity, respectively. Consider smooth scattering data settling down to a possibly extremal Kerr solution on…

Non-scattering-data dispersion-manageability conjecture. Given such $u_0$, both of the following assertions hold: (i) for fixed $t_0,t_1$, there exist $\gamma_+=\gamma_+(u_0)$ and…

Let $N\in\mathbb N_{\geq1}$, let $\phi$ solve $\Box_{\mathbb R^{2N+1,1}}\phi=0$ with no incoming radiation, and let…

Let $\Omega\subset\mathbb{R}^3$ be the exterior domain of a smooth, compact, strictly convex obstacle, and consider the defocusing nonlinear Schrödinger equation with Dirichlet bou…

Consider the defocusing nonlinear Schrödinger equation … in $\mathbb{R}^d$, with $d\geq 1$ and $\alpha\geq 4/d$. Set $s_c=d/2-2/\alpha$, and let…

Let $H_{\mu,\nu}$ be the operator parametrized by $\mu,\nu\geq0$, let $H_{\rm D}$ be the Dirichlet reference operator, and let $W_-(H_{\mu,\nu},H_{\rm D})$ denote the incoming wave…

Layered-scattering determination conjecture. The averages $\mathcal L^{\mathsf{av}}_{0\leq j\leq\lceil k/2\rceil}$ determine the intrinsic global invariants…

Let $\phi_{\mathcal H^+}$ be a suitable finite-energy scalar-field profile on the event horizon of a spherically symmetric Einstein–Maxwell–Klein–Gordon black hole, and let…

Let $V$ be a super-exponentially decreasing potential, let $\widehat V$ have completely regular growth, and let $B(V)$ be its Breit–Wigner series. Assume that $V$ is not compactly…

Let $V$ be a super-exponentially decreasing potential, so that its Fourier–Laplace transform $\widehat V$ is entire, and let $F$ be its Froese function, … Assume that $\widehat V$…

Breit–Wigner convergence conjecture. The series $B(V)$ converges if and only if $V$ is compactly supported.

Let $\boldsymbol{\eta}$ be a profile, let $C\in\mathcal{F}$ be a polynomial in fields and their derivatives, and let $\Psi,\Psi'\in\mathcal{D}_2$. A renormalization scheme for time…

For a defocusing nonlinear Schrödinger equation or nonlinear wave equation, let $u$ be a solution whose critical Sobolev norm remains bounded in time. Bounded critical Sobolev norm…

Ground state conjecture. The solution extends uniquely to a global solution $u\in C(\mathbb{R}:\dot{H}^1(\mathbb{R}^d))$ of

Let $\Gamma$ be a $C^4$-smooth, asymptotically straight planar curve. Let $\Sigma_{\alpha,\Gamma}(k)$ be the on-shell scattering matrix at energy $k^2$, and let $S_\Gamma$ be the o…

Suppose that $A$ and $S-S_\infty$ are Schwartz-class functions for some constant $S_\infty\in\mathbb{R}$, and that $k\in\mathbb{C}\setminus\{0\}$ admits a global classical solution…

Let ${\mathcal H}_k$ denote the Hilbert space associated with the incoming $(k,l)$ scattering sector, and let ${\mathcal D}_l$ denote the corresponding outgoing space. An operator…

Generic resonance abundance conjecture. For an open and dense set of $g\in\mathcal{G}(M)$, there is an infinite number of resonances outside any such strip. The conjecture asserts…

Let $M$ be a cusp manifold and let $\mathcal{G}(M)$ denote the space of $C^\infty$ metrics on $M$ for which $(M,g)$ is a cusp manifold with negative sectional curvature. The theore…

Let the obstacles be planar scatterers, and let the scattered field be defined in the exterior of the obstacles and on their boundaries. Assume that the scattered field is analytic…