45 problems

Let $(M,g_1)$ and $(M,g_2)$ be compact Riemannian manifolds with smooth boundary, with $\dim(M)\geq 3$. For each metric, let $C_{g_i}$ denote the Cauchy data set of harmonic functi…

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 $\Omega\subset\mathbb{R}^3$ be a compact domain, and let $\epsilon\in C^2(\mathcal{S}^2,\Omega)$ be an elastic strain field in an isotropic medium that satisfies equilibrium an…

Mohan–Pandey conjecture. The following implications hold:

Freiman–Lev conjecture. One has

The BC method concerns the recovery of information in inverse boundary value problems (BVPs), including hyperbolic, elliptic, and parabolic problems. BC method unification conjectu…

Let $\mathcal{E}$ be a well-connected network with an odd number of boundary nodes and response matrix $M_R(\mathcal{E})$. Let $A(\mathcal{E})$ be the matrix constructed from this…

Modified anisotropic Calderón conjecture at nonzero frequency. If

Anisotropic Calderón conjecture at zero frequency. If

Consider generic stiffness tensor fields $c_i$, density fields $\rho_i$, and Riemannian metrics $g_i$ in the setting of the stated Riemannian anisotropic elasticity problem, and le…

Let $E(3)$ be the space of three-dimensional stiffness tensors, let $F$ map a stiffness tensor to its slowness polynomial, and write the target as $\mathbb{R}[p_1,p_2,p_3]$. The sl…

Let $n\geqslant3$, let $y\in[1/C_\infty,1/C_0)$, and let $\widetilde{C}_1\leqslant\cdots\leqslant\widetilde{C}_{n+1}$ be the nondecreasing reordering of the parameters…

Let $n\geq 2$, let $G=C_n \oplus C_n$, let $k\in [0,n-1]$, and let $S$ be a sequence of terms from $G$ with length $|S|=\operatorname{D}(G)+k-1=2n-2+k$ having no nonempty zero-sum…

Wilson-loop injectivity conjecture. If $(M,g)$ is a closed odd-dimensional Riemannian manifold with Anosov geodesic flow, then the Wilson loop operator

Analytic-conjugacy rigidity conjecture. Any two $3$-scatterer systems that are analytically conjugate to one another in a neighborhood of the homoclinic orbit must be isometric. Eq…

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…

Let $D \subset \mathbb{R}^m$, with $m \geqslant 2$, be a bounded domain with sufficiently smooth boundary, and let $r>0$ satisfy $|B_r|=|D|$. A function $u$ is panharmonic if it be…

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

Michel's conjecture. All simple manifolds are boundary rigid.

Let $\tau=(\tau_1,\ldots,\tau_n)\in\mathbb{R}^n_{>0}$, and define … Let $\operatorname{Stab}(\tau)$ be the subgroup of $S_n$ consisting of coordinate permutations that fix $\tau$.…

Generic-injectivity conjecture. The recovery of a set of $n$ complex numbers from $n+1$ power sums with coprime powers is unique: for generic $z\in\mathbb{C}^n$, the fiber…

Let ${\bf B}_{m+\ell}$ be partitioned as … and define … Here $\widetilde{{\bf B}}_\ell$ is the corresponding projected bidiagonal block produced by the recycled hybrid projection m…

Let $m=2k+1$, let $c_1,\ldots,c_m$ be specified weights, let $\gamma_1,\ldots,\gamma_m$ denote ray directions, and let $\mathcal{S}$ be the corresponding star transform. A configur…

Higher-order Compton scattering smoothing conjecture. For each scattering order $i$, $g_i$ is approximated by a Fourier integral operator $\mathcal{L}_i(\cdot,n_e)$ of order

Let $d/2\leq p<\infty$, let $\Omega$ be a bounded domain with Lipschitz boundary, and let $H_c^{s,p}(\Omega)$ denote the compactly supported Sobolev space of potentials. For a pote…