34 problems

For every solution of the Calogero–Moser derivative nonlinear Schrödinger equation with initial data $u_0\in L^2_+(\mathbb R)$ belonging to the stated natural critical initial-data…

For $n\ge 2$, fix an incident direction $\theta_0\in\mathbb S^{n-1}$. Determine whether a compactly supported potential $q$ is uniquely determined by its far-field pattern…

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…