235 problems

A simple Riemannian manifold is a compact Riemannian manifold with strictly convex boundary and no conjugate points, such that every pair of points is joined by a unique minimizing…

Let $D\subset\mathbb{R}^2$ be a simply connected domain, and understand equality of domains modulo sets of measure zero. The Pompeiu property is the property characterized in the s…

Well-definedness conjecture. The experimental conductivity $\gamma_{\mathrm{exp}}$ is well defined for every conductivity $\gamma\in\mathcal{G}^R$.

Let $(M_1,g_1)$ and $(M_2,g_2)$ be Riemannian manifolds with mutual boundary, and let $\Lambda_{g_1}$ and $\Lambda_{g_2}$ denote their Dirichlet-to-Neumann maps for harmonic functi…

Let $(\mathsf{M},g)$ and $(\mathsf{M}',g')$ be the spacetimes under consideration. For each $k\in\boldsymbol{\mathrm{N}}$, let $\nu^k$ and $\nu'^k$ be the corresponding probability…

Freiman–Lev conjecture. One has

Coercivity conjecture. The coercivity condition is generally satisfied: there exist constants $c_{\mathcal{H}_{{K}^E}},c_{\mathcal{H}_{{K}^A}}>0$ such that

Let $n \geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. Suppose that $\gamma_1,\gamma_2 \in W^{1,n}(\Omega)$ are uniformly elliptic, and let…

Boundary rigidity conjecture. Compact simple manifolds of any dimension are boundary rigid.

Let $\Omega\subset\mathbb{R}^d$ be a bounded domain, let $\gamma$ be the conductivity, and assume $d\geq 3$. The Dirichlet-to-Neumann data determine $\gamma$ uniquely. Brown's conj…

Let $(M,g)$ be a compact Riemannian manifold with boundary of dimension $n\geq 3$. Its scattering relation records, for geodesics leaving the boundary, when and where they next mee…

Scattering rigidity conjecture. The scattering relation for non-trapping manifolds determines the metric uniquely up to the natural obstruction.

Let $Omega$ be an inclusion of fixed volume in a medium and let its electrical polarization tensor be $M$. The matrix $M$ measures the leading dipolar perturbation caused by a unif…

MSE conjecture. The mean squared error of the improved estimator satisfies

Generic nonlinear reconstruction conjecture. The set $\mathfrak{O}$ of $\delta\in\mathfrak{D}$ such that

Let $\hat H^0=-\frac{\hbar^2}{2\mu}\Delta+V$ be a Schrödinger operator with an infinitely growing potential $V$, and let $\hat H_\alpha({\bf q})$ denote its point perturbation at p…

Jollivet–Sharafutdinov inverse Steklov spectral conjecture. The metrics $g_1$ and $g_2$ have the same Steklov spectrum if and only if they are $\sigma$-isometric. This conjecture i…

Let $M$ be the compact manifold under consideration, let $f,g$ be functions on $M$, and let $I$ be a missing spectral window with complement $I^c$ consisting of the observed freque…

Dual-degeneracy explanation. We conjecture that this phenomenon is due to the dual degeneracy of the problem.

Let $N>0$, and let $m_i:=\int_0^{\delta}v^i g(v)\,\mathrm{d}v$ for $i\in\{1,2,3\}$. The functions $b$, $u^{(N)}(t,x)$, $u^{(N)}_{\partial}(t)$, and the initial datum $n_0(x)$ be as…

Consider the outer optimization iterates for the inverse mean field game problem, constrained by the discrete mean field game system, and let the feasible region denote the set of…

Let $\Omega$ and the given function $z$ be smooth as in Theorem, and let $u$ be a weak solution of the Dirichlet problem. The nonlocal Neumann datum $\mathcal N_s u$ is conjectured…

A coefficient inverse problem (CIP) is an inverse problem for recovering coefficients of a differential equation from boundary measurements; here the approach combines a convexific…

Extension conjecture. The paper conjectures that its type of analysis can be extended to such conditional generative models.

Over-regularization conjecture. The lower bound imposed on the $\mathcal{A}_i$ in the regularization analysis could be replaced by requiring the $\mathcal{A}_i$ to be larger than t…